Self-Similarity Exercises

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  1. Escher prints:
    1. Discuss the symmetries of Whirlpools (Visions of Symmetry, p. 250).
    2. Discuss the symmetries of Smaller and Smaller (Visions of Symmetry, p. 253).
  2. For each part, describe all symmetries:
    1. The golden triangle spiral.
    2. This crop circle, found May 21, 2003 in Wiltshire, England.
    3. This drawing of calcarina clavigera, a microorganism.
  3. Pattern for exercise 3.
    Make two patterns based on the square pattern shown at right, similar to Escher's Regular Division of the Plane by Similar Figures sketches.
  4. 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.
  5. 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.
  6. What’s going on in Division? What is dividing, and is there any pattern to it?
  7. Describe what’s happening in Fish and Scales. Compare with Print Gallery.
  8. For each of these, draw iterations 0-4.
    1. The initiator is a square, and the transformation is a dilation by 1/2 toward the upper left corner.
      Iteration #0 Iteration #1
      Iter-square-ex0.svg Iter-square-ex1.svg
    2. 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
      Iter-triangle-ex0.svg Iter-triangle-ex1.svg
    3. 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
      Iter-circle-ex0.svg Iter-circle-ex1.svg Iter-circle-ex2.svg
  9. Iteration 3 is shown. Draw iterations 0,1, 2 and 4.
    Iter-rectangle-ex3.svg
  10. Draw the missing iteration:
    Iter-boxfractal.svg
  11. Describe the initiator and the transformation that is iterated:
    1. Dilation-triangles.svg
    2. Pentaflake.svg
  12. List five things in nature that display self-similarity.
  13. The Face of War. Salvador Dali, 1940.
  14. Dali’s The Face of War is an example of self-similarity.
    1. Describe the initiator and the tranformation that is iterated
    2. How many iterations are in the image?
  15. Droste brand cocoa.
  16. 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.
    1. Describe the initiator and the tranformations that are iterated
    2. How many copies of the nurse are in the image?
  17. Another example of the Droste effect: La vache qui rit brand cheese. Explain why the logo is a fractal.
  18. The Sierpinski Triangle appears in African art. Draw three iterations of the Sierpinski triangle. File:African-sierpinski.jpg
  19. 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: Fibonacci-rectangles.svg
    Make a table showing the side lengths of each rectangle. Calculate the ratio of the long side to the short side for each rectangle.
  20. 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!)
  21. 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.

Instructor:Self-Similarity Exercises Solutions