Regular polyhedra in the 3-torus

TL;DR

This paper classifies regular polyhedra in the 3-torus by analyzing rank 3 lattices preserved by finite orthogonal groups, linking it to the classification of regular polyhedra in 3-space.

Contribution

It provides a novel classification of regular polyhedra in the 3-torus based on lattice and symmetry group analysis, extending known classifications in 3-space.

Findings

Classification of rank 3 lattices preserved by finite orthogonal groups

Derivation of regular polyhedra in the 3-torus from lattice analysis

Connection established between 3-torus polyhedra and 3-space polyhedra

Abstract

In this paper we discuss the classification rank $3$ lattices preserved by finite orthogonal groups of isometries and derive from it the classification of regular polyhedra in the $3$-dimensional torus. This classification is highly related to the classification of regular polyhedra in the $3$-space.