Sphere packing

Builds on Euclidean space Closed ball Lattice (group)

Definition

Let $n \ge 1$ and let $r > 0$ . A sphere packing in the Euclidean space $\R^n$ with radius $r$ is a set $C \subset \R^n$ of centres such that $\abs{x - y} \ge 2r$ for all distinct $x, y \in C$ . The packing is the family of closed balls $B(x, r)$ with $x \in C$ ; their interiors are pairwise disjoint. The packing is a lattice packing if $C$ is a lattice, or a translate of one.

Remarks

Example: in $\R^2$ take $C = \Z^2$ and $r = \tfrac12$ . Two distinct integer points are at distance at least $1 = 2r$ , so the discs of radius $\tfrac12$ centred at them form a lattice packing, the square packing. Non-example: $C = \Z^2$ with $r = 0.6$ ; the points $(0, 0)$ and $(1, 0)$ are at distance $1 < 1.2$ , so their discs overlap. For a lattice $\Lambda$ with shortest nonzero vector of length $m$ , the largest radius is $r = m/2$ . Some authors allow balls of different radii; the term "sphere packing" then refers to any family of balls with disjoint interiors.

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Closed ball concept
Euclidean space object
Lattice (group) concept
Sphere packing definition

Tag	`T4PGC6NH56`
URI	`m2://T4PGC6NH56`
Kind	definition
Writing	W0 stub
MSC	52C17, 11H31, 05B40
Defines	Sphere packing
Uses	Euclidean space, Closed ball, Lattice (group)

Sphere packing

Definition

Remarks

Reading path

Further reading

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