---
tag: BWSN2JAMTK
kind: concept
name: Covering density
msc: [52C17, 52C15, 11H31]
sources:
  - url: https://en.wikipedia.org/wiki/Covering_problems
---

## Summary

The covering density of a covering of space by bodies measures, on average, how many times each point is covered.
It is at least $1$, and the covering problem asks for the smallest possible value, the covering density of the body.
It is the counterpart of the [[packing density|4P5PSWADJD]], with overlaps allowed and gaps forbidden.
For discs in the plane the optimum is $2\pi/\sqrt{27} \approx 1.2092$, given by the hexagonal lattice arrangement.

## Explanations

### L1 human

Suppose you must light a large square with lamps that each light a disc, and every spot must be lit.
The discs must overlap somewhere, and the covering density measures how much light is wasted on average.
For equal discs, the best is to place the centres in a triangular grid, so that each disc is cut to a regular hexagon by its neighbours.
Even then about $21\%$ of the light is spent on overlaps.