Geometric realizations for Dyck's regular map on a surface of genus 3

Klein's and Dyck's regular maps on Riemann surfaces of genus 3 were one impetus for the theory of regular maps, automorphic functions, and algebraic curves. Recently a polyhedral realization inE ³ of Klein's map was discovered 18], thereby underlining the strong analogy to the icosahedron. In this paper we show that Dyck's map can be realized inE ³ as a polyhedron of Kepler-Poinsot-type, i.e., with maximal symmetry and minimal self-intersections. Furthermore some closely related polyhedra and a Kepler-Poinsot-type realization of Sherk's regular map of genus 5 are discussed.