Editing Lattice (group) (section)

==Symmetry considerations and examples==
A lattice is the [[symmetry group]] of discrete [[translational symmetry]] in ''n'' directions. A pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself.<ref>{{Cite web |title=Symmetry in Crystallography Notes |url=http://xrayweb.chem.ou.edu/notes/symmetry.html |access-date=2022-11-06 |website=xrayweb.chem.ou.edu}}</ref> As a group (dropping its geometric structure) a lattice is a [[Finitely-generated abelian group|finitely-generated]] [[free abelian group]], and thus isomorphic to <math>\mathbb{Z}^n</math>.

A lattice in the sense of a 3-[[dimension]]al array of regularly spaced points coinciding with e.g. the [[atom]] or [[molecule]] positions in a [[crystal]], or more generally, the orbit of a [[Group action (mathematics)|group action]] under translational symmetry, is a translation of the translation lattice: a coset, which need not contain the origin, and therefore need not be a lattice in the previous sense.

A simple example of a lattice in <math>\mathbb{R}^n</math> is the subgroup <math>\mathbb{Z}^n</math>. More complicated examples include the [[E8 lattice]], which is a lattice in <math>\mathbb{R}^{8}</math>, and the [[Leech lattice]] in <math>\mathbb{R}^{24}</math>.   The [[period lattice]] in <math>\mathbb{R}^2</math> is central to the study of [[elliptic functions]], developed in nineteenth century mathematics; it generalizes to higher dimensions in the theory of [[abelian function]]s.  Lattices called [[root lattice]]s are important in the theory of [[simple Lie algebra]]s; for example, the E8 lattice is related to a Lie algebra that goes by the same name.