The Classification of Quasi-Regular Polyhedra of Genus 2

Reinhard Franz and Daniel Huson

Abstract:
The method of chamber systems is used to provide a complete
list of all possible tessellations of the closed, orientable
surface of genus 2 by (topological) n-gons and m-gons
(n,m > 2) satisfying a certain local symmetry condition.
Using a computer program it is shown that
(up to homeomorphism) there are precisely 379 such
quasi-regular polyhedra. S. Bilinski constructed  the first
polyhedron of the considered type for each of the 17
possible combinations of m- and n-gons
using geometrical methods. It is
the intention of the authors to demonstrate the usefulness
and suitability of chamber systems in dealing with problems
of the above type.

Keywords:
Census of planar maps, tessellations, polyhedra

Appeared in:
Discrete and Computational Geometry, 7:347--357 (1992)