On Kepler's Small Stellated Dodecahedron

We give a new geometric computation for the Jacobian of the Riemann surface of genus 4 associated to Kepler’s small stellated dodecahedron. Starting with Threlfall’s description, we introduce several other flat conformal geometries on this surface which are related to holomorphic 1-forms. They allow us to show that the Jacobian is isogeneous to a fourfold product of a single elliptic curve whose lattice constant can be determined in two essentially different ways, yielding an unexpected relation between hypergeometric integrals. We also obtain a new platonic tessellation of the surface. Figure 1: The small stellated dodecahedron