Geometry | Part IB, 2003

State and prove the Gauss-Bonnet formula for the area of a spherical triangle. Deduce a formula for the area of a spherical nn-gon with angles α1,α2,,αn\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}. For what range of values of α\alpha does there exist a (convex) regular spherical nn-gon with angle α\alpha ?

Let Δ\Delta be a spherical triangle with angles π/p,π/q\pi / p, \pi / q and π/r\pi / r where p,q,rp, q, r are integers, and let GG be the group of isometries of the sphere generated by reflections in the three sides of Δ\Delta. List the possible values of (p,q,r)(p, q, r), and in each case calculate the order of the corresponding group GG. If (p,q,r)=(2,3,5)(p, q, r)=(2,3,5), show how to construct a regular dodecahedron whose group of symmetries is GG.

[You may assume that the images of Δ\Delta under the elements of GG form a tessellation of the sphere.]

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