# The Golden Ratio

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|                                        | |ImageName=The Golden Ratio                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |                                                          | |ImageName=The Golden Ratio                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
| -                                      | |Image=Goldenrectangleappwarp.jpg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 | +                                                        | |Image=Goldenratio.jpg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | |ImageIntro=The '''golden number,''' often denoted by lowercase Greek letter "phi", is <br /><math>{\varphi}=\frac{1 + \sqrt{5}}{2} = 1.61803399...</math>. <br />                                                                                                                                                                                                                                                                                                                                                                                                                                                | +                                                        | |ImageIntro=The '''golden number,''' often denoted by lowercase Greek letter "phi", is <br />                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
| -                                      | The term '''golden ratio''' refers to any ratio which has the value phi.  The image to the right is a warped representation of dividing and subdividing a rectangle into the golden ratio.  The result is [[Field:Fractals|fractal-like]]. This page explores real world applications for the golden ratio, common misconceptions about the golden ratio, and multiple derivations of the golden number.                                                                                                                                                                                                          | +                                                        | :<math> \frac{a+b}{a} = \frac{a}{b} \equiv \varphi,</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
| -                                      | |ImageDescElem=The golden number, approximately 1.618, is called golden because many geometric figures involving this ratio are often said to possess special beauty. Be that true or not, the ratio has many beautiful and surprising mathematical properties. The Greeks were aware of the golden ratio, but did not consider it particularly significant with respect to aesthetics.  It was not called the "divine" proportion until the 15th century, and was not called "golden" ratio until the 18th century.  <br />                                                                                      | +                                                        | where the Greek letter [[Phi (letter)|phi]] (<math>\varphi</math>) represents the golden ratio. Its value is:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
| -                                      | Since then, it has been claimed that the golden ratio is the most aesthetically pleasing ratio, and claimed that this ratio has appeared in architecture and art throughout history.  Among the most common such claims are that the Parthenon and Leonardo Da Vinci's Mona Lisa use the golden ratio. Even more esoteric claims propose that the golden ratio can be found in the human facial structure, the behavior of the stock market, and the Great Pyramids. <br />                                                                                                                                       | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | However, such claims have been criticized in scholarly journals as wishful thinking or sloppy mathematical analysis.  Additionally, there is no solid evidence that supports the claim that the golden rectangle is the most aesthetically pleasing rectangle.                                                                                                                                                                                                                                                                                                                                                    | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        | <math>{\varphi}=\frac{1 + \sqrt{5}}{2} = 1.61803399...</math>. <br />                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        | The term '''golden ratio''' refers to any ratio which has the value phi.  The image to the right illustrates dividing and subdividing a rectangle into the golden ratio.  This page explores how the Golden Ratio can be observed and found in the arts, mathematics, and nature.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        | |ImageDescElem===The Golden Ratio as an Irrational Number==                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        | ==The Golden Ratio in the Arts==                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
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| -                                      | ===Misconceptions about the Golden Ratio===                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       | +                                                        | ===Parthenon===                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | Many rumors and misconceptions surround the golden ratio. There have been many claims that the golden ratio appears in art and architecture.  In reality, many of these claims involve warped images and large margins of error.  One claim is that the Great Pyramids exhibit the golden ratio in their construction.  This belief is illustrated to the right.                                                                                                                                                                                                                                                  | +                                                        | :[[Image:ParthenonDIAG.gif]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
| -                                      | [[Image:Finalpyramid1.jpg|Markowsky has determined the above dimensions to be incorrect.|thumb|400px|left]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       | +                                                        | There is an abundance of artists who consciously used the Golden Ratio from as long ago as 400 BCE. Once such example is the Greek sculptor Phidias, who built the Parthenon. The exterior dimensions of the Parthenon form the golden rectangle, and the golden rectangle can also be found in the space between the columns.                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
| -                                      | [[Image:Monalisa01.jpg|Does the Mona Lisa exhibit the golden ratio?|thumb|400px|right]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
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| -                                      | In his paper, ''Misconceptions about the Golden Ratio,'' George Markowsky disputes this claim, arguing that the dimensions assumed in the picture are not anywhere close to being correct.  Another belief is that a series of [[The Golden Ratio#Jump2|golden rectangles]] appears in the ''Mona Lisa''.                                                                                                                                                                                                                                                                                                         | +                                                        | ===Vitruvian Man===                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
