Regular polytopes, sphere packings and Apollonian sections

TL;DR

This paper investigates the geometric and arithmetic properties of sphere packings derived from regular polytopes, identifying crystallographic cases, and analyzing their sections and M"obius spectra across dimensions.

Contribution

It proves all integral polytopes are crystallographic, classifies 11 such polytopes in any dimension, and introduces Apollonian sections to connect packings with regular 4-polytopes.

Findings

All integral polytopes are crystallographic.

There are exactly 11 crystallographic regular polytopes in any dimension.

The M"obius spectrum of each regular polytope is computed.

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.