# Harter-Heighway Dragon

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| -                                      | |ImageName=Harter-Heighway Dragon Curve (3D- twist)                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | |ImageName=Harter-Heighway Dragon Curve                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        | |Image=DragonCurve.jpg                                                                                                                                                                                                                                                                                                                                                                                                                                                            |                                                     | |Image=DragonCurve.jpg                                                                                                                                                                                                                                                                                                                                                                                                     |
| -                                      | |ImageIntro=This image is an artistic rendering of the Harter-Heighway Curve (also called the Dragon Curve), which is a fractal. This curve is an iterated function system and is often referred to as the Jurassic Park Curve, because it garnered popularity after being drawn and alluded to in the novel Jurassic Park by Michael Crichton (1990).                                                                                                                            | +                                                   | |ImageIntro=This image is an artistic rendering of the Harter-Heighway Curve (also called the Dragon Curve), which is a fractal. It is often referred to as the Jurassic Park Curve because it garnered popularity after being drawn and alluded to in the novel Jurassic Park by Michael Crichton (1990).                                                                                                                 |
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| -                                      | [[Image:DragonCurve_Construction.png|600px|thumb|First 5 iterations of the Harter-Heighway Curve|left]]                                                                                                                                                                                                                                                                                                                                                                           |                                                     |                                                                                                                                                                                                                                                                                                                                                                                                                            |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                   | This fractal is described by a <balloon title="A crooked line, like the the one used in this fractal, can be considered a curve."> curve</balloon> that undergoes a repetitive process (called an [[Iterated Functions|iterated process]]). To begin the process, the curve has a basic segment of a straight line.                                                                                                        |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                   | Then at each iteration,                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                   | :*Each line is replaced with two line segments at an angle of 90 degrees (other angles can be used to make fractals that look slightly different).                                                                                                                                                                                                                                                                         |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                   | :*Each line is rotated alternatively to the left or to the right of the line it is replacing.                                                                                                                                                                                                                                                                                                                              |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                   | [[Image:DragonCurve_Construction.png|900px|thumb|Base Segment and First 5 iterations of the Harter-Heighway Curve|center]]                                                                                                                                                                                                                                                                                                 |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                   | [[Image:DragonCurve_int15.gif|thumb|200px|right|15th iteration]]                                                                                                                                                                                                                                                                                                                                                           |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                   | The Harter-Heighway Dragon is created by iteration of the curve process described above, and is thus a type of fractal known as '''iterated function systems'''. This process can be repeated infinitely, and the perimeter or length of the dragon is in fact infinite. However, if you look to the image at the right, a 15th iteration of the Harter-Heighway Dragon is already enough to create an impressive fractal. |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                   | An interesting property of this curve is that although the corners of the fractal seem to touch at various points, the curve never actually crosses over itself. Also, the curve exhibits '''self-similarity''' when iterated infinitely because as you look look closer and closer at the curve, the magnified parts of the curve continue to look like the larger curve.                                                 |
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| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                   | To learn another method to create the Harter-Heighway Dragon, click [http://sierra.nmsu.edu/morandi/coursematerials/JurassicParkFractal.html here]                                                                                                                                                                                                                                                                         |
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| -                                      | This fractal is described by a curve that undergoes an iterated process. To begin the process, the curve starts out as a line as the base segment. Each iteration replaces each line with two line segments at an angle of 90 degrees (other angles can be used to make various looking fractals), with each line being rotated alternatively to the left or to the right of the line it is replacing. To learn more about [[Iterated Functions|iterated functions]], click here. | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                            |
| -                                      | [[Image:DragonCurve_int15.gif|thumb|200px|right|15th iteration]]                                                                                                                                                                                                                                                                                                                                                                                                                  | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                            |
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| -                                      | The Harter-Heighway Dragon is created by iteration of the curve process described above. This process can be repeated infinitely, and  the perimeter of the dragon is in fact infinite. However, if you look to the image at the right, a 15th iteration of the Harter-Heighway Dragon is already enough to create an impressive fractal.                                                                                                                                         | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                            |
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| -                                      | The perimeter of the Harter-Heighway curve increases by<math>\sqrt{2}</math> with each repetition of the curve.                                                                                                                                                                                                                                                                                                                                                                   | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                            |
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| -                                      | An interesting property of this curve is that the curve never crosses itself. Also, the curve exhibits ''self-similarity'' because as you look closer and closer at the curve, the curve continues to look like the larger curve.                                                                                                                                                                                                                                                 | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                            |
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|                                        | |HighSchool=Yes                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |                                                     | |HighSchool=Yes                                                                                                                                                                                                                                                                                                                                                                                                            |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                   | {{hide|1=                                                                                                                                                                                                                                                                                                                                                                                                                  |
