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Regular polytopes, sphere packings and Apollonian sections

Iván Rasskin

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 Möbius spectrum of every regular polytope.

Regular polytopes, sphere packings and Apollonian sections

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 Möbius spectrum of every regular polytope.

Paper Structure

This paper contains 25 sections, 13 theorems, 68 equations, 39 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Main contributions
  3. Organization of the paper
  4. Acknowledgements:
  5. Figures:
  6. Preliminaries on sphere packings and regular polytopes
  7. Inversive coordinates
  8. Polytopal sphere packings
  9. Crystallographic and integral polytopes
  10. Regular polytopes
  11. The fundamental symmetries and basis of regular polytopes
  12. The full symmetry group of the regular Apollonian arrangements
  13. The Platonic crystallographic packings
  14. Geometry and arithmetic of the regular polytopal sphere packings
  15. The regular crystallographic packings
  16. ...and 10 more sections

Key Result

Lemma 1

Let $\mathcal{P}$ be an edge-scribable $(d+1)$-polytope with $d\ge2$. Then $\mathcal{P}$ is crystallographic if and only if, for any two dual spheres $S_f$, $S_{f'}$ of any polytopal $d$-sphere packing $\mathcal{S}_\mathcal{P}$, we have $| \langle S_f,S_{f'}\rangle |\in \lbrace \frac{\sqrt n}{2}\,|

Figures (39)

  • Figure 1: An integral hypercubic crystallographic packing after 0, 1, 2 and 3 iterations. The numbers are the bends of the spheres.
  • Figure 2: (Left) The integral hypercubic crystallographic packing $\mathscr P_{\{4,3,3\}}(0,0,1,2)$. (Right) The integral 24-cell crystallographic packing $\mathscr P_{\{3,4,3\}}(0,0,1,2)$.
  • Figure 3: An edge-scribed realization (left) and a canonical realization (right) of a $4$-pyramid.
  • Figure 4: (Top) Three polyhedra with the spherical caps corresponding to their vertices. (Below) The arrangement projection of the three polyhedra. The last two are packings but only the third one is polytopal.
  • Figure 5: (Left) A cube with a fundamental domain of its symmetry group (in dark gray), a fundamental basis ($v_1,v_2,v_3,v_4$) and the walls of the fundamental symmetries (in blue). (Right) A cubic circle packing with the corresponding fundamental basis (in red) and the fundamental symmetries (in blue).
  • ...and 34 more figures

Theorems & Definitions (25)

  • Conjecture 1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 1
  • Lemma 2
  • proof
  • Theorem 1
  • Theorem 2
  • proof
  • ...and 15 more