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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.
Cable car on Broadway just north of 2nd Street looking south, Los Angeles, c. 1893–1895 Above image zoomed out, Los Angeles, c. 1893–1895 The Women's Christian Temperance Union building, also known as Temperance Temple, at Temple and Fort (now Broadway) streets, with a Temple Street Cable Railway car, 1890
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 the study of Diophantine geometry, the square lattice of points with integer coordinates is often referred to as the Diophantine plane. In mathematical terms, the Diophantine plane is the Cartesian product Z × Z {\displaystyle \scriptstyle \mathbb {Z} \times \mathbb {Z} } of the ring of all integers Z {\displaystyle \scriptstyle \mathbb {Z} } .
A unimodular lattice is even or type II if all norms are even, otherwise odd or type I. The minimum of a positive definite lattice is the lowest nonzero norm. Lattices are often embedded in a real vector space with a symmetric bilinear form. The lattice is positive definite, Lorentzian, and so on if its vector space is.
In six-dimensional Euclidean geometry, the 6-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 6-simplex, rectified 6-simplex, and birectified 6-simplex facets. These facet types occur in proportions of 1:1:1 respectively in the whole honeycomb.
Lattice (group), for the case where M is a Z-module embedded in a vector space V over the field of real numbers R, and the Euclidean metric is used to describe the lattice structure References [ edit ]
In general terms, ideal lattices are lattices corresponding to ideals in rings of the form [] / for some irreducible polynomial of degree . [1] All of the definitions of ideal lattices from prior work are instances of the following general notion: let be a ring whose additive group is isomorphic to (i.e., it is a free -module of rank), and let be an additive isomorphism mapping to some lattice ...