|                                        | ==Properties==                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |                                                     | ==Properties==                                                                                                                                                                                                                                                                                                                                                                                                             |
| -                                      | {{Hide|1=                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |                                                     |                                                                                                                                                                                                                                                                                                                                                                                                                            |
| -                                      | '''Perimeter''' {{hide|1=                                                                                                                                                                                                                                                                                                                                                                                                                                                         |                                                     |                                                                                                                                                                                                                                                                                                                                                                                                                            |
| -                                      | [[Image:DragonCurve_basic.png|thumb|right|First iteration in detail]]                                                                                                                                                                                                                                                                                                                                                                                                             |                                                     |                                                                                                                                                                                                                                                                                                                                                                                                                            |
| -                                      | The perimeter of the Harter-Heighway curve increases by a factor of <math>\sqrt{2}</math> for each iteration.                                                                                                                                                                                                                                                                                                                                                                     |                                                     |                                                                                                                                                                                                                                                                                                                                                                                                                            |
| -                                      | For example, if the first iteration is split up into two isosceles triangles, the ratio between the base segment and first iteration is: <math>\frac{2s\sqrt{2}}{2s} = \sqrt{2}</math>                                                                                                                                                                                                                                                                                            |                                                     |                                                                                                                                                                                                                                                                                                                                                                                                                            |
| -                                      | }}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |                                                     |                                                                                                                                                                                                                                                                                                                                                                                                                            |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                   | The Harter-Heighway Dragon curve has several different properties we can derive.                                                                                                                                                                                                                                                                                                                                           |
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| -                                      | '''Number of Sides''' {{hide|1=                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                   | ===Perimeter===                                                                                                                                                                                                                                                                                                                                                                                                            |
| -                                      | The number of sides (<math>N_k</math>) of the Harter-Heighway curve for any degree of iteration (''k'') is given by <math>N_k = 2^k\,</math>, where the "sides" of the curve refer to alternating slanted lines of the fractal.                                                                                                                                                                                                                                                   | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                            |
| -                                      | }}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                            |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                   | [[Image:DragonCurve_basic.png|thumb|right|1st iteration of the Harter-Heighway Dragon]]                                                                                                                                                                                                                                                                                                                                    |
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| -                                      | '''Fractal Dimension''' {{hide|1=                                                                                                                                                                                                                                                                                                                                                                                                                                                 |                                                     |                                                                                                                                                                                                                                                                                                                                                                                                                            |
| -                                      | [[Image:DragonCurveDimension.png|right|thumb|225px|2nd iteration of the Harter-Heighway Dragon]]                                                                                                                                                                                                                                                                                                                                                                                  |                                                     |                                                                                                                                                                                                                                                                                                                                                                                                                            |
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| -                                      | The [[Fractal Dimension]] of the Harter-Heighway Curve can also be calculated using the equation: <math>\frac{logN}{loge}</math>                                                                                                                                                                                                                                                                                                                                                  | +                                                   | The perimeter of the Harter-Heighway curve increases by a factor of <math>\sqrt{2}</math> for each iteration.                                                                                                                                                                                                                                                                                                              |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                   | For example, if you look at the picture to the right, the straight red line shows the fractal as its base segment and the black crooked line shows the fractal at its first iteration.                                                                                                                                                                                                                                     |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |                                                     |                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                   | If the first iteration is split up into two triangles, the ratio of the perimeter of the first iteration over the base segment is:                                                                                                                                                                                                                                                                                         |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                   | ::<math>\frac{s\sqrt{2} + s\sqrt{2}}{s + s} = \frac{2s\sqrt{2}}{2s} = \frac{\sqrt{2}}{1}</math>                                                                                                                                                                                                                                                                                                                            |