| -                                      | However, the placing of the golden rectangles seems arbitrary. Markowsky also disputes the belief that the human body exhibits the golden ratio.  To read more, [http://www.math.nus.edu.sg/aslaksen/teaching/maa/markowsky.pdf click here!]                                                                                                                                                                                                                                                                                                                                                                      | +                                                        | :[[Image:Vitruvian man mixed.jpeg|350px]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        | Another instance in which the Golden ratio appears is in Leonardo Da Vinci’s drawing of the Vitruvian Man. Da Vinci’s picture of man’s body fits the approximation of the golden ratio very closely. This picture is considered to a depiction of a perfectly proportioned human body. The Golden Ratio in this picture is the distance from the naval to the top of his head, divided by the distance from the soles of the feet to the top of the head.                                                                                                                                                                                                                                                                                                                                |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        | ===Music===                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        | Some people have argued that the Golden Ratio in music produces aesthetically pleasing sounds. Major sixth and the minor sixth chords are considered to be the most pleasing intervals, and subsequently, they are the intervals related to the Golden Ratio. The standard tuning tone used is an A and it vibrates at 440 vibrations per second. A major sixth interval below that would be a C, which has a frequency of 264 vibrations per second. The ratio of the two frequencies reduces to 5/3, which is the ratio of two Fibonacci numbers. Similarly, a minor sixth can be obtained from a high C, which has a frequency of 528 vibrations per second, and an E, which is 330 vibrations per second. This ratio reduces to 8/5, which is also a ratio of two Fibonacci numbers. |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        | :[[Image:Chromaticscales.gif|350px]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
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| -                                      | ====''What do you think?''====                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    | +                                                        | The Golden Ratio manifests in the idea that music is proportionally balanced. However, it is an area of debate, because some scholars claim that these appearances are purely coincidental, or a result of number juggling from aficionados. However, many contemporary artists have intentionally used the Golden Ratio in their pieces. Composer, mathematician and teacher Joseph Schillinger believed that music could be based entirely on mathematical formulation. He developed a System of Musical Composition in which successive notes in a melody were followed by Fibonacci intervals when counted in half steps. Half steps are the smallest intervals possible, and it is the closest note that can be played higher or lower.                                             |
| -                                      | George Markowsky argues that, like the ''Mona Lisa,'' the Parthenon does not exhibit a series of golden rectangles (discussed below).  Do you think the Parthenon was designed with the golden ratio in mind or is the image below simply a stretch of the imagination?                                                                                                                                                                                                                                                                                                                                           | +                                                        | For example, if C were the first note in a composition, the following note would be half a step higher, because one is a Fibonacci number. So, the second note would be a D flat. Schillinger would alternate between moving up and down in intervals. Thus, the third note would be two half steps (because two is the next Fibonacci number), down from the D flat, which would be B flat. The next Fibonacci number after two is three, so the next note would be three half steps higher than the last. Three half steps higher than B flat is E flat. Schillinger believed that these notes convey the same sense of harmony as the phyllotactic ratios found in leaves.                                                                                                            |
| -                                      | :[[Image:Golden ratio parthenon.jpg|300px]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       | +                                                        | |ImageDesc===Fibonacci Numbers==                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
| -                                      | <ref>[http://lotsasplainin.blogspot.com/2008/01/wednesday-math-vol-8-phi-golden-ratio.html "Parthenon"], Retrieved on 16 May 2012.</ref>                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                        | One of the greatest breakthroughs regarding the Golden Ratio came when its relation to Fibonacci numbers, also known as the Fibonacci sequence, was discovered. Fibonacci numbers can also be found in the arts, and nature. The Fibonacci sequence is the series of numbers,                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        | 0,1,1,2,3,5,8,13,21,34,55,89,144,233…                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        | The next term in the Fibonacci sequence, starting from the third, is determined by adding the previous two terms together. The Fibonacci sequence is related to the Golden Ratio because as the sequence grows, the ratio of consecutive terms gradually approaches the Golden Ratio. For example here are the ratios of the successive numbers in the Fibonacci sequence:  <br/>                                                                                                                                                                                                                                                                                                                                                                                                        |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        | 1/1=1.000000 <br/>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        | 2/1=2.000000 <br/>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        | 3/2=1.500000 <br/>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        | 5/3=1.666666 <br/>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        | 8/5=1.600000 <br/>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        | 13/8=1.625000 <br/>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        | 21/13=1.615385 <br/>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        | 34/21=1.619048 <br/>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        | 55/34=1.617647 <br/>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        | 89/55=1.618182 <br/>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        | 144/89=1.717978 <br/>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        | 233/144=1.618056 <br/>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        | 377/233=1.618026 <br/>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        | 610/377=1.618037 <br/>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        | 987/610=1.618033  <br/>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
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| -                                      | ==A Geometric Representation==                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |                                                          |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
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| -                                      | ===The Golden Ratio in a Line Segment===                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                        | ==The Golden Ratio in Nature==                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
