---
tag: HF32CBN5MA
kind: concept
name: Sphere packing
msc: [52C17, 11H31, 05B40]
sources:
  - url: https://en.wikipedia.org/wiki/Sphere_packing
same_as:
  wikidata: Q900117
  wikipedia: Sphere packing
---

## Summary

A sphere packing is an arrangement of balls in space that do not overlap.
The classical question asks for the largest fraction of space that a packing of equal balls can fill, its [[packing density|4P5PSWADJD]].
The answer is known in dimensions $1$, $2$, $3$, $8$ and $24$.
In the plane it is the hexagonal packing, and in space the face-centred cubic packing of the [[Kepler conjecture|AFMBQT06ZV]].
In dimensions $8$ and $24$ it is given by the [[E8 lattice|2CXH65EKY7]] and the [[Leech lattice|XETV3657DG]].
Sphere packings are linked with error-correcting codes, lattices, number theory and crystallography.

## Explanations

### L1 human

Think of oranges stacked in a crate.
A sphere packing is any way of placing equal balls so that none of them overlap; they may touch.
Stacking layers of oranges so that each orange sits in a hollow of the layer below fills about $74\%$ of space.
The same question makes sense in the plane with coins, and in higher dimensions with balls that cannot be drawn but can be described by coordinates.