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| In [[mathematics]], the ''n''-dimensional '''integer lattice''' (or '''cubic lattice'''), denoted '''Z'''<sup>''n''</sup>, is the [[lattice (group)|lattice]] in the [[Euclidean space]] '''R'''<sup>''n''</sup> whose lattice points are [[n-tuple|''n''-tuple]]s of [[integer]]s. The two-dimensional integer lattice is also called the [[square lattice]], or grid lattice. '''Z'''<sup>''n''</sup> is the simplest example of a [[root lattice]]. The integer lattice is an odd [[unimodular lattice]].
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| ==Automorphism group==
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| The [[automorphism group]] (or group of [[congruence relation|congruence]]s) of the integer lattice consists of all [[permutation]]s and sign changes of the coordinates, and is of order 2<sup>''n''</sup> ''n''<nowiki>!</nowiki>. 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]]
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| :<math>(\mathbb Z_2)^n \rtimes S_n</math>
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| where the [[symmetric group]] ''S''<sub>''n''</sub> acts on ('''Z'''<sub>2</sub>)<sup>''n''</sup> by permutation (this is a classic example of a [[wreath product]]).
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| 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.
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| ==Diophantine geometry==
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| In the study of [[Diophantine geometry]], the square lattice of points with integer coordinates is often referred to as the '''Diophantine plane'''. In mathematical terms, the Diophantine plane is the [[Cartesian product]] <math>\scriptstyle\mathbb{Z}\times\mathbb{Z}</math> of the ring of all integers <math>\scriptstyle\mathbb{Z}</math>. The study of [[Erdős–Diophantine graph|Diophantine figures]] focuses on the selection of nodes in the Diophantine plane such that all pairwise distances are integer.
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| ==Coarse geometry==
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| In [[coarse structure|coarse geometry]], the integer lattice is coarsely equivalent to [[Euclidean space]].
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| ==See also==
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| *[[Regular grid]]
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| ==References==
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| {{refimprove|date=August 2013}}
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| * {{cite book|authors=Olds, C.D. et al|title=The Geometry of Numbers