Instructor:Self-Similarity Exercises Solutions

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Self-Similarity Exercises

    1. Whirlpools has a C2 symmetry which interchanges the two colors. In addition, each "whirlpool" has a dilation by a factor of 1/2, a dilation-rotation with a 90° rotation and dilation factor of aboout 80%, and finally a dilation-reflection that interchanges colors.
    2. Smaller and Smaller has C4 symmetry overall. Ignoring details at the edges, there is a dilation (central similarity) by a factor of 1/2 which takes black lizards to black lizards and interchanges the red and white lizards.
    1. The spiral limits to a point which is the center point for a dilation-rotation by about 110° with a scale factor of about 1.6 (the golden ratio)
    2. The crop circle has overall C6 symmetry, and a dilation-reflection with a scale factor of about 60%.
    3. Though the spiral stops after two times around, the organism suggests a dilation-rotation of order 12 with a scale factor around 95%.
  1. Make two patterns
  2. Square-limit-scaffold.svg
  3. Sketch101-scaffold.svg. Notice the top pattern of six large squares is repeated twice on a smaller scale below itself.
  4. Lizards are dividing. On the outside, lizards are large. Moving towards the center, the large lizards are "torn", and then split into two smaller lizards. The print has C2 symmetry (note Escher's two signatures), but no other pattern to the divisions is apparent.
  5. Scales of big fish turn into small fish which become bigger fish which has scales which are small fish - there is a cycle of containment. This is also the theme of Print Gallery, where the gallery contains a man looking at a picture of a town which has the gallery in it.
  6. Iteration #4 is shown. a. Iter-square-ex4.svg b. Iter-triangle-ex4.svg c. Iter-circle-ex4.svg
  7. 0 1 2 3 4
    Iter-rectangle-ex0.svg Iter-rectangle-ex1.svg Iter-rectangle-ex2.svg Iter-rectangle-ex3.svg Iter-rectangle-ex4.svg
  8. Iter-boxfractal-ans.svg
  9. a. Initiator is an equilateral triangle, transformation is a dilation-rotation with scale factor of 1/2 and rotation by 180°. b. Initiator is a pentagon, five transformations put copies of the pentagon along each of its sides, then overall rescaling to maintain size. You could also say that five "cuts" are made in each pentagon at each step.
  10. Trees, snail shells, river systems, lungs, ferns for example.
  11. Initiator is the face, three transformations shrink and put a copy in each eye and one in the mouth. At least three iterations are shown.
  12. Initiator is the nurse with tray, cup and box. Two transformations are iterated, one puts a copy on the box and the other on the cup. There are at least seven nurses.
  13. The logo is a fractal because it is self similar - there are smaller copies of the whole cheese on each of the cow's earrings.