Towards a Geometric Understanding of the 4-Dimensional Point Groups

TL;DR

This paper classifies finite 4D point groups through geometric analysis, introducing a new classification for toroidal groups and developing a parameterization of great circles on the 3-sphere.

Contribution

It provides a comprehensive classification of 4D orthogonal groups, including a novel approach for toroidal groups based on their action on invariant tori.

Findings

Classification of finite 4D orthogonal groups achieved

New parameterization of great circles on the 3-sphere developed

Connection to Hopf fibrations established

Abstract

We classify the finite groups of orthogonal transformations in 4-space, and we study these groups from the viewpoint of their geometric action, using polar orbit polytopes. For one type of groups (the toroidal groups), we develop a new classification based on their action on an invariant torus, while we rely on classic results for the remaining groups. As a tool, we develop a convenient parameterization of the oriented great circles on the 3-sphere, which leads to (oriented) Hopf fibrations in a natural way.