Editing Lattice (group) (section)

==Dividing space according to a lattice==
A lattice <math>\Lambda</math> in <math>\mathbb{R}^n</math> thus has the form
:<math>\Lambda = \biggl\{ \sum_{i=1}^n a_i v_i \mathbin{\bigg\vert} a_i \in\mathbb{Z} \biggr\},</math><!-- must use \mid instead of \vert with manual spacing, but it's currently broken -->
where <math display="inline">
\{v_1, v_2, \ldots, v_n\}
</math> is a basis for <math>\mathbb{R}^n</math>. Different bases can generate the same lattice, but the [[absolute value]] of the [[determinant]] of the [[Gram matrix]] of the vectors <math display="inline">
v_i
</math> is uniquely determined by <math>\Lambda</math> and denoted by d(<math>\Lambda</math>). If one thinks of a lattice as dividing the whole of <math>\mathbb{R}^n</math> into equal [[polyhedron|polyhedra]] (copies of an ''n''-dimensional [[parallelepiped]], known as the ''[[fundamental region]]'' of the lattice), then d(<math>\Lambda</math>) is equal to the ''n''-dimensional [[volume]] of this polyhedron. This is why d(<math>\Lambda</math>) is sometimes called the '''covolume''' of the lattice. If this equals 1, the lattice is called [[unimodular lattice|unimodular]].