On the Size of Equifacetted Semi-Regular Polytopes

TL;DR

This paper explores the diversity of semi-regular abstract polytopes, demonstrating that equifacetted semi-regular polytopes can have arbitrarily many symmetry orbits, contrasting with classical convex polytope theory.

Contribution

It proves that equifacetted semi-regular abstract polytopes can possess an arbitrarily large number of automorphism orbits, revealing unexpected phenomena in their symmetry structure.

Findings

Equifacetted semi-regular polytopes can have arbitrarily many flag orbits.

The symmetry properties differ significantly from classical convex polytopes.

The results highlight the richness of semi-regular abstract polytope theory.

Abstract

Unlike the situation in the classical theory of convex polytopes, there is a wealth of semi-regular abstract polytopes, including interesting examples exhibiting some unexpected phenomena. We prove that even an equifacetted semi-regular abstract polytope can have an arbitrary large number of flag orbits or face orbits under its combinatorial automorphism group.