Instructor:Wallpaper Exercises Solutions

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Wallpaper Exercises

  1. Draw reflections and glides on A-C. Draw rotations on D-F.
    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
  2. (Cosmati patterns) A. pmg, B. pm, C. p6m, D. p4m
  3. (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.
  4. (Printable sheet) A. cm, B. pmg, C. cmm, D. p6m, E. p4g, F. p31m
  5. 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:
    • <math>C_1, C_2, C_3, C_4, C_6 </math> correspond to p1, p2, p3, p4, and p6 respectively.
    • <math> D_1 kk </math>, <math> D_1 gg </math> and <math> D_1 kg </math> correspond to pm, pg and cm
    • <math> D_2 kkkk </math>, <math> D_2 gggg </math>, <math> D_2 kkgg </math> and <math> D_2 kgkg </math> correspond to pmm, pgg, pmg and cmm
    • <math> D_3^* </math>, <math> D_3^O </math> correspond to p3m1 and p31m
    • <math> D_4^* </math>, <math> D_4^O </math> correspond to p4m and p4g
    • <math> D_6 </math> corresponds to p6m
    Note that this problem is very challenging to students!
  6. 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.
  7. Sketch a p4m pattern.
  8. Some common patterns:
    cmm pmm pgg p4g p2 pmg
    Bricks-cmm.jpg Bricks-pmm.jpg Bricks-pgg.jpg Bricks-p4g.jpg Bricks-p2.jpg Bricks-pmg.jpg
    See also http://plato.acadiau.ca/courses/educ/reid/Geometry/brick/plane1.html
  9. Use P to create a. p1; b. pm; c. p4 and d. another.