Any compact metric space X supports the following map f. Given p in X, the point f(p) is defined to be the point in X farthest from p. Typically, f is single-valued and one can do dynamics. I will give a complete description of what happens when X is the regular dodecahedron equipped with its intrinsic metric. In this case f is piecewise algebraic, and every orbit accumulates on the 1-skeleton of a certain tiling of X by 180 convex quadrilaterals. I will demonstrate the result with a computer program I wrote, and explain the key features in the proof.