Sphere packing in five dimensions

Equal circles leave gaps when packed on a flat surface; equal balls do the same in ordinary space. Mathematicians can ask the same question in five-dimensional space, where a position has five coordinates instead of three. How much of that space can equal balls occupy without overlapping? Finding a very dense arrangement is one task. Proving that no arrangement could be denser is the unsolved part.[1]

How much five-dimensional space can congruent balls occupy when their interiors must be disjoint? The target is maximum asymptotic density over all packings in $\mathbb{R}^{5}$: the occupied-volume fraction is assessed at arbitrarily large scales. Restricting the centres to a lattice, a repeating grid generated by five independent directions, would narrow the problem. Known constructions supply lower bounds. A proof of optimality requires a universal upper bound meeting an achieved density, including for nonlattice arrangements. Counting balls that can touch one chosen ball is a different, local question.[1]

Excellent constructions lack a matching bound ruling out every denser arrangement.[1]

The July 2026 refereed constructions remain conjecturally optimal. Local kissing configurations and lattice-only bounds do not settle arbitrary global packings.[1]

A design can look excellent while leaving open the possibility that another arrangement does better. This problem asks for the actual geometric limit. It also asks what a proof must control when the competing arrangements need not repeat in a regular grid.[1]

The gap between a construction and a universal bound is a gap in what we know geometry permits. Closing it would explain why irregular or differently structured packings cannot beat the optimum. New arrangements at the conjectured density add a second issue: identifying the limit need not uniquely identify the packing that achieves it.[1]