Instructor:Wallpaper Exercises Solutions

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Wallpaper Exercises

  1. a. p2; b. p4g; c. p3; d. pg; e. p3m1; f. p1; g. p4.
  2. 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!
  3. 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
  4. Sketch a p4m pattern.
  5. a. Early sketches have rotation (p2, pgg). Later sketches have no rotation (p1, pg);
    b. Because four legged mammals only look natural in one orientation, with feet facing down. See Visions of Symmetry pg 77-78.
  6. Four Legged Mammals
    Sketch Group Sketch Group
    Sketch #1 (Lion) p2 Sketch #61 (Weird) pg
    Sketch #2 (Lion) pgg Sketch #66 (Pegasus) pg
    Sketch #6 (Camel) p2 Sketch #67 (Horseman) pg
    Sketch #7 (Squirrel) p2 Sketch #76 (Bird/Horse) pg
    Sketch #8 (Horse) p2 Sketch #78 (Pegasus) pg
    Sketch #16 (Dog) pgg Sketch #83 (Odd stuff) p1
    Sketch #97 (Dog) pg
    Sketch #105 (Pegasus) p1
    Sketch #129 (Fish/horse) p1
    Sketch #130 (Fish/horse) pg
  7. Because they can be shown in any orientation, they look natural in symmetry patterns which have rotation symmetry. They can naturally be used as decoration on floors, walls, or celings since we are used to seeing them from different angles. In addition, these animals need no visible means of support. (See Visions of Symmetry pg. 77)
  8. Choose a sketch and compare with others.
  9. Use P to create a. p1; b. pm; c. p4 and d. another.
  10. A. pmg, B. pm, C. p6m, D. p4m
  11. 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.
  12. a. The cultures are: Japanese (1), the Orthodox church (2) and the Moorish culture is represented by a pattern from Cordoba (3) and from the Alhambra (the rest); b. The parallelogram, rectangle, square, triangle, rhombus, and hexagon; c. Top row: p1, p4 and bottom row p2, pg, pgg; d. The top row ones are likely easiest to explain. The tessellations with the glide reflections are likely the most difficult to explain.
  13. 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.