Lattice

The ordinary meaning of lattice is the basis for several technical usages

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• In nontechnical usage, a lattice is a structure of crossed laths with open spaces left between them; see also lattice tower, lattice steel pylon, lattice bridge.
• In one mathematical usage, a lattice is a partially ordered set (poset) in which any two elements have a supremum and an infimum. The Hasse diagrams of these posets look (in some simple cases) like the lattices of ordinary language. These lattices can also be defined as algebraic structures.
• In another mathematical usage, a lattice is a discrete subgroup of Rⁿ that spans Rⁿ as a real vector space, or a translate of a discrete subgroup. The elements of a lattice are regularly spaced, reminiscent of the intersection points of a lath lattice. See also below, and unimodular lattice, Leech lattice, Niemeier lattice, lattice point problems.

:Such a lattice arises from translational symmetry. For each point in Rn we have a set of corresponding points, called a lattice, that consists of all translates of the original point. If the original point is the origin of coordinates, this lattice is a discrete subgroup of Rn. In any case, it is a translate of a discrete subgroup.

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• The concept is used in materials science, in which a lattice is a 3-dimensional array of regularly spaced points coinciding with the atom or molecule positions in a crystal.
• In computational physics a lattice is an n-dimensional geometrical structure of sites, connected by bonds, which represent positions which may be occupied by atoms, molecules, electrons, spins, etc. For an article dealing with the formal representation of such structures see Lattice Geometries. Different types of lattice models are used in physics, mathematics, and finance.
• In digital signal processing, lattice filters are electronic filters with a special recursive structure.
• In music lattices are often used to model pitch space, especially just intonation. See: TonalSoft.com: Lattices Text and diagrams © 1998 by Joseph L. Monzo.
• Lattice Corporation made some of the first compilers for PCs.
• Lattice Semiconductor http://www.latticesemi.com is a company which makes FPGAs and related electronics products.
• In baking, various pastries and pies can be made with a lattice pattern, and special cutters are available to facilitate the creation of such patterns.