| -                                      | [[Image:Golden_segment.jpg|400px]][[Image:Animation2.gif]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
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| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        | ===Spirals & Phyllotaxis===                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
| -                                      | The golden number can be defined using a line segment divided into two sections of lengths a and b.  If a and b are appropriately chosen, the ratio of a to b is the same as the ratio of a + b to a and both ratios are equal to φ.  The line segment above (left) exhibits the golden proportion. The line segments above (right) are also examples of the golden ratio.  In each case,                                                                                                                                                                                                                         | +                                                        | [[Image:Sunflower head.jpeg|px200]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
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| -                                      | <math>\frac{{\color{Red}\mathrm{red}}+\color{Blue}\mathrm{blue}}{{\color{Blue}\mathrm{blue}} }= \frac{{\color{Blue}\mathrm{blue}} }{{\color{Red}\mathrm{red}} }= \varphi . </math>                                                                                                                                                                                                                                                                                                                                                                                                                                | +                                                        | Spirals are abundant concerning phyllotaxis, which describes the way leaves are arranged on a plant stem. In 92 percent of Norway spruce cones, the spirals were found to appear in rows five and eight rows. They appeared in rows of four and seven in six percent, and four and six in four percent. In addition, the number of right-handed spirals appears to be equal to the number of left-handed spirals. The arrangements of the spirals in these spruce cones are found as the following pairs of rows: 2/3,3/5,5/8,8/13,13/21,21/34,34/55,55/89, and 89/144. These numbers are all numbers that belong to the Fibonacci Series.                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        | [[Image:Pinecone3.gif]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
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| -                                      | <div id="Jump2"></div>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                        | A similar phenomenon can be observed in sunflower heads. A sunflower head has both clockwise and counterclockwise spirals. The numbers of the spirals in a sunflower usually depend on the size of the sunflower. However, the ratios of the spirals that usually occur are 89/55,144/89, and even 233/144. Once again, all of the numbers in these ratios are consecutive Fibonacci numbers.                                                                                                                                                                                                                                                                                                                                                                                            |
| -                                      | ===The Golden Rectangle===                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | A '''golden rectangle''' is any rectangle whose sides are proportioned in the golden ratio.  When the sides lengths are proportioned in the golden ratio the rectangle is said to possess the '''golden proportions.'''  A golden rectangle has sides of length <math>\varphi \times r</math> and <math>1 \times r</math> where <math>r</math> can be any constant.  Remarkably, when a square is cut off of the golden rectangle, the remaining rectangle also exhibits the golden proportions.  This continuing pattern is visible in the golden rectangle below.                                               | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | :[[Image:Coloredfinalrectangle1.jpg]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
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| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        | |AuthorName=Joyce Han                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
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| -                                      | ===Triangles===                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | [[Image:1byrrectangle1.jpg|500px]][[Image:Pentagon_final.jpg|300px]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
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| -                                      | The golden number, φ, is used to construct the '''golden triangle,''' an isoceles triangle that has legs of length <math>\varphi \times r</math> and base length of <math>1 \times r</math> where <math>r</math> can be any constant. It is above and to the left.  Similarly, the '''golden gnomon''' has base <math>\varphi \times r</math> and legs of length <math>1 \times r</math>. It is shown above and to the right.   These triangles can be used to form regular pentagons (pictured above) and <balloon title="A pentagram is a five pointed star made with 5 straight strokes">pentagrams.</balloon> | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
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| -                                      | The pentgram below, generated by the golden triangle and the golden gnomon, has many side lengths proportioned in the golden ratio.                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | :[[Image:Star1.jpg]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | :::<math>\frac{{\color{SkyBlue}\mathrm{blue}} }{{\color{Red}\mathrm{red}} } = \frac{{\color{Red}\mathrm{red}} }{{\color{Green}\mathrm{green}} } = \frac{{\color{Green}\mathrm{green}} }{{\color{Magenta}\mathrm{pink}} } = \varphi . </math>                                                                                                                                                                                                                                                                                                                                                                      | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | These triangles can be used to form [[Field:Fractals| fractals]] and are one of the only ways to tile a plane using '''pentagonal symmetry'''. Pentagonal symmetry is best explained through example.  Below, we have two fractal examples of pentagonal symmetry.  Images that exhibit pentagonal symmetry have five symmetry axes.  This means that we can draw five lines from the image's center, and all resulting divisions are identical.                                                                                                                                                                  | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | :[[Image:Penta1.jpg|400px]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | :[[Image:Pent111.jpg|400px]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | |ImageDesc==Mathematical Representations of the Golden Ratio=                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | ==An Algebraic Derivation of Phi==                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | {{SwitchPreview|ShowMessage=Click to expand|HideMessage=Click to hide|PreviewText=How can we derive the value of φ from its characteristics as a ratio? We may algebraically solve for the ratio (φ) by observing that ratio satisfies the following property by definition:                                                                                                                                                                                                                                                                                                                                      | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | :<math>\frac{b}{a} = \frac{a+b}{b} = \varphi</math>|FullText=                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | Let <math> r </math> denote the ratio :                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | :<math>r=\frac{a}{b}=\frac{a+b}{a}</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | So                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | :<math>r=\frac{a+b}{a}=1+\frac{b}{a}</math> which can be rewritten as                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | :<math>1+\cfrac{1}{a/b}=1+\frac{1}{r}</math> thus,                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | :<math>r=1+\frac{1}{r}</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | Multiplying both sides by <math>r</math>, we get                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | :<math> {r}^2=r+1</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | which can be written as:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | :<math>r^2 - r - 1 = 0 </math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | Applying the <balloon title="load:myContent">quadratic formula</balloon>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <span id="myContent" style="display:none">                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | An equation, <math>\frac{-b \pm \sqrt {b^2-4ac}}{2a}</math>, which produces the solutions for equations of form <math>ax^2+bx+c=0</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | </span>, we get <math>r = \frac{1 \pm \sqrt{5}} {2}</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | The ratio must be positive because we can not have negative line segments or side lengths.  Because the ratio has to be a positive value,                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | :<math>r=\frac{1 + \sqrt{5}}{2} = 1.61803399... =\varphi</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | |NumChars=500}}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | ==Continued Fraction Representation and [[Fibonacci sequence|Fibonacci Sequences]]==                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | The golden ratio can also be written as what is called a '''continued fraction,'''a fraction of infinite length whose denominator is a quantity plus a fraction, which latter fraction has a similar denominator, and so on. This is done by using <balloon title="Recursion is the method of substituting an equation into itself">recursion</balloon>.                                                                                                                                                                                                                                                          | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | {{SwitchPreview|ShowMessage=Click to expand|HideMessage=Click to hide|PreviewText=   |FullText=We have already solved for φ using the following equation:                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <math>{\varphi}^2-{\varphi}-1=0</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | We can add one to both sides of the equation to get                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <math>{\varphi}^2-{\varphi}=1</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | Factoring this gives                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <math> \varphi(\varphi-1)=1 </math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | Dividing by <math>\varphi</math> gives us                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <math>\varphi -1= \cfrac{1}{\varphi }</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | Adding 1 to both sides gives                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <math>\varphi =1+ \cfrac{1}{\varphi }</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | Substitute in the entire right side of the equation for <math> \varphi </math> in the bottom of the fraction.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <math>\varphi = 1 + \cfrac{1}{1 + \cfrac{1}{\varphi } }</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | Substituting in again,                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <math>\varphi = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{\varphi}}}</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <math>\varphi = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{\cdots}}}</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | Continuing this substation forever, the last infinite form is a continued fraction                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | If we evaluate truncations of the continued fraction by evaluating only part of the continued fraction, replacing <math>\varphi</math> by 1, we produce the ratios between consecutive terms in the [[Fibonacci sequence]].                                                                                                                                                                                                                                                                                                                                                                                       | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <math>\varphi \approx 1 + \cfrac{1}{1} = 2</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <math>\varphi \approx 1 + \cfrac{1}{1+\cfrac{1}{1}} = 3/2</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <math>\varphi \approx 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1} } } = 5/3</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <math>\varphi \approx 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1+\cfrac{1}{1}}}} = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{2}}} =1 + \cfrac{1}{1 + \cfrac{2}{3}} = 8/5</math>                                                                                                                                                                                                                                                                                                                                                                                                                                  | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | Thus we discover that the golden ratio is approximated  in the Fibonacci sequence.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <math>1,1,2,3,5,8,13,21,34,55,89,144...\,</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <table style="position:relative;left:50px">                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <tr>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <td align="right"><math>1/1</math></td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <td align="center"><math>=</math></td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <td align="left"><math>1</math></td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | </tr>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <tr>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <td align="right"><math>2/1</math></td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <td align="center"><math>=</math></td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <td align="left"><math>2</math></td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | </tr>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <tr>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <td align="right"><math>3/2</math></td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <td align="center"><math>=</math></td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <td align="left"><math>1.5</math></td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | </tr>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <tr>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <td align="right"><math>8/5</math></td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <td align="center"><math>=</math></td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <td align="left"><math>1.6</math></td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | </tr>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <tr>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <td align="right"><math>13/8</math></td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <td align="center"><math>=</math></td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <td align="left"><math>1.625</math></td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | </tr>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <tr>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <td align="right"><math>21/13</math></td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <td align="center"><math>=</math></td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <td align="left"><math>1.61538462...</math></td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | </tr>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <tr>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <td align="right"><math>34/21</math></td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <td align="center"><math>=</math></td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <td align="left"><math>1.61904762...</math></td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | </tr>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <tr>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <td align="right"><math>55/34</math></td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <td align="center"><math>=</math></td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <td align="left"><math>1.61764706...</math></td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | </tr>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <tr>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <td align="right"><math>89/55</math></td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <td align="center"><math>=</math></td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <td align="left"><math>1.61818182...</math></td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | </tr>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | </table>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <!--Alan, here's your old stuff just in case I messed up the numbers or something. --Maria                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <math>2/1 = 2, 3/2 = 1.5, 8/5 = 1.6, 13/8 = 1.625, 21/13 = 1.61538462...,\,</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <math>34/21 = 1.61904762..., 55/34=1.61764706..., 89/55 = 1.61818182..., ...\,</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | -->                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <math>\varphi = 1.61803399...\,</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | As you go farther along in the Fibonacci sequence, the ratio between the consecutive terms approaches the golden ratio. Many real-world applications of the golden ratio are related to the Fibonacci sequence. For more real-world applications of the golden ratio [[Fibonacci Numbers|click here!]]                                                                                                                                                                                                                                                                                                            | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | In fact, we can prove that the ratio between terms in the Fibonacci sequence approaches the golden ratio by using mathematical [[Induction]].                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | {{Switch|link1=Click to show proof|link2=Click to hide proof|1= |2=                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | Since we have already shown that                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <math> \varphi = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{\cdots}}} </math>,                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | we only need to show that each of the terms in the continued fraction is the ratio of Fibonacci numbers as shown above.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | First, let                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | :<math> x_1=1</math>,                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | :<math> x_2=1+\frac{1}{1}=1+\frac{1}{x_1} </math>,                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | :<math> x_3= 1+\frac{1}{1+\frac{1}{1}}=1+\frac{1}{x_2} </math> and so on so that                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | :<math> x_n=1+\frac{1}{x_{n-1}} </math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | These are just the same truncated terms as listed above. Let's also denote the terms of the Fibonacci sequence as                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | :<math> s_n=s_{n-1}+s_{n-2} </math> where <math>s_1=1</math>,<math>s_2=1</math>,<math>s_3=2</math>,<math>s_4=3</math> etc.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <br>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | We want to show that                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | :<math> x_n=\frac{s_{n+1}}{s_n} </math> for all n.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | First, we establish our [[Induction|base case]]. We see that                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | :<math> x_1=1=\frac{1}{1}=\frac{s_2}{s_1} </math>, and so the relationship holds for the base case.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | Now we assume that                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | :<math> x_k=\frac{s_{k+1}}{s_{k}} </math> for some <math> 1 \leq k < n </math> (This step is the [[Induction|inductive hypothesis]]).  We will show that this implies that                                                                                                                                                                                                                                                                                                                                                                                                                                        | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | :<math> x_{k+1}=\frac{s_{(k+1)+1}}{s_{k+1}}=\frac{s_{k+2}}{s_{k+1}} </math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <br><br>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | By our assumptions about ''x<sub>1</sub>'',''x<sub>2</sub>''...''x<sub>n</sub>'', we have                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | :<math> x_{k+1}=1+\frac{1}{x_k} </math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | By our inductive hypothesis, this is equivalent to                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | :<math>x_{k+1}=1+\frac{1}{\frac{s_{k+1}}{s_{k}}}</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | Now we only need to complete some simple algebra to see                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | :<math> x_{k+1}=1+\frac{s_k}{s_{k+1}} </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | :<math> x_{k+1}=\frac{s_{k+1}+s_k}{s_{k+1}} </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | Noting the definition of <math>s_n=s_{n-1}+s_{n-2}</math>, we see that we have                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <math> x_{k+1}=\frac{f_{k+2}}{f_{k+1}} </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | So by the principle of mathematical induction, we have shown that the terms in our continued fraction are represented by ratios of consecutive Fibonacci numbers.                                                                                                                                                                                                                                                                                                                                                                                                                                                 | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | The exact continued fraction is                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | :<math> x_{\infty} = \lim_{n\rightarrow \infty}\frac{f_{n+1}}{f_n} =\varphi </math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | }}|NumChars=75}}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | ==Proof of the Golden Ratio's Irrationality==                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | {{SwitchPreview|ShowMessage=Click to expand|HideMessage=Click to hide|PreviewText=     |FullText=                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | Remarkably, the Golden Ratio is  <balloon title="A number is irrational if it cannot be expressed as the fraction between two integers.">irrational</balloon>, despite the fact that we just proved that is approximated by a ratio of Fibonacci numbers.                                                                                                                                                                                                                                                                                                                                                         | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | We will use the method of <balloon title="A method of proving a statement true by assuming that it's false and showing this assumption would logically lead to a statement that is already known to be untrue.">contradiction</balloon>  to prove that the golden ratio is irrational.                                                                                                                                                                                                                                                                                                                            | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | Suppose <math>\varphi </math> is rational. Then it can be written as fraction in lowest terms <math> \varphi = b/a</math>, where a and b are integers.                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | Our goal is to find a different fraction that is equal to <math> \varphi </math> and is in lower terms. This will be our contradiction that will show that <math> \varphi </math> is irrational.                                                                                                                                                                                                                                                                                                                                                                                                                  | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | First note that the definition of <math> \varphi = \frac{b}{a}=\frac{a+b}{b} </math> implies that <math> b > a </math> since clearly <math> b+a>b </math> and the two fractions must be equal.                                                                                                                                                                                                                                                                                                                                                                                                                    | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <br>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | Now, since we know                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | :<math> \frac{b}{a}=\frac{a+b}{b} </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | we see that <math> b^2=a(a+b) </math> by cross multiplication. Foiling this expression gives us <math> b^2=a^2+ab </math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | Rearranging this gives us <math> b^2-ab=a^2 </math>, which is the same as :<math> b(b-a)=a^2 </math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | Dividing both sides of the equation by ''a(b-a)'' gives us                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | :<math> \frac{b}{a}=\frac{a}{b-a} </math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | Since <math> \varphi=\frac{b}{a} </math>, this means <math> \varphi=\frac{a}{b-a} </math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | Since we have assumed that a and b are integers, we know that b-a must also be an integer. Furthermore, since <math> a<b </math>, we know that <math> \frac{a}{b-a} </math> must be in lower terms than <math> \frac{b}{a} </math>.                                                                                                                                                                                                                                                                                                                                                                               | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | Since we have found a fraction of integers that is equal to <math> \varphi </math>, but is in lower terms than <math> \frac{b}{a} </math>, we have a contradiction: <math> \frac{b}{a} </math> cannot be a fraction of integers in lowest terms. Therefore <math> \varphi </math> cannot be expressed as a fraction of integers and is irrational.                                                                                                                                                                                                                                                                | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | |NumChars=75}}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | :*Markowsky. “Misconceptions about the Golden Ratio.” College Mathematics Journal. Vol 23, No 1 (1992). pp 2-19.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
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| -                                      | |other=Algebra, Geometry                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | |AuthorName=azavez1                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | |SiteName=The Math Forum                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | |SiteURL=http://mathforum.org/mathimages/index.php/Image:180px-Pentagram-phi.svg.png                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | +                                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        | |Field=Algebra                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |                                                          | |Field=Algebra                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
|                                        | |Field2=Geometry                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |                                                          | |Field2=Geometry                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
| -                                      | |References=<references />                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |                                                          |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | |ToDo=-animation?                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |                                                          |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
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| -                                      | http://www.metaphorical.net/note/on/golden_ratio                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |                                                          |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | http://www.mathopenref.com/rectanglegolden.html                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |                                                          |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        | |InProgress=Yes                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |                                                          | |InProgress=Yes                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | |HideMME=No                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |                                                          |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
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| -                                      | |Field=Algebra                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |                                                          |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
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## Current revision

The Golden Ratio
Fields: Algebra and Geometry
Image Created By: Joyce Han

The Golden Ratio

The golden number, often denoted by lowercase Greek letter "phi", is
$\frac{a+b}{a} = \frac{a}{b} \equiv \varphi,$

where the Greek letter phi ($\varphi$) represents the golden ratio. Its value is:

${\varphi}=\frac{1 + \sqrt{5}}{2} = 1.61803399...$.
The term golden ratio refers to any ratio which has the value phi. The image to the right illustrates dividing and subdividing a rectangle into the golden ratio. This page explores how the Golden Ratio can be observed and found in the arts, mathematics, and nature.

# Basic Description

==The Golden Ratio as an Irrational Number==

## The Golden Ratio in the Arts

### Parthenon

There is an abundance of artists who consciously used the Golden Ratio from as long ago as 400 BCE. Once such example is the Greek sculptor Phidias, who built the Parthenon. The exterior dimensions of the Parthenon form the golden rectangle, and the golden rectangle can also be found in the space between the columns.

### Vitruvian Man

Another instance in which the Golden ratio appears is in Leonardo Da Vinci’s drawing of the Vitruvian Man. Da Vinci’s picture of man’s body fits the approximation of the golden ratio very closely. This picture is considered to a depiction of a perfectly proportioned human body. The Golden Ratio in this picture is the distance from the naval to the top of his head, divided by the distance from the soles of the feet to the top of the head.

### Music

Some people have argued that the Golden Ratio in music produces aesthetically pleasing sounds. Major sixth and the minor sixth chords are considered to be the most pleasing intervals, and subsequently, they are the intervals related to the Golden Ratio. The standard tuning tone used is an A and it vibrates at 440 vibrations per second. A major sixth interval below that would be a C, which has a frequency of 264 vibrations per second. The ratio of the two frequencies reduces to 5/3, which is the ratio of two Fibonacci numbers. Similarly, a minor sixth can be obtained from a high C, which has a frequency of 528 vibrations per second, and an E, which is 330 vibrations per second. This ratio reduces to 8/5, which is also a ratio of two Fibonacci numbers.

The Golden Ratio manifests in the idea that music is proportionally balanced. However, it is an area of debate, because some scholars claim that these appearances are purely coincidental, or a result of number juggling from aficionados. However, many contemporary artists have intentionally used the Golden Ratio in their pieces. Composer, mathematician and teacher Joseph Schillinger believed that music could be based entirely on mathematical formulation. He developed a System of Musical Composition in which successive notes in a melody were followed by Fibonacci intervals when counted in half steps. Half steps are the smallest intervals possible, and it is the closest note that can be played higher or lower. For example, if C were the first note in a composition, the following note would be half a step higher, because one is a Fibonacci number. So, the second note would be a D flat. Schillinger would alternate between moving up and down in intervals. Thus, the third note would be two half steps (because two is the next Fibonacci number), down from the D flat, which would be B flat. The next Fibonacci number after two is three, so the next note would be three half steps higher than the last. Three half steps higher than B flat is E flat. Schillinger believed that these notes convey the same sense of harmony as the phyllotactic ratios found in leaves.

# A More Mathematical Explanation

[Click to view A More Mathematical Explanation]

## Fibonacci Numbers

One of the greatest breakthroughs regarding the Golden Ratio came when its rela [...]

[Click to hide A More Mathematical Explanation]

## Fibonacci Numbers

One of the greatest breakthroughs regarding the Golden Ratio came when its relation to Fibonacci numbers, also known as the Fibonacci sequence, was discovered. Fibonacci numbers can also be found in the arts, and nature. The Fibonacci sequence is the series of numbers, 0,1,1,2,3,5,8,13,21,34,55,89,144,233… The next term in the Fibonacci sequence, starting from the third, is determined by adding the previous two terms together. The Fibonacci sequence is related to the Golden Ratio because as the sequence grows, the ratio of consecutive terms gradually approaches the Golden Ratio. For example here are the ratios of the successive numbers in the Fibonacci sequence:
1/1=1.000000
2/1=2.000000
3/2=1.500000
5/3=1.666666
8/5=1.600000
13/8=1.625000
21/13=1.615385
34/21=1.619048
55/34=1.617647
89/55=1.618182
144/89=1.717978
233/144=1.618056
377/233=1.618026
610/377=1.618037
987/610=1.618033

## The Golden Ratio in Nature

### Spirals & Phyllotaxis

Spirals are abundant concerning phyllotaxis, which describes the way leaves are arranged on a plant stem. In 92 percent of Norway spruce cones, the spirals were found to appear in rows five and eight rows. They appeared in rows of four and seven in six percent, and four and six in four percent. In addition, the number of right-handed spirals appears to be equal to the number of left-handed spirals. The arrangements of the spirals in these spruce cones are found as the following pairs of rows: 2/3,3/5,5/8,8/13,13/21,21/34,34/55,55/89, and 89/144. These numbers are all numbers that belong to the Fibonacci Series.

``` 
```

A similar phenomenon can be observed in sunflower heads. A sunflower head has both clockwise and counterclockwise spirals. The numbers of the spirals in a sunflower usually depend on the size of the sunflower. However, the ratios of the spirals that usually occur are 89/55,144/89, and even 233/144. Once again, all of the numbers in these ratios are consecutive Fibonacci numbers.

# Teaching Materials

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Leave a message on the discussion page by clicking the 'discussion' tab at the top of this image page.
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