Kissing number

Builds on Euclidean space Closed ball Euclidean distance

Definition

Let $n \ge 1$ . A kissing configuration in the Euclidean space $\R^n$ is a finite set $X \subset \R^n$ such that:

$\abs{x} = 2$ for every $x \in X$ ;
$\abs{x - y} \ge 2$ for all distinct $x, y \in X$ , where $\abs{\cdot}$ is the Euclidean distance.

The kissing number $\tau(n)$ is the largest size of a kissing configuration in $\R^n$ .

Remarks

The conditions say that the closed balls of radius $1$ centred at the points of $X$ touch the unit ball at $0$ and have pairwise disjoint interiors. Example: in $\R^2$ , the six points $2(\cos(k\pi/3), \sin(k\pi/3))$ for $k = 0, \dots, 5$ have norm $2$ , and neighbours are at distance exactly $2$ , so they form a kissing configuration; hence $\tau(2) \ge 6$ . Non-example: seven points of norm $2$ in $\R^2$ cannot form one. Two of them make an angle less than $60^\circ$ at $0$ , and are then at distance less than $2$ . Equivalently, $\tau(n)$ is the largest number of points on the unit sphere $S^{n-1}$ with pairwise angles at least $60^\circ$ . The lattice kissing number is the largest size of such a configuration when $X$ is the set of shortest nonzero vectors of a lattice, scaled to length $2$ .

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Closed ball concept
Euclidean distance concept
Euclidean space object
Kissing number definition

Tag	`HWC3SQGCQ0`
URI	`m2://HWC3SQGCQ0`
Kind	definition
Writing	W0 stub
MSC	52C17, 05B40, 11H31
Defines	Kissing number
Uses	Euclidean space, Closed ball, Euclidean distance

Kissing number

Definition

Remarks

Reading path

Further reading

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