---
tag: S360JBEYKD
kind: concept
name: Circle packing
msc: [52C15, 52C26, 05B40]
sources:
  - url: https://en.wikipedia.org/wiki/Circle_packing
same_as:
  wikidata: Q5121501
  wikipedia: Circle packing
---

## Summary

A circle packing is an arrangement of circles in the plane, or on a surface, whose discs do not overlap.
One studies dense packings of equal circles, where the hexagonal packing is optimal by [[Thue's theorem|3P246RHXMF]].
One also studies packings with prescribed tangencies, where the [[circle packing theorem|GR1DDT1B6B]] realizes every planar graph.
Circle packings give a discrete analogue of [[conformal maps|0YYT36R4YD]], and fractal examples such as the [[Apollonian gasket|8SETG842RX]].

## Explanations

### L1 human

Lay coins on a table so that no two coins overlap; that is a circle packing.
The coins may all have the same size, as in a tray of identical cans, or different sizes.
If the coins are equal, the honeycomb arrangement, where each coin touches six others, wastes the least space.