The regular Grünbaum polyhedron of genus 5

G. Gévay 1 , E. Schulte 2 ,  and J. M. Wills 3
  • 1 Bolyai Institute, University of Szeged, 6720 Szeged, Hungary
  • 2 Department of Mathematics, Northeastern University, Boston, MA, 02115, USA
  • 3 Mathematics Institute, University of Siegen, 57068 Siegen, Germany


We discuss a polyhedral embedding of the classical Fricke-Klein regular map of genus 5 in ordinary space E3. This polyhedron was originally discovered by Grünbaum in 1999, but was recently rediscovered by Brehm andWills. We establish isomorphism of the Grünbaum polyhedron with the Fricke-Klein map, and confirm its combinatorial regularity. The Grünbaum polyhedron is among the few currently known geometrically vertex-transitive polyhedra of genus g ≥ 2, and is conjectured to be the only vertex-transitive polyhedron in this genus range that is also combinatorially regular. We also contribute a new vertex-transitive polyhedron, of genus 11, to this list, as the 7th known example. In addition we show that there are only finitely many vertex-transitive polyhedra in the entire genus range g ≥ 2.

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Advances in Geometry is a mathematical journal which publishes original research articles of excellent quality in the area of geometry. Geometry is a field of long-standing tradition and eminent importance. The study of space and spatial patterns is a major mathematical activity, and geometric ideas and the geometric language permeate all of mathematics.