---
tag: WN9V7WSJKF
kind: concept
name: Lattice (group)
msc: [11H06]
specializes: [0WJX8MGG37]
sources:
  - url: https://en.wikipedia.org/wiki/Lattice_(group)
same_as:
  wikidata: Q1096885
  wikipedia: Lattice (group)
---

## Summary

A lattice in $\R^n$ is the set of all integer combinations of $n$ linearly independent vectors.
Equivalently, it is a discrete subgroup of $\R^n$ that spans $\R^n$.
As a group it is [[free abelian|0WJX8MGG37]] of rank $n$, and its covolume, the volume of a fundamental parallelepiped, measures how dense it is.
Lattices are the basic objects of the geometry of numbers, through [[Minkowski's theorem|0K387S2PKQ]], and they appear in number fields, in crystallography and in cryptography.

## Explanations

### L1 human

Picture the points with integer coordinates in the plane: a regular grid.
Any grid obtained by taking two independent arrows and all their integer combinations is a lattice.
Squares, slanted parallelograms and the hexagonal pattern of a honeycomb all come from lattices.
For example, the vectors $(1, 0)$ and $(1/2, \sqrt{3}/2)$ generate the hexagonal lattice.
Different pairs of arrows can generate the same lattice, but the area of the parallelogram they span is always the same.