Partitioning the triangles of the cross polytope into surfaces

TL;DR

This paper proves that the 2-skeleton of the cross polytope can be decomposed into symmetric surfaces of genus at most one, revealing new geometric and combinatorial structures with high symmetry.

Contribution

It provides a constructive proof of decomposing the 2-skeleton of the cross polytope into highly symmetric surfaces, and characterizes the structure for certain dimensions.

Findings

Decomposition of the 2-skeleton into genus 0 or 1 surfaces.

Existence of transitive automorphism groups on these surfaces.

Special cases where the 2-skeleton of the simplex forms symmetric tori and Möbius strips.

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 $g \leq 1$, each with a transitive automorphism group given by the vertex transitive $\mathbb{Z}_{2k}$-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.