Lattice (group) - Dividing Space According To A Lattice

Dividing Space According To A Lattice

A typical lattice Λ in thus has the form

where {v₁, ..., v_n} is a basis for . Different bases can generate the same lattice, but the absolute value of the determinant of the vectors v_i is uniquely determined by Λ, and is denoted by d(Λ). If one thinks of a lattice as dividing the whole of into equal polyhedra (copies of an n-dimensional parallelepiped, known as the fundamental region of the lattice), then d(Λ) is equal to the n-dimensional volume of this polyhedron. This is why d(Λ) is sometimes called the covolume of the lattice.