Ford Spheres in the Clifford-Bianchi Setting

TL;DR

This paper generalizes Ford spheres to hyperbolic spaces associated with Clifford-Bianchi groups, establishing their properties and connections to classical number theory, and explores their geometric and algebraic structure.

Contribution

It introduces a new class of Ford spheres in higher-dimensional hyperbolic spaces linked to Clifford-Bianchi groups, extending classical concepts and analyzing their geometric properties.

Findings

Ford spheres are integral and have disjoint interiors.

They intersect tangentially when they do intersect.

Connectedness is established under Clifford-Euclidean conditions.

Abstract

We define Ford Spheres $\mathcal{P}$ in hyperbolic $n$-space associated to Clifford-Bianchi groups $PSL_2(O)$ for $O$ orders in rational Clifford algebras associated to positive definite, integral, primitive quadratic forms. For $\mathcal{H}^2$ and $\mathcal{H}^3$ these spheres correspond to the classical Ford circles and Ford spheres (these are non-maximal subsets of classical Apollonian packings). We prove the Ford spheres are integral, have disjoint interiors, and intersect tangentially when they do intersect. If we assume that $O$ is Clifford-Euclidean then $\mathcal{P}$ is also connected. We also give connections to Dirichlet's Theorem and Farey fractions. In a discussion section, we pose some questions related to existing packings in the literature.