Papers › The Farthest Point Map on the Regular Dodecahedron
The Farthest Point Map on the Regular Dodecahedron
Richard Evan Schwartz
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Let X be the regular dodecahedron, equipped with its intrinsic path metric. Given p ∈X let G(p)=-q where q is the point on X which maximizes the distance to p. (Generically, G is single-valued.) We give a complete description of the map G and as a consequence show that the ω-limit set of G is the 1-skeleton of a subdivision of X into 180 convex quadrilaterals. G is a piecewise bi-quadratic map, and each algebraic piece is defined by a straight line construction involving a rhombus. The rhombi involved have the same shapes as the ones in the Penrose tiling. Our proof is computer-assisted but rigorous.
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