The regular Grünbaum polyhedron of genus 5

G. Gévay 1 , E. Schulte 2  und 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

Abstract

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.

Artikel kaufen
Erhalten sie sofort unbegrenzten Zugriff auf den Artikel.
Anmelden
Haben Sie den Zugang bereits erworben? Melden Sie sich bitte an.


oder
Zugriff über Ihre Institution

Zeitschrift + Hefte

Suche