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https://en.wikipedia.org/wiki/Lattice_(group)
Lattice (group)
Symmetry considerations and examples
A simple example of a lattice in {\displaystyle \mathbb {R} ^{n}}\mathbb {R} ^{n} is the subgroup {\displaystyle \mathbb {Z} ^{n}}\mathbb {Z} ^{n}. More complicated examples include the E8 lattice, which is a lattice in {\displaystyle \mathbb {R} ^{8}}{\mathbb {R}}^{{8}}, and the Leech lattice in {\displaystyle \mathbb {R} ^{24}}{\mathbb {R}}^{{24}}. The period lattice in {\displaystyle \mathbb {R} ^{2}}\mathbb {R} ^{2} is central to the study of elliptic functions, developed in nineteenth century mathematics; it generalises to higher dimensions in the theory of abelian functions. Lattices called root lattices are important in the theory of simple Lie algebras; for example, the E8 lattice is related to a Lie algebra that goes by the same name.

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