Quaternion Orders and Sphere packings

TL;DR

This paper introduces a new class of sphere packings derived from quaternion algebra analogs of Bianchi groups, expanding the understanding of integral crystallographic packings.

Contribution

It develops quaternion algebra-based analogs of Bianchi groups and constructs new sphere packings similar to Apollonian packings.

Findings

Constructed new sphere packings from quaternion groups

Established connections between quaternion orders and crystallographic packings

Extended the theory of Apollonian packings to higher dimensions

Abstract

We introduce an analog of Bianchi groups for rational quaternion algebras and use it to construct sphere packings that are analogs of the Apollonian circle packing known as integral crystallographic packings.