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Variations on five-dimensional sphere packings

Henry Cohn, Isaac Rajagopal

TL;DR

The paper investigates the structure of optimal sphere packings and kissing configurations in low-to-mid dimensions, focusing on five- and nine-dimensional cases. It extends Szöllősi's $Q_5$ kissing configuration by constructing a new non-isometric $R_5$ and embedding these into five-dimensional sphere packings via a four-coloring of $\Delta_1$–$\Delta_2$ tilings on $\mathbb{R}^2$, stacked over $D_3$. The work identifies two uniform 5D packings (one $2$-periodic and one $4$-periodic), completes a near-full classification of 5D uniform packings containing the known kissing configurations, and presents a new 9D kissing configuration by layer modification of the Leech–Sloane configuration. Together, these results illustrate the richness of layer-based constructions, reveal both the reach and limits of current five-dimensional classifications, and motivate further exploration of high-dimensional analogues.

Abstract

We analyze Szöllősi's recent construction of a conjecturally optimal five-dimensional kissing configuration and produce a new such configuration, the fourth to be discovered. We construct five-dimensional sphere packings from these configurations, which augment Conway and Sloane's list of conjecturally optimal packings. We also construct a new kissing configuration in nine dimensions. None of these constructions improves on the known records, but they provide geometrically distinct constructions achieving these records.

Variations on five-dimensional sphere packings

TL;DR

The paper investigates the structure of optimal sphere packings and kissing configurations in low-to-mid dimensions, focusing on five- and nine-dimensional cases. It extends Szöllősi's kissing configuration by constructing a new non-isometric and embedding these into five-dimensional sphere packings via a four-coloring of – tilings on , stacked over . The work identifies two uniform 5D packings (one -periodic and one -periodic), completes a near-full classification of 5D uniform packings containing the known kissing configurations, and presents a new 9D kissing configuration by layer modification of the Leech–Sloane configuration. Together, these results illustrate the richness of layer-based constructions, reveal both the reach and limits of current five-dimensional classifications, and motivate further exploration of high-dimensional analogues.

Abstract

We analyze Szöllősi's recent construction of a conjecturally optimal five-dimensional kissing configuration and produce a new such configuration, the fourth to be discovered. We construct five-dimensional sphere packings from these configurations, which augment Conway and Sloane's list of conjecturally optimal packings. We also construct a new kissing configuration in nine dimensions. None of these constructions improves on the known records, but they provide geometrically distinct constructions achieving these records.

Paper Structure

This paper contains 9 sections, 3 theorems, 24 equations, 4 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Five-dimensional kissing configurations
  3. Six-dimensional kissing configurations
  4. From kissing configurations to sphere packings
  5. Packings that fiber over $D_3$
  6. Uniform packings
  7. Symmetries of uniform packings
  8. Higher dimensions
  9. Nine-dimensional kissing configurations

Key Result

Theorem 2.1

There are at least four non-isometric kissing configurations of $40$ points in five dimensions, namely $D_5$, $L_5$, $Q_5$, and $R_5$.

Figures (4)

  • Figure 1.1: The centers of the spheres in the face centered cubic kissing arrangement on the left, and the centers of the spheres in the hexagonal close packing kissing arrangement on the right. To produce the arrangement on the right from the one on the left, the top layer is deleted and the bottom layer is reflected across the central layer (the hexagon).
  • Figure 4.1: The four possible ways to arrange triangles $\Delta_1$ and $\Delta_2$ around a point, labeled $a$ through $d$. In arrangements $a$ and $b$, two of the neighboring points are labeled $p$ and $q$.
  • Figure 4.2: The uniform packings with kissing configuration $Q_5$ (on the left) and $R_5$ (on the right).
  • Figure 4.3: The irregular tetrahedron and octahedron with $\Delta_1$ facets, unfolded into nets.

Theorems & Definitions (7)

  • Theorem 2.1
  • Conjecture 3.1
  • Conjecture 4.1
  • Conjecture 4.2
  • Theorem 4.3
  • proof
  • Theorem 5.1