Lattice (group)

In mathematics, especially in geometry and group theory, a lattice in $\mathbb {R} ^{n}$ $\mathbb {R} ^{n}$ is a subgroup of $\mathbb {R} ^{n}$ $\mathbb {R} ^{n}$ which is isomorphic to $\mathbb {Z} ^{n}$ $\mathbb {Z} ^{n}$ , and which spans the real vector space $\mathbb {R} ^{n}$ $\mathbb {R} ^{n}$ . In other words, for any basis of $\mathbb {R} ^{n}$ $\mathbb {R} ^{n}$ , the subgroup of all linear combinations with integer coefficients of the basis vectors forms a lattice. A lattice may be viewed as a regular tiling of a space by a primitive cell.