---
tag: 4P5PSWADJD
kind: concept
name: Packing density
msc: [52C17, 05B40, 11H31]
sources:
  - url: https://en.wikipedia.org/wiki/Packing_density
same_as:
  wikidata: Q18385933
  wikipedia: Packing density
---

## Summary

The packing density of a packing is the fraction of space covered by its bodies.
It is measured inside large balls and passed to the limit, since space is infinite.
The packing density of a body is the largest density of a packing of its congruent copies.
For equal balls this is the optimal [[sphere packing|HF32CBN5MA]] density, known exactly only in dimensions $1$, $2$, $3$, $8$ and $24$.

## Explanations

### L1 human

Fill a huge box with equal coins laid flat, as tightly as you can, and compute the share of the box that the coins cover.
Making the box larger and larger, this share settles near one number, the density of the arrangement.
Coins in a square grid cover $\pi/4 \approx 78.5\%$ of the plane.
Coins in a hexagonal pattern cover $\pi/\sqrt{12} \approx 90.7\%$, and no arrangement does better.