#####
Séminaire Lotharingien de Combinatoire, B22e (1989).

[Formerly: Publ. I.R.M.A. Strasbourg, 1990, 414/S-22, p.
71-79.]

# Reinhard Franz and Daniel Huson

# The Classification of Quasi-Regular Polyhedra of Genus 2

**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
tessellations. S. Bilinski constructed the first tessellation
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.

The paper has been finally published under the same title in
*Discrete Comput. Geom.* **7** (1992),
347-357.