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Five lattices in the Euclidean plane. There are five 2D lattice types as given by the crystallographic restriction theorem. Below, the wallpaper group of the lattice is given in IUCr notation, Orbifold notation, and Coxeter notation, along with a wallpaper diagram showing the symmetry domains. Note that a pattern with this lattice of ...
Unimodular lattices are equal to their dual lattices, and for this reason, unimodular lattices are also known as self-dual. Given a pair (m,n) of nonnegative integers, an even unimodular lattice of signature (m,n) exists if and only if m−n is divisible by 8, but an odd unimodular lattice of signature (m,n) always exists. In particular, even ...
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 Leech lattice can be explicitly constructed as the set of vectors of the form 2 −3/2 (a 1, a 2, ..., a 24) where the a i are integers such that + + + and for each fixed residue class modulo 4, the 24 bit word, whose 1s correspond to the coordinates i such that a i belongs to this residue class, is a word in the binary Golay code.
This article summarizes the classes of discrete symmetry groups of the Euclidean plane. The symmetry groups are named here by three naming schemes: International notation, orbifold notation, and Coxeter notation. There are three kinds of symmetry groups of the plane: 2 families of rosette groups – 2D point groups; 7 frieze groups – 2D line ...
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 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 ...
Lattices such as this are used - for example - in the Flory–Huggins solution theory In mathematical physics , a lattice model is a mathematical model of a physical system that is defined on a lattice , as opposed to a continuum , such as the continuum of space or spacetime .