Difference between revisions of "Instructor:Spherical Geometry Exercises Solutions"

1. Yes, every point on the sphere has exactly one antipodal points.
2. Although tessellations of the plane suggest infinity because they can be continued forever, Escher felt the necessity of an edge harmed the effect. Escher says that as you turn the ball, the neverending series of motifs suggests infinity. On the other hand, there are only finitely many motifs on the ball. Which is more compelling?
3. There are really two valid choices here: 1) A is between B and C if A is on a geodesic segment joining B and C, or 2) A is between B and C if A is on the short geodesic segment joining B and C. In both cases, St. Louis is between the poles. In case 1, the north pole is between the south pole and St. Louis, but not in case 2.
4. Draw the picture.
5. Draw the picture.
6. Draw the picture (it should look a lot like question 4's picture).
7. a. 90°; b. 1/8; c. 90°; d. 1/8; e. 36°; f. 1/20; g. 90°; h. 1/16; i. 90°; j. 180°
8. 720°
9. 720°
10. 300°
11. $\frac{(n+2)180^\circ}{n}$
12. Draw a geodesic segment connecting two corners of the quadrilateral. This splits the quadrilateral into two triangles. The sum of angles in the quadrilateral is the sum of the angles in the two triangles, which is larger than 180° + 180° = 360°.
13. The north and south edges of Colorado are not geodesics - they are made from parallels. This means Colorado is not a quadrilateral, it has curved edges.
14. Draw the picture.
15. A biangle with corner angle $\theta$ covers $\theta/360^\circ$ of the sphere.
16. a. They are close, since the diameter is 10 and circumference should be $10\pi$, about 31.4 cubits; b. On a sphere, the diameter of a circle with diameter $d$ is less than $d\times\pi$, so it could possibly be built; c. A sphere of diameter about 19.1 cubits (a pretty small planet!).
17. No. Any three points on the sphere give a flat triangle, but it's possible to pick four points on a sphere that don't lie at the corners of a flat quadrilateral.
18. Build the models!
19. Tetrahedrons.
20. Octahedrons.
21. 12.
22. 60.
23. Draw the pictures.
24. The regular tessellation by squares (it's self-dual).
25. a. Tetrahedron; b. Octahedron; c. Cube; d. Icosahedron; e. Dodecahedron
26. He's relating duality of polyhedron to gender roles. In particular, he (arbitrarily) puts the "female" solids inside the "male" solids because they are subservient. He's also got to reach to include the tetrahedron, which is self-dual, and therefore a hermaphrodite (!??). Good thing Kepler lived around the turn of the 16th century, because this sort of gender stereotyping was typical then.