Towards a proof of the 24-cell conjecture

TL;DR

This paper reviews approaches to solving key open problems in 4D sphere packings, including the 24-cell conjecture and the uniqueness of densest arrangements, aiming to advance understanding of optimal packings.

Contribution

It discusses potential methods for proving the 24-cell conjecture and addresses the open problem of the uniqueness of maximum kissing arrangements in 4D.

Findings

Highlights the importance of the 24-cell conjecture for densest packings

Proposes approaches to prove the 24-cell conjecture

Clarifies the relationship between the conjecture and the D4 lattice

Abstract

This review paper is devoted to the problems of sphere packings in 4 dimensions. The main goal is to find reasonable approaches for solutions to problems related to densest sphere packings in 4-dimensional Euclidean space. We consider two long-standing open problems: the uniqueness of maximum kissing arrangements in 4 dimensions and the 24-cell conjecture. Note that a proof of the 24-cell conjecture also proves that the checkerboard lattice packing D4 is the densest sphere packing in 4 dimensions.