Wallpaper Exercises

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  1. Decide which of the 17 symmetry groups each of these Escher drawings has. You can ignore colors.
    1. Sketch #6 (Camels)
    2. Sketch #13 (Dragonflies)
    3. Sketch #21 (Men)
    4. Sketch #67 (Horsemen)
    5. Sketch #69 (Birds, fish, turtles)
    6. Sketch #105 (Pegasus)
    7. Sketch #119 (Fish)
  2. G. Polya came up with his own names for the 17 wallpaper groups. His picture, with the names, is on page 23 of Visions of Symmetry (and see also Escher's sketches on the following pages). Figure out the crystallography names for his 17 patterns, and make a chart showing the correspondence. (Note that Polya's <math>D_{2}kkkk</math> is colored, and he considers the color preserving symmetry group)
  3. Find four different patterns used for laying bricks (look around). Sketch them on graph paper, and decide which symmetry group each one has.
  4. Sketch an interesting pattern with symmetry group p4m.
  5. Flip through Escher's regular division notebook, pages 116-229 of Visions of Symmetry. Find ten sketches featuring four-legged mammals. (A four-legged mammal has four legs, and is a mammal - horse, dog, pegasus, lion, etc. etc. No people, fish, lizards.)
    1. Find the wallpaper symmetry group for each of the ten sketches (see Visions of Symmetry page 330). How does Escher's choice of symmetry group change between the early (low-numbered) prints and the later (high-numbered) prints?
    2. Explain why he made this deliberate change.
    3. Do you find the patterns in the later sketches more satisfying?
  6. Explain why Escher predominantly chose to use birds, fish, and lizards in his patterns.
  7. Choose a sketch from the regular division notebook (pages 116-229 of Visions of Symmetry) that you particularly like. Explain what you like about it, and compare with similar sketches that don't work as well.
  8. In the border pattern section we constructed some borderpatterns by picking a motif and then reflecting, rotating and glide reflecting the motif until we had a pattern with the desired symmetries. Use the letter P as your starting point: LetterP.jpg
    1. The symmetry group p1 has only translations in its symmetry group. Create a p1 tessellation using the letter P.
    2. The symmetry group pm has only translations and reflections in its symmetry group. Create a pm tessellation using the letter P.
    3. The symmetry group p4 has only translations and 4-fold rotations in its symmetry group. Create a p4 tessellation using the letter P.
    4. Pick one other symmetry group and create a tessellation with that group.
  9. The Cosmati were a family of artisans in Medieval Rome who laid beautiful tile floors throughout the city, notably in many of the churches of the period. The examples below are pictures taken at Santa Maria in Cosmedin by Blake Mellor. Find the symmetry groups of these tilings. (Click on the pictures for a larger image.)

  10. The ancient Egyptians decorated their tombs with some interesting patterns. The examples below are pictures taken from Thierry Benderitters site osirisnet. Find the symmetry groups of these tilings. (Click on the pictures for a larger image.)
  11. In Visions of Symmetry pg 31-36 there is a discussion of how Escher thought about tessellations, or as he called them: "regelmatige vlakverdelingen" (regular divisions of the plane). Escher used three posters when lecturing about his work on these regular divisions of the plane.
    1. The first poster shows examples of tessellations from various cultures. What cultures are represented?
    2. The second poster represents the geometric tessellations that Escher used as a starting point to create his more intricate tessellations. What are the six "primitive" tessellations?
    3. The third poster explains and demonstrates the geometric motions that preserve shape: translation, rotation and glide reflection. What are the symmetry groups of the 5 examples Escher gives?
    4. Visions of Symmetry talks about the explanations of the 5 tessellations in this third poster. The tessellation with only tessellations is the easiest to understand. After reading the explanations, how would you rank the level of diffuculty of the other tessellations? Think about explaining the drawings to someone not in this class. This should give you some idea of how you would rank the explanations.

Instructor:Wallpaper Exercises Solutions (Instructors only).