# Lattice Points

Lattice Points

In mathematics, especially in geometry and group theory, a lattice in is a discrete subgroup of which spans the real vector space . Every lattice in can be generated from a basis for the vector space by forming all linear combinations with integer coefficients. A lattice may be viewed as a regular tiling of a space by a primitive cell.

Lattices have many significant applications in pure mathematics, particularly in connection to Lie algebras, number theory and group theory. They also arise in applied mathematics in connection with coding theory, in cryptography because of conjectured computational hardness of several lattice problems, and are used in various ways in the physical sciences. For instance, in materials science and solid-state physics, a lattice is a synonym for the "frame work" of a crystalline structure, a 3-dimensional array of regularly spaced points coinciding with the atom or molecule positions in a crystal. More generally, lattice models are studied in physics, often by the techniques of computational physics.

### Other articles related to "lattice points, lattice, points, point, lattices, lattice point":

Lattice (group) - Lattice Points in Convex Sets
... volume of a symmetric convex set S to the number of lattice points contained in S ... The number of lattice points contained in a polytope all of whose vertices are elements of the lattice is described by the polytope's Ehrhart polynomial ... See also Integer points in polyhedra ...
Parallel-TEBD - Implementation
... This function describes a N-point lattice which we would like to compute on P different compute nodes ... This means that if we distribute the lattice points evenly among the compute nodes (the easiest scenario), an even number of lattice points 2k is assigned to each compute node ... Indexing the lattice points from 0 to N-1 (note that the usual indexing is 1,N) and the compute nodes from 0 to P-1, the lattice points would be distributed as follows ...
Lattice Points - Lattices in General Vector-spaces
... Whilst we normally consider lattices in this concept can be generalized to any finite dimensional vector space over any field ... Then the R lattice in V generated by B is given by Different bases B will in general generate different lattices ... of T is in - the unit group of elements in R with multiplicative inverses) then the lattices generated by these bases will be isomorphic since T induces an isomorphism between the two lattices ...
E8 Lattice - Lattice Points
... The E8 lattice is a discrete subgroup of R8 of full rank (i.e ... It can be given explicitly by the set of points Γ8 ⊂ R8 such that all the coordinates are integers or all the coordinates are half-integers (a mixture of integers and half-i ... It is not hard to check that the sum of two lattice points is another lattice point, so that Γ8 is indeed a subgroup ...
Crystallographic Restriction Theorem - Dimensions 2 and 3 - Lattice Proof
... symmetry in dimension 2 or 3 must move a lattice point to a succession of other lattice points in the same plane, generating a regular polygon of ... We might call this a proof in the style of Busby Berkeley, with lattice vectors rather than pretty ladies dancing and swirling in geometric patterns.) Now consider an 8-fold rotation, and the displacement ... If a displacement exists between any two lattice points, then that same displacement is repeated everywhere in the lattice ...

### Famous quotes containing the word points:

The three main medieval points of view regarding universals are designated by historians as realism, conceptualism, and nominalism. Essentially these same three doctrines reappear in twentieth-century surveys of the philosophy of mathematics under the new names logicism, intuitionism, and formalism.
Willard Van Orman Quine (b. 1908)