**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.

Read more about Lattice Points: Symmetry Considerations and Examples, Dividing Space According To A Lattice, Lattice Points in Convex Sets, Computing With Lattices, Lattices in Two Dimensions: Detailed Discussion, Lattices in Three Dimensions, Lattices in Complex Space, In Lie Groups, Lattices in General Vector-spaces

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

**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 ...

... 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**...

**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 ...

... 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)