Coxeter arrangements in three dimensions

TL;DR

This paper proves that in three dimensions, hyperplane arrangements with all regions isometric are necessarily Coxeter arrangements, providing a precise characterization in three dimensions.

Contribution

It establishes a characterization of three-dimensional Coxeter arrangements based on the isometry of their regions, and discusses affine arrangements with similar properties.

Findings

Three-dimensional arrangements with isometric regions are Coxeter arrangements.

Characterization of 3D Coxeter arrangements via isometric regions.

Identification of affine arrangements with isometric regions that are not reflection arrangements.

Abstract

Let ${\mathcal A}$ be a finite real linear hyperplane arrangement in three dimensions. Suppose further that all the regions of ${\mathcal A}$ are isometric. We prove that ${\mathcal A}$ is necessarily a Coxeter arrangement. As it is well known that the regions of a Coxeter arrangement are isometric, this characterizes three-dimensional Coxeter arrangements precisely as those arrangements with isometric regions. It is an open question whether this suffices to characterize Coxeter arrangements in higher dimensions. We also present the three families of affine arrangements in the plane which are not reflection arrangements, but in which all the regions are isometric.