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In geometry and group theory, a lattice in the real coordinate space is an infinite set of points in this space with the properties that coordinate-wise addition or subtraction of two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in the space is within some maximum distance of a lattice point.
In geometry and mathematical group theory, a unimodular lattice is an integral lattice of determinant 1 or −1. For a lattice in n-dimensional Euclidean space, this is equivalent to requiring that the volume of any fundamental domain for the lattice be 1. The E 8 lattice and the Leech lattice are two famous examples.
The A n root lattice – that is, the lattice generated by the A n roots – is most easily described as the set of integer vectors in R n+1 whose components sum to zero. The A 2 root lattice is the vertex arrangement of the triangular tiling. The A 3 root lattice is known to crystallographers as the face-centered cubic (or cubic close packed ...
The Leech lattice Λ 24 is the unique lattice in 24-dimensional Euclidean space, E 24, with the following list of properties: It is unimodular; i.e., it can be generated by the columns of a certain 24×24 matrix with determinant 1. It is even; i.e., the square of the length of each vector in Λ 24 is an even integer.
This category concerns lattices, sets of regularly placed points in a Euclidean space; equivalently discrete subgroups of translation groups or finitely generated free abelian groups. For topics concerning partially ordered sets with join and meet operations, see Lattice (order) or Category:Lattice theory.
In mathematics, the n-dimensional integer lattice (or cubic lattice), denoted , is the lattice in the Euclidean space whose lattice points are n-tuples of integers. The two-dimensional integer lattice is also called the square lattice , or grid lattice.
In mathematics, a perfect lattice (or perfect form) is a lattice in a Euclidean vector space, that is completely determined by the set S of its minimal vectors in the sense that there is only one positive definite quadratic form taking value 1 at all points of S. Perfect lattices were introduced by Korkine & Zolotareff (1877).
The term kagome lattice was coined by Japanese physicist Kôdi Husimi, and first appeared in a 1951 paper by his assistant Ichirō Shōji. [7] The kagome lattice in this sense consists of the vertices and edges of the trihexagonal tiling. Despite the name, these crossing points do not form a mathematical lattice.