Wallpaper Exercises

From EscherMath

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 D₂kkkk 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.

Instructor:Wallpaper Exercises Solutions (Instructors only).