The Eisenlohr-Farris Algorithm for fully transitive polyhedra

TL;DR

This paper introduces a constructive method for classifying fully transitive polyhedra in three-dimensional space based on their symmetry groups, enabling systematic generation of such polyhedra.

Contribution

It presents a novel algorithm for generating all fully transitive polyhedra in D space using conjugation classes of crystallographic groups, with a practical example.

Findings

Generated a new fully transitive polyhedron in D space.

Provided a comprehensive classification method for fully transitive polyhedra.

Demonstrated the application of the method in Euclidean space D.

Abstract

The purpose of this note is to present a method for classifying three-dimensional polyhedra in terms of their symmetry groups. This method is constructive and it is described in terms of the conjugation classes of crystallographic groups in $\mathbb{E}^3$. For each class of groups $\Gamma$ the method can generate without duplication all polyhedra in three-dimensional space on which $\Gamma$ acts fully-transitively. It was proposed by J. M. Eisenlohr and S. L. Farris for generating every fully transitive polyhedra in $\mathbb{E}^d$. We also illustrate how the method can be applied in the euclidean space $\mathbb{E}^3$ by generating a new fully transitive polyhedron.