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.