Kepler’s small stellated dodecahedron as a Riemann surface

Matthias Weber

Vol. 220 (2005), No. 1, 167–182

DOI: 10.2140/pjm.2005.220.167

Abstract

We provide a new geometric computation for the Jacobian of the Riemann surface of genus 4 associated to the small stellated dodecahedron. Starting with Threlfall’s description, we introduce other flat conformal geometries on this surface which are related to holomorphic 1-forms. They allow us to show that the Jacobian is isogenous 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.

Keywords

Jacobians, flat structures, small stellated dodecahedron

Mathematical Subject Classification 2000

Primary: 30F30

Milestones

Received: 3 June 1999

Revised: 27 January 2004

Accepted: 30 March 2004

Published: 1 May 2005