Course:Math 124 F07 Bart Homework 2

Due Friday September 21

Instructions: On this assignment collaborative work is permitted. You may discuss the problems with other students, but the work you turn in must be your own (i.e. written up by you). You may of course use hints and suggestions given during class or office hours. The solutions you turn in must be

   * Neatly written up (using complete sentences), on one side of each page.
   * Labeled appropriately with the problem number.
   * Stapled (if there is more than one page). 

1. Copy (or print out) these simple frieze patterns, and mark all symmetries for each pattern:
   1. 
   2. 

   3. 

2. Use this motif to draw seven frieze patterns, one with each symmetry group:
3. Find the four colorful strip patterns in the top left corner of Visions of Symmetry Page 13. For each pattern, decide which frieze symmetry group it has. You can ignore colors.
4. What is the symmetry group of the frieze pattern on Visions of Symmetry Page 42?
5. Explain why a frieze pattern can have only 180° rotations.
6. 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 $D_{2}kkkk$ is colored, and he considers the color preserving symmetry group)
7. Find four different patterns used for laying bricks (look around). Sketch them on graph paper, and decide which symmetry group each one has.
8. 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?