You can use the Geogebra Program to do this assignment. Click here to download it. Use the Download button unless you already have java installed on your computer. You should use the template here to enter your answers in. Just paste your images from Geogebra into the boxes.
You can also just use a compass and straightedge. You will probably want to copy your work at each stage.
Part I: Triangles.
Part II. Squares.
Mark the points where the perpendicular lines meet the two circle at the top and connect the four points to complete the square. you can hide the perpendicular lines.
3. Mark the midpoints if the bottom and top of the square, using the center line of the vesica pisces. You may now erase everything but the square. Save at this point, or make a copy. You will need this again later.
Find the midpoints of the two sides of the square. You will have to use the line bisector in Geogebra. If doing it by hand you can measure to find an approximate midpoint, or, better, use these instructions to bisect the lines manually.
Now connect the diagonal points of the square. Make a smaller square by connecting the four midpoints of the lines that make the original square. Now connect the diagonals of this smaller square. Where have you seen this diagram before?
4. Make a smaller square inside the second square by connecting its midpoints (marked by the diagonals of the original square) in the same manner as above. Now make a fourth larger square outside the original square. Extend the two midpoint lines of the original square, and construct a line at the top left corner that is parallel to the diagonal until that line meets the extended midpoint lines. Repeat for the other three corners.
How is each square related to the diagonal of the next smaller square? What are the relationships between the sizes of the four squares?
Start with a square again, from step 3 above. Draw the diagonal from point A to D. Draw the lines perpendicular to that diagonal AD at points A and D. Then draw a Vesica Pisces with a circle center A, radius AD and a circle center D, radius DA.
Mark where the circles intersect the perpendicular lines and complete the square built on the diagonal AD, by making the segments DF, FE, and EA.
6. Now make another diagonal CB in the original square. and repeat all the steps from number 5 for this diagonal to create another square based upon it, CBHG.
How many squares do you see?
Indicate the relationship between the areas and the sides of the different size squares.
|Unit square ABDC||1||1|
III. The Golden Section and the Pentagon:
1. Construct a Golden rectangle.
Start with a square. Use the regular polygon tool or start from a copy of the square constructed above.
Bisect the bottom of the square and then continue that bottom segment in both directions.
Draw a circle with center E and radius ED to intersect the bottom line. You are inscribing the square in a semi-circle. Mark the point where the circle hits the line F. The line AF is cut by B in the golden section.
Mark the other point where the circle hits the bottom line G. Extend CD in both directions. Raise perpendiculars up at F and G. Mark the two points where these hit line CD, H and I. GHIF is a Square Root of 5 rectangle and ACIF and GHDB are Golden Rectangles.
2. The Golden Spiral.
Start with a Golden rectangle ACIF above. Note that BDIF is also a golden rectangle.
Measure out on DB and IF a length equal to DI. Mark these points J and K. Make the square JDIK
BJKF is also a Golden Rectangle.
Repeat the procedure a couple more times.
Draw an arc from A to D with radius BA. Then do the same with the next smaller square: an arc from D to K with radius JD.
Continue with the next smaller square and so on as far down as you can get. This is the Logarithmic or golden spiral.
Start with a line divided in a Golden section, such as ABF from above. You can also reconstruct one using the square root of 5 rectangle method from above.
Draw circle with center A and radius AB and another circle with center B and radius BA.
Now draw a circle with center A and radius AF. Then another circle with center B with the same radius AF. (You will have to measure AF and use the circle with determined radius function in Geogebra) Mark the points where the two large circles intersect each other and the two small circles. Connect each of these points with each other and with AB to make the pentagon.
4. Pentagram Star.
Start with the pentagon. You may erase all the guidelines. Connect each vertex with the one directly opposite it. This will give you a pentagram star inside the pentagon.
You can repeat this process again
within the internal
Extend each of the sides of the original pentagon to make a larger pentagram outside.
How many instances of the golden relationship can you find between the parts of the pentagram?
IV. The Platonic Solids:
Start with a vesica pisces divided into 4 triangles as in part I above:
Remove the circles and fold you have a tetrahedron.
Extend the vesica pisces to six circles and use it to trace out these 6 squares:
The same pattern with 5 circles will give the octahedron:
Use the same pattern with 8 circles for the icosahedron:
Just hand in the drawn or printed templates.
Extra Credit ONLY
You can cut them out and actually construct the solids for extra credit. If you like , you can print out these already drawn templates to make your models.
1. Measure at least 5 commonly used man made objects. Can you find at least one that has proportions in the golden section?
2. Look at some pictures of art works or architecture. Find at least 5 that have prominent features in the golden section.List their names and the features measured. You might also find this template useful in looking at larger objects. You may find this table helpful in reporting your results
|Name||Feature||Artist (if known)||Web address (if from web) or source||Image
3. Look at some natural objects and parts of natural objects (any living thing, part of a living thing, or highly organized inorganic object, such as a crystal. Don't use anything that has been reshaped by Man, such a cut gem or carved wood.) Find and list at least 8 examples (at least 4 should be from different objects, not parts of the same object) of the golden section. Did you find any organized natural objects whose main divisions were not in the golden proportion? List them. You may find this table helpful in reporting your results
|Object||Part in golden proportion||Image or drawing (optional)||web address:(if from web) or source|
© 2006 David Banach
This work is licensed under a Creative Commons License.