Lattice Points - Dividing Space According To A Lattice

Dividing Space According To A Lattice

A typical lattice Λ in thus has the form

\Lambda = \left.\left\{ \sum_{i=1}^n a_i v_i \; \right\vert \; a_i \in\Bbb{Z} \right\}

where {v1, ..., vn} is a basis for . Different bases can generate the same lattice, but the absolute value of the determinant of the vectors vi 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.

Read more about this topic:  Lattice Points

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