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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.
The 17 wallpaper groups, with finite fundamental domains, are given by International notation, orbifold notation, and Coxeter notation, classified by the 5 Bravais lattices in the plane: square, oblique (parallelogrammatic), hexagonal (equilateral triangular), rectangular (centered rhombic), and rhombic (centered rectangular).
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.
A lattice, often taken to be a lattice in -dimensional Euclidean space or the -dimensional torus if the lattice is periodic. Concretely, Λ {\displaystyle \Lambda } is often the cubic lattice . If two points on the lattice are considered 'nearest neighbours', then they can be connected by an edge, turning the lattice into a lattice graph .
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).
In mathematics, a eutactic lattice (or eutactic form) is a lattice in Euclidean space whose minimal vectors form a eutactic star. This means they have a set of positive eutactic coefficients c i such that (v, v) = Σc i (v, m i) 2 where the sum is over the minimal vectors m i. "Eutactic" is derived from the Greek language, and means "well ...
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 . A related three dimensional structure formed by the vertices and edges of the quarter cubic honeycomb , filling space by regular tetrahedra and truncated tetrahedra , has ...
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.