Instructor:Hyperbolic Geometry Exercises Solutions

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Revision as of 22:42, 29 April 2008 by Bryan (talk | contribs) (Fix #12)
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Hyperbolic Geometry Exercises

  1. Four geodesics that don't cross. Hyp-ex1-sol.svg
  2. Four geodesics through the center. Hyp-ex2-sol.svg
  3. Three geodesics through point, parallel to given geodesic. Hyp-ex3-sol.svg
  4. Triangle with three 5° angles. Hyp-ex4-sol.svg
  5. 90°-5°-5° triangle. Hyp-ex5-sol.svg
  6. Right angled pentagon. Hyp-ex6-sol.svg
  7. 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.
  8. 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.
  9. Only b, c, and d, the Circle Limit artworks.
    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>. Hyp-ex12d-sol.svg.
    5. Defect 280°. Area <math>\frac{14}{9}\pi</math>.
  10. a) {3,10}; b) All are 36°; c) 72°; d) <math>2\pi/5</math>; e) Draw {10,3}.
  11. a) {6,4}; b) All are 90°; c) 180°; d) <math>\pi</math>; e) Draw {4,6}.
  12. a) 180°; b)Largest area <math>\pi</math>, smallest area 0.
  13. Largest area </math>2\pi</math>, smallest area 0.
  14. 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>