Complete the two equilateral triangles formed by the points. Extend the sides of the top triangle down to meet the circles.
Complete the two new triangles formed by the point of intersection of these extended lines with the circles. You can erase all the lines except the large triangle now.
Connect each midpoint with the vertex across from it. Where have you seen this triangle figure before? It may help to erase the circles.
Where have you seen this triangle figure before? It may help to erase the circles.
Part II. Squares.
Start with a Vesica Pisces and draw a horizontal line connecting the midpoints. Then draw a line perpendicular to the line at its two endpoints..
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. 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.
5a..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 in the table below.
Unit square ABDC
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.
3. 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 Euklid) 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.
5. 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.
6. You can repeat this process again within the internal pentagon.
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? Indicate how many and where you find them.
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:
PART V: Finding the Golden Section
Required Measurements. You must do the minimum. You can get extra points by doing more and finding really interesting things.
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
Artist (if known)
Web address (if from web)
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