# Self-Similarity Exercises

From EscherMath

- Escher prints:
- Discuss the symmetries of Whirlpools (Visions of Symmetry, p. 250).
- Discuss the symmetries of Smaller and Smaller (Visions of Symmetry, p. 253).

- For each part, describe all symmetries:
- The golden triangle spiral.
- This crop circle, found May 21, 2003 in Wiltshire, England.
- This drawing of calcarina clavigera, a microorganism.

- Make two patterns based on the square pattern shown at right, similar to Escher's Regular Division of the Plane by Similar Figures sketches.
- Escher's Square Limit is based on the same geometric scaffolding as Regelmatige vlakverdeling, Plate VI. Find four copies of the Plate VI geometry in the Square Limit geometry and draw a sketch to show where they are. Note that the Plate VI geometry is “house” shaped, and the Square Limit is a square, so draw a square and put (at least) four “houses” in it.
- Sketch the underlying geometric scaffolding for Sketch #101 (Division). Use graph paper, and make a 45°-45°-90° triangle for each lizard. As a hint, start with a rectangle that is 16 squares wide by 8 squares high for the top row of four lizards.
- What’s going on in Division? What is dividing, and is there any pattern to it?
- Describe what’s happening in Fish and Scales. Compare with Print Gallery.
- For each of these, draw iterations 0-4.
- The initiator is a square, and the transformation is a dilation by 1/2 toward the upper left corner.
Iteration #0 Iteration #1 - The initiator is a 45°-45°-90° triangle, and the transformation is a dilation-rotation, turning 45° clockwise and dilating by a factor of $ 1/\sqrt{2} $.
Iteration #0 Iteration #1 - The initiator is a circle, and two transformations are iterated. They are dilations by 1/2 towards the leftmost and rightmost points of the circle.
Iteration #0 Iteration #1 Iteration #2

- The initiator is a square, and the transformation is a dilation by 1/2 toward the upper left corner.
- Iteration 3 is shown. Draw iterations 0,1, 2 and 4.

- Draw the missing iteration:
- Describe the initiator and the transformation that is iterated:
- List five things in nature that display self-similarity.
- Dali’s The Face of War is an example of self-similarity.
- Describe the initiator and the tranformation that is iterated
- How many iterations are in the image?

- The Droste effect is a term for a picture that would realistically contain an image of itself. The term
comes from a well-known example, the design for the cocoa boxes for the Dutch brand Droste.
- Describe the initiator and the tranformations that are iterated
- How many copies of the nurse are in the image?

- Another example of the Droste effect: La vache qui rit brand cheese. Explain why the logo is a fractal.
- The Sierpinski Triangle appears in African art. Draw three iterations of the Sierpinski triangle. File:African-sierpinski.jpg
- On a fresh piece of graph paper, draw one small square. Draw another square next to it on the right, making a 2x1 rectangle. Now draw another square along the long edge of the 2x1 rectangle, making a 3x2 rectangle. Continue this process, spiraling outward, until you're out of room on the page:

Make a table showing the side lengths of each rectangle. Calculate the ratio of the long side to the short side for each rectangle. - In the previous problem, the rectangles changed shape less and less as the process continued (the ratios of long to short sides didn't change much). Suppose you want a rectangle that stays exactly the exactly the same shape when a square is attached: File:Golden-rectangle.svg This means the side ratios must be equal: $ \frac{x}{1} = \frac{1+x}{x} $. Solve this equation for $ x $. (Cross multiply and use the quadratic formula!)
- Calculate the ratio of your height to the height of your navel. Do the same for four friends. Compare your results with the previous two problems.