Instructor:Wallpaper Exercises Solutions

Wallpaper Exercises

1. Draw reflections and glides on A-C. Draw rotations on D-F.
2. 1. Sketch #6 (Camels):p2
   2. Sketch #13 (Dragonflies): p4g
   3. Sketch #21 (Men): p3
   4. Sketch #67 (Horsemen): pg
   5. Sketch #69 (Birds, fish, turtles): p3m1
   6. Sketch #105 (Pegasus): p1
   7. Sketch #119 (Fish): p4
3. (Cosmati patterns) A. pmg, B. pm, C. p6m, D. p4m
4. (Egypt tombs) A. p1, or cm or even cmm depending on how closely you look at the details. B. p4m, C. cmm, D. Not a periodic pattern.
5. (Printable sheet) A. cm, B. pmg, C. cmm, D. p6m, E. p4g, F. p31m
6. Note that C stands for "cyclic" and D for "dihedral". Polya / Escher used the number to stand for the highest degree of rotational symmetry. Similarly k stands for mirror line, and g stands for glide-reflection. With that in mind we find that:
   • $C_{1},C_{2},C_{3},C_{4},C_{6}$ correspond to p1, p2, p3, p4, and p6 respectively.
   • $D_{1}kk$, $D_{1}gg$ and $D_{1}kg$ correspond to pm, pg and cm
   • $D_{2}kkkk$, $D_{2}gggg$, $D_{2}kkgg$ and $D_{2}kgkg$ correspond to pmm, pgg, pmg and cmm
   • $D_{3}^{*}$, $D_{3}^{O}$ correspond to p3m1 and p31m
   • $D_{4}^{*}$, $D_{4}^{O}$ correspond to p4m and p4g
   • $D_{6}$ corresponds to p6m
   Note that this problem is very challenging to students!
7. a. No 4 and 6; b. No 4 and 3; c. Yes 3 and 6; d. Yes 2 and 3; e. No perp and 6; f. No perp and 3.
8. Sketch a p4m pattern.
9. Some common patterns:

   cmm	pmm	pgg	p4g	p2	pmg

   See also http://plato.acadiau.ca/courses/educ/reid/Geometry/brick/plane1.html
10. Use P to create a. p1; b. pm; c. p4 and d. another.