Integer lattice - Citizendia

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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, especially in Geometry and Group theory, a lattice in R n is a Discrete subgroup of In Mathematics, a tuple is a Sequence (also known as an "ordered list" of values called the components of the tuple The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French The two-dimensional integer lattice is also called the square lattice, or grid lattice. In Mathematics, the square lattice is one of the five two-dimensional lattice types Zⁿ is the simplest example of a root lattice. This article discusses root systems in mathematics For root systems of Plants see Root. The integer lattice is an odd unimodular lattice. In Mathematics, a unimodular lattice is a lattice of Discriminant 1 or &minus1

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!. In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself In several fields of Mathematics the term permutation is used with different but closely related meanings As a matrix group it is given by the set of all n×n signed permutation matrices. In Mathematics, a matrix group is a group G consisting of invertible matrices over some field K, usually fixed In Mathematics, a generalized permutation matrix (or monomial matrix) is a matrix with the same nonzero pattern as a Permutation matrix, i This group is isomorphic to the semidirect product

$(\mathbb Z_2)^n \rtimes S_n$

where the symmetric group S_n acts on (Z₂)ⁿ by permutation (this is a classic example of a wreath product). In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying 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. In Mathematics, a dihedral group is the group of symmetries of a Regular polygon, including both rotations and reflections A regular Octahedron has 24 rotational (or orientation-preserving symmetries and a total of 48 symmetries including transformations that combine a reflection and a rotation

Diophantine geometry

In the study of Diophantine geometry, the square lattice of points with integer coordinates is often referred to as the Diophantine plane. In Mathematics, a Diophantine equation is an indeterminate Polynomial Equation that allows the variables to be Integers only In mathematical terms, the Diophantine plane is the Cartesian product $\scriptstyle\mathbb{Z}\times\mathbb{Z}$ of the ring of all integers $\scriptstyle\mathbb{Z}$ . Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory. The study of Diophantine figures focusses on the selection of nodes in the Diophantine plane such that all pairwise distances are integer. In Diophantine geometry, an Erdős-Diophantine graph, named after Paul Erdős and Diophantus of Alexandria, is a Complete graph with vertices

Automorphism group

Diophantine geometry

See also