---
tag: B408FZYXF9
kind: concept
name: Kissing number
msc: [52C17, 05B40, 11H31]
sources:
  - url: https://en.wikipedia.org/wiki/Kissing_number
same_as:
  wikidata: Q1371530
  wikipedia: Kissing number
---

## Summary

The kissing number $\tau(n)$ is the largest number of non-overlapping unit balls in $\R^n$ that can all touch one central unit ball.
It is known exactly only in dimensions $1$, $2$, $3$, $4$, $8$ and $24$, where it equals $2$, $6$, $12$, $24$, $240$ and $196560$.
The problem is a close relative of [[sphere packing|HF32CBN5MA]] and of the theory of [[spherical codes|SDCX4WDP4F]].

## Explanations

### L1 human

Put one coin on a table and ask how many equal coins can touch it without overlapping each other.
The answer is six, and they fit perfectly around it.
For balls in space the answer is twelve, but here there is some room to spare, which made the question hard.
In higher dimensions the numbers grow very quickly and are mostly unknown.