| Lattice (group) |
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In Mathematics , especially in Geometry and Group Theory , a lattice in '''R'''''n'' is a Discrete Subgroup of '''R'''''n'' which Spans the Real Vector Space '''R'''''n''. Every lattice in '''R'''''n'' can be generated from a Basis for the vector space by forming all Linear Combination s with Integral coefficients. Lattices have many significant applications in pure mathematics, particularly in connection to Lie Algebra s, number theory and group theory. They also arise in applied mathematics in connection with Coding Theory , and are used in various ways in the physical sciences; for instance 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 . It also occurs in computational physics, in which 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 . Quite general Lattice Model s are used in Physics . SYMMETRY CONSIDERATIONS AND EXAMPLES A lattice is the Symmetry Group of discrete Translational Symmetry in ''n'' directions. A pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself. A lattice in the sense of a 3-, which need not contain the origin, and therefore need not be a lattice in the previous sense. A simple example of a lattice in R''n'' is the subgroup '''Z'''''n''. A more complicated example is the Leech Lattice , which is a lattice in R24. The Period Lattice in R2 is central to the study of Elliptic Functions , developed in Nineteenth Century mathematics; it generalises to higher dimensions in the theory of Abelian Function s. DIVIDING SPACE ACCORDING TO A LATTICE A typical lattice Λ in R''n'' thus has the form : |
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