1. Let X be a non-empty, closed octagonal surface (meaning that every face of X is an octagon), in which 3 octagons meet

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1 Let X Be A Non Empty Closed Octagonal Surface Meaning That Every Face Of X Is An Octagon In Which 3 Octagons Meet 1 (52.59 KiB) Viewed 33 times

1. Let X be a non-empty, closed octagonal surface (meaning that every face of X is an octagon), in which 3 octagons meet at every vertex. (a) Show that the Euler characteristic satisfies 8X(X) = -V, where V is the number of vertices. Deduce that X cannot be realised as a subset of R3 with convex interior. [4 marks] (b) Assume that each face of X is a regular octagon. Compute the curvature of a vertex. Recall the Gauss-Bonnet formula for a closed polygonal surface, and show that it holds for X by direct verification. [3 marks] (c) Prove that x(x) < -2. [3 marks]