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| -                                      | As seen from the image of the second iteration of the curve, there are two new curves that arise during the iteration and N = 2.                                                                                                                                                                                                                                                                                                                                                  | +                                                   | ===Number of Sides===                                                                                                                                                                                                                                                                                                                                                                                                      |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                   | [[Image:DragonCurve_Sides.png|900px|center]]                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                   | The number of sides (<math>N_k</math>) of the Harter-Heighway curve for any degree of iteration (''k'') is given by <math>N_k = 2^k\,</math>, where the "sides" of the curve refer to alternating slanted lines of the fractal.                                                                                                                                                                                            |
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| -                                      | Also, the ratio of the lengths of each new curve to the old curve is: <math>\frac{2s\sqrt{2}}{2s} = \sqrt{2} </math>, so e = <math>\sqrt{2}</math>.                                                                                                                                                                                                                                                                                                                               | +                                                   | For example, the third iteration of this curve should have a total number of sides <math>N_3 = 2^3 = 8\,</math>.                                                                                                                                                                                                                                                                                                           |
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| -                                      | Thus, the fractal dimension is <math>\frac{logN}{loge} = \frac{log2}{log\sqrt{2}} = 2 </math>, and it is a space-filling curve.                                                                                                                                                                                                                                                                                                                                                   | +                                                   | ===Fractal Dimension===                                                                                                                                                                                                                                                                                                                                                                                                    |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                   | The [[Fractal Dimension]] of the Harter-Heighway Curve can also be calculated using the equation: <math>\frac{logN}{loge}</math>.                                                                                                                                                                                                                                                                                          |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                   | Let us use the second iteration of the curve as seen below to calculate the fractal dimension.                                                                                                                                                                                                                                                                                                                             |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                   | [[Image:DragonCurveDimension1.png|center]]                                                                                                                                                                                                                                                                                                                                                                                 |
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| -                                      | }}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                | +                                                   | There are two new curves that arise during the iteration so that <math>N = 2\,</math>.                                                                                                                                                                                                                                                                                                                                     |
| -                                      | }}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                            |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                   | Also, the ratio of the lengths of each new curve to the old curve is: <math>\frac{4(s\sqrt{2})}{4(s)} = \sqrt{2} </math>, so that <math>e = \sqrt{2}</math>.                                                                                                                                                                                                                                                               |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                   | Thus, the fractal dimension is <math>\frac{logN}{loge} = \frac{log2}{log\sqrt{2}} = 2 </math>, and it is a <balloon title="A space-filling curve in 2-dimensions is a curve with a fractal dimension of exactly 2. This means that the curve touches every point in the unit square."> space-filling curve</balloon>.                                                                                                      |
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|                                        | ==Changing the Angle==                                                                                                                                                                                                                                                                                                                                                                                                                                                            |                                                     | ==Changing the Angle==                                                                                                                                                                                                                                                                                                                                                                                                     |
| -                                      | {{hide|1=                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |                                                     |                                                                                                                                                                                                                                                                                                                                                                                                                            |
| -                                      | The Harter-Heighway curve iterates with a 90 degree angle. However, if the angle is changed, new curves can be created:                                                                                                                                                                                                                                                                                                                                                           |                                                     |                                                                                                                                                                                                                                                                                                                                                                                                                            |
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| -                                      | <gallery caption="" widths="300px" heights="200px" perrow="3">                                                                                                                                                                                                                                                                                                                                                                                                                    | +                                                   | The Harter-Heighway curve iterates with a 90 degree angle; however, if the angle is changed, new curves can be created. The following fractals are the result of 13 iterations.                                                                                                                                                                                                                                            |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                   | <gallery caption="" widths="305px" heights="205px" perrow="3">                                                                                                                                                                                                                                                                                                                                                             |
|                                        | Image:CurveAngle85.jpg|'''Curve with angle 85'''                                                                                                                                                                                                                                                                                                                                                                                                                                  |                                                     | Image:CurveAngle85.jpg|'''Curve with angle 85'''                                                                                                                                                                                                                                                                                                                                                                           |
|                                        | Image:CurveAngle100.jpg|'''Curve with angle 100'''                                                                                                                                                                                                                                                                                                                                                                                                                                |                                                     | Image:CurveAngle100.jpg|'''Curve with angle 100'''                                                                                                                                                                                                                                                                                                                                                                         |
|                                        | Image:CurveAngle110.jpg|'''Curve with angle 110'''                                                                                                                                                                                                                                                                                                                                                                                                                                |                                                     | Image:CurveAngle110.jpg|'''Curve with angle 110'''                                                                                                                                                                                                                                                                                                                                                                         |
|                                        | </gallery>                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |                                                     | </gallery>                                                                                                                                                                                                                                                                                                                                                                                                                 |
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|                                        | |other=Algebra                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |                                                     | |other=Algebra                                                                                                                                                                                                                                                                                                                                                                                                             |
|                                        | |AuthorName=SolKoll                                                                                                                                                                                                                                                                                                                                                                                                                                                               |                                                     | |AuthorName=SolKoll                                                                                                                                                                                                                                                                                                                                                                                                        |
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|                                        | |Field=Dynamic Systems                                                                                                                                                                                                                                                                                                                                                                                                                                                            |                                                     | |Field=Dynamic Systems                                                                                                                                                                                                                                                                                                                                                                                                     |
|                                        | |Field2=Fractals                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |                                                     | |Field2=Fractals                                                                                                                                                                                                                                                                                                                                                                                                           |
| -                                      | |FieldLinks=                                                                                                                                                                                                                                                                                                                                                                                                                                                                      | +                                                   | |FieldLinks=:To read about an alternate method of creating the Harter-Heighway Dragon http://sierra.nmsu.edu/morandi/coursematerials/JurassicParkFractal.html                                                                                                                                                                                                                                                              |
|                                        | |References=                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |                                                     | |References=                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        | Wikipedia, [http://en.wikipedia.org/wiki/Dragon_curve Wikipedia's Dragon Curve page]                                                                                                                                                                                                                                                                                                                                                                                              |                                                     | Wikipedia, [http://en.wikipedia.org/wiki/Dragon_curve Wikipedia's Dragon Curve page]                                                                                                                                                                                                                                                                                                                                       |
|                                        | Cynthia Lanius, [http://math.rice.edu/~lanius/frac/jurra.html Cynthia Lanius' Fractals Unit: A Jurassic Park Fractal]                                                                                                                                                                                                                                                                                                                                                             |                                                     | Cynthia Lanius, [http://math.rice.edu/~lanius/frac/jurra.html Cynthia Lanius' Fractals Unit: A Jurassic Park Fractal]                                                                                                                                                                                                                                                                                                      |
| -                                      | |ToDo=*An animation for the showing the fractal being drawn gradually through increasing iterations                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | |ToDo=                                                                                                                                                                                                                                                                                                                                                                                                                     |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                   | An animation of the fractal being drawn gradually through increasing iterations (a frame for each individual iteration)                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                   | Also, an animation that draws the curve at the 13 or so iteration, but slowly to show that the curve never crosses itself.                                                                                                                                                                                                                                                                                                 |
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## Current revision

Harter-Heighway Dragon Curve
Fields: Dynamic Systems and Fractals
Image Created By: SolKoll
Website: Wikimedia Commons

Harter-Heighway Dragon Curve

This image is an artistic rendering of the Harter-Heighway Curve (also called the Dragon Curve), which is a fractal. It is often referred to as the Jurassic Park Curve because it garnered popularity after being drawn and alluded to in the novel Jurassic Park by Michael Crichton (1990).

# Basic Description

This fractal is described by a curve that undergoes a repetitive process (called an iterated process). To begin the process, the curve has a basic segment of a straight line.

Then at each iteration,

• Each line is replaced with two line segments at an angle of 90 degrees (other angles can be used to make fractals that look slightly different).
• Each line is rotated alternatively to the left or to the right of the line it is replacing.

Base Segment and First 5 iterations of the Harter-Heighway Curve
15th iteration

The Harter-Heighway Dragon is created by iteration of the curve process described above, and is thus a type of fractal known as iterated function systems. This process can be repeated infinitely, and the perimeter or length of the dragon is in fact infinite. However, if you look to the image at the right, a 15th iteration of the Harter-Heighway Dragon is already enough to create an impressive fractal.

An interesting property of this curve is that although the corners of the fractal seem to touch at various points, the curve never actually crosses over itself. Also, the curve exhibits self-similarity when iterated infinitely because as you look look closer and closer at the curve, the magnified parts of the curve continue to look like the larger curve.

# A More Mathematical Explanation

Note: understanding of this explanation requires: *Algebra

## Properties

The Harter-Heighway Dragon curve has several different properties we can derive.

### Perimeter

1st iteration of the Harter-Heighway Dragon

The perimeter of the Harter-Heighway curve increases by a factor of $\sqrt{2}$ for each iteration. For example, if you look at the picture to the right, the straight red line shows the fractal as its base segment and the black crooked line shows the fractal at its first iteration.

If the first iteration is split up into two triangles, the ratio of the perimeter of the first iteration over the base segment is:

$\frac{s\sqrt{2} + s\sqrt{2}}{s + s} = \frac{2s\sqrt{2}}{2s} = \frac{\sqrt{2}}{1}$

### Number of Sides

The number of sides ($N_k$) of the Harter-Heighway curve for any degree of iteration (k) is given by $N_k = 2^k\,$, where the "sides" of the curve refer to alternating slanted lines of the fractal.

For example, the third iteration of this curve should have a total number of sides $N_3 = 2^3 = 8\,$.

### Fractal Dimension

The Fractal Dimension of the Harter-Heighway Curve can also be calculated using the equation: $\frac{logN}{loge}$.

Let us use the second iteration of the curve as seen below to calculate the fractal dimension.

There are two new curves that arise during the iteration so that $N = 2\,$.

Also, the ratio of the lengths of each new curve to the old curve is: $\frac{4(s\sqrt{2})}{4(s)} = \sqrt{2}$, so that $e = \sqrt{2}$.

Thus, the fractal dimension is $\frac{logN}{loge} = \frac{log2}{log\sqrt{2}} = 2$, and it is a space-filling curve.

## Changing the Angle

The Harter-Heighway curve iterates with a 90 degree angle; however, if the angle is changed, new curves can be created. The following fractals are the result of 13 iterations.

Curve with angle 85  Curve with angle 100  Curve with angle 110

# Teaching Materials

There are currently no teaching materials for this page. Add teaching materials.

# About the Creator of this Image

SolKoll is interested in fractals, and created this image using an iterated function system (IFS).

# Related Links

### Additional Resources

To read about an alternate method of creating the Harter-Heighway Dragon http://sierra.nmsu.edu/morandi/coursematerials/JurassicParkFractal.html

# References

Wikipedia, Wikipedia's Dragon Curve page Cynthia Lanius, Cynthia Lanius' Fractals Unit: A Jurassic Park Fractal

# Future Directions for this Page

An animation of the fractal being drawn gradually through increasing iterations (a frame for each individual iteration) Also, an animation that draws the curve at the 13 or so iteration, but slowly to show that the curve never crosses itself.

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