# Locally optimal 2-periodic sphere packings

## TL;DR

This paper extends Voronoi's method to enumerate all locally optimal 2-periodic sphere packings in various dimensions, showing that in dimensions 3, 4, and 5, no such packing exceeds the density of optimal lattices.

## Contribution

It generalizes Voronoi's enumeration algorithm to 2-periodic packings and implements it for dimensions up to 6, providing new insights into local optimality.

## Key findings

- No 2-periodic packing exceeds the density of optimal lattices in dimensions 3, 4, and 5.
- Partial enumeration performed in dimension 6.
- Method successfully enumerates all locally optimal 2-periodic packings when finitely many exist.

## Abstract

The sphere packing problem is an old puzzle. We consider packings with m spheres in the unit cell (m-periodic packings). For the case m = 1 (lattice packings), Voronoi proved there are finitely many inequivalent local optima and presented an algorithm to enumerate them, and this computation has been implemented in up to d = 8 dimensions. We generalize Voronoi's method to m > 1 and present a procedure to enumerate all locally optimal 2-periodic sphere packings in any dimension, provided there are finitely many. We implement this computation in d = 3, 4, and 5 and show that no 2-periodic packing surpasses the density of the optimal lattices in these dimensions. A partial enumeration is performed in d = 6.