Regular polytopes, sphere packings and Apollonian sections
Combinatorics
2024-10-14 v3 Group Theory
Metric Geometry
Number Theory
Abstract
In this paper, we explore the geometry and the arithmetic of a family of polytopal sphere packings induced by regular polytopes in any dimension. We prove that every integral polytope is crystallographic, and we show that there are 11 crystallographic regular polytopes in any dimension. After introducing the notion of Apollonian section, we determine which Platonic crystallographic packings emerge as cross-sections of the Apollonian arrangements of the regular 4-polytopes. Additionally, we compute the M\"obius spectrum of every regular polytope.
Keywords
Cite
@article{arxiv.2109.00655,
title = {Regular polytopes, sphere packings and Apollonian sections},
author = {Iván Rasskin},
journal= {arXiv preprint arXiv:2109.00655},
year = {2024}
}