In mathematics, The n-dimensional integer lattice (or cubic lattice), denoted Zⁿ, is the lattice in the Euclidean space Rⁿ whose lattice points are n-tuples of integers. The two-dimensional integer lattice is also called the square lattice, or grid lattice. Zⁿ is the simplest example of a root lattice. The integer lattice is an odd unimodular lattice. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... See lattice for other meanings of this term, both within and without mathematics. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... See also tuple (music) as in duple and triple. ... The integers are commonly denoted by the above symbol. ... Upright square tiling. ... In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. ... In mathematics, a unimodular lattice is a lattice of discriminant 1 or −1. ...

Automorphism group

The automorphism group (or group of congruences) of the integer lattice consists of all permutations and sign changes of the coordinates, and is of order 2ⁿ n!. As a matrix group it is given by the set of all n×n signed permutation matrices. This group is isomorphic to the semidirect product In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... As an abstract term, congruence means similarity between objects. ... Permutation is the rearrangement of objects or symbols into distinguishable sequences. ... In mathematics, a matrix group is a group G consisting of invertible matrices over some field K, usually fixed in advance, with operations of matrix multiplication and inversion. ... In abstract algebra, a semidirect product describes a particular way in which a group can be put together from two subgroups. ...

where the symmetric group S_n acts on (Z₂)ⁿ by permutation (this is a classic example of a wreath product). In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ... In mathematics, the wreath product of group theory is a specialized product of two groups, based on a semidirect product. ...

For the square lattice, this is the group of the square, or the dihedral group of order 8; for the three dimensional cubic lattice, we get the group of the cube, or octahedral group, of order 48. This article may be confusing for some readers, and should be edited to enhance clarity. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

See also

• Regular grid