Supplemental material to the article "Partitions of the triangles of the cross polytope into surfaces''

TL;DR

This paper provides a constructive proof that the 2-skeleton of the k-dimensional cross polytope can be decomposed into closed surfaces of genus at most 1 with symmetric automorphism groups, and describes similar decompositions for simplices.

Contribution

It introduces a new constructive method for decomposing the 2-skeletons of cross polytopes into symmetric surfaces and extends results to simplices for specific dimensions.

Findings

Decomposition of the 2-skeleton of the k-dimensional cross polytope into genus ≤ 1 surfaces.

Existence of highly symmetric tori and Möbius strips in the 2-skeleton of certain simplices.

Automorphism groups acting transitively on the decomposed surfaces.

Abstract

We present a constructive proof, that there exists a decomposition of the 2-skeleton of the k-dimensional cross polytope \beta^k into closed surfaces of genus \leq 1, each with a transitive automorphism group given by the vertex transitive Z_{2k}-action on \beta^k. Furthermore we show, that for each k \equiv 1,5(6) the 2-skeleton of the (k-1)-simplex is a union of highly symmetric tori and M\"obius strips.