Instructor:Hyperbolic Geometry Exercises Solutions

Hyperbolic Geometry Exercises

1. Many ways to draw. Here's one:
2. Many ways to draw. Here's one:
3. Many ways to draw. Here's one:
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7. 180°
8. Fish in Circle Limit I alternate directions along geodesics, so that they are alternately head to head and tail to tail. In Circle Limit III, fish along the white lines all face the same direction, as if they could all be swimming forwards.
9. He's talking about Circle Limit III. He needed to print each of four colors plus black. Because of the fourfold rotation symmetry, he only carved a block to print ¼ of the circle. So, (five colors) × (four impressions each) gives 20 impressions total.
10. • Sketch #45 (Angels and Devils): a) 8; b) 4; c) 45°-45°-90°; d) 180°; e) Euclidean
    • Sphere with Angels and Devils: a) 6; b) 4; c) 60°-60°-90°; d) 210°; e) Spherical
    • Circle Limit IV (Heaven and Hell): a) 8; b) 6; c) 45°-45°-60°; d) 150°; e) Hyperbolic
11. Only b, c, and d, the Circle Limit artworks.
12. 1. Defect 165°. Area <math>\frac{11}{12}\pi</math>.
    2. Defect 80°. Area <math>\frac{4}{9}\pi</math>.
    3. Defect 90°. Area <math>\frac{\pi}{2}</math>.
    4. Defect 90°. Area <math>\frac{\pi}{2}</math>. .
    5. Defect 280°. Area <math>\frac{14}{9}\pi</math>.
13. a) {6,4}; b) All are 90°; c) 180°; d) <math>\pi</math>
19. a) 6, 12, 18, 24, 30, 6n; b) 5, 5, 1, 0, 0, 0; c) 7, 21, 63, 189, 567, <math>7\cdot 3^{n-1}</math>