Regular polytopes, sphere packings and Apollonian sections

Iván Rasskin

Published 2021-09-02Version 1

In this paper, we explore the connection between polytopes and sphere packings. After introducing the notion of Apollonian section, we prove our main result, that every integral tetrahedral, octahedral or cubical Apollonian circle packing is contained as an Apollonian section of an integral orthoplicial Apollonian section. This result implies that the set of curvatures in every integral tetrahedral, octahedral or cubical Apollonian circle packing is contained in the set of curvatures of an integral orthoplicial Apollonian sphere packing. We also prove that cross polytopes in dimension 3 and above and the 24-cell are M\"obius unique, we give a new set of non edge-scribable 4-polytopes, and we obtain a generalization of the octahedral Descartes' Theorem of Guettler and Mallows in every dimension.