A lattice An in n-dimensional Euclidean space En consists of the aggregate of all points with coordinates (xx,…, xn), where
![](//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20160109033009780-0575:S2040618500034912_inline1.gif?pub-status=live)
for some real ars (r, s = 1,…, n), subject to the condition ∥ αrs ∥nn ╪ 0. The determinant Δn of Λn, is denned by the relation
, the sign being chosen to ensure that Δn > 0.
If A1…, An are the n points of Λn having coordinates (a11, a21…, anl),…, (a1n, a2n,…, ann), respectively, then every point of Λn may be expressed in the form
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS2040618500034912/resource/name/S2040618500034912_eqnU1.gif?pub-status=live)
and Ai,…, An, together with the origin O, are said to generate Λn. This particular set of generating points is not unique; it may be proved that a necessary and sufficient condition that n points of Λn should generate the lattice is that the n × n determinant formed by their x coordinates should be ±Δn, or, equivalently, that the n×n determinant formed by their corresponding u-coordinates should be ±1.