Preserving and Increasing Symmetries of Polyhedral Maps

TL;DR

This paper studies local symmetry-preserving operations on polyhedral maps, identifying which can increase symmetry, especially for maps of various genera, with detailed results for specific inflation factors and well-known operations.

Contribution

It provides a comprehensive analysis of symmetry-increasing operations on polyhedral maps, including classifications based on genus and specific operations like Goldberg-Coxeter and leapfrog.

Findings

Operations with inflation factor ≤6 can increase symmetry under certain conditions.

Goldberg-Coxeter and leapfrog operations can increase symmetry for specific classes of maps.

Complete characterization of symmetry-preserving and increasing operations for maps of various genera.

Abstract

In this article we investigate the question which local symmetry preserving operations can not only preserve, but also increase the symmetry of a polyhedral map. Often operations that can increase symmetry, can nevertheless not do so for polyhedral maps of every genus. So for maps that can increase symmetry, we also investigate for which genera they can do so. We give complete answers for operations with inflation factor at most 6 (that is: that increase the number of edges by a factor of at most 6) and for the chemically relevant Goldberg-Coxeter operations and the leapfrog operation.