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
Famous quotes containing the word points:
“The dominant metaphor of conceptual relativism, that of differing points of view, seems to betray an underlying paradox. Different points of view make sense, but only if there is a common co-ordinate system on which to plot them; yet the existence of a common system belies the claim of dramatic incomparability.”
—Donald Davidson (b. 1917)