Symmetries of 3-polytopes with fixed edge lengths

TL;DR

This paper investigates symmetries of 3-polytopes that preserve both combinatorial structure and edge lengths, providing conditions under which these symmetries are realizable as isometries, using Cauchy's rigidity theorem in a novel way.

Contribution

It introduces a sufficient condition for realizing edge-length preserving combinatorial symmetries as geometric isometries of 3-polytopes.

Findings

Identifies a simple criterion for symmetry realization

Uses Cauchy's rigidity theorem in a novel proof approach

Enhances understanding of polytope symmetry structures

Abstract

We consider an interesting class of combinatorial symmetries of polytopes which we call \emph{edge-length preserving combinatorial symmetries}. These symmetries not only preserve the combinatorial structure of a polytope but also map each edge of the polytope to an edge of the same length. We prove a simple sufficient condition for a polytope to realize all edge-length preserving combinatorial symmetries by isometries of ambient space. The proof of this condition uses Cauchy's rigidity theorem in an unusual way.