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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 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 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 ...
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, 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.
The 17520 norm 8 lattice points fall into two classes (two orbits under the action of the E 8 automorphism group): 240 are twice the norm 2 lattice points while 17280 are 3 times the shallow holes surrounding the origin. A hole in a lattice is a point in the ambient Euclidean space whose distance to the nearest lattice point is a local maximum.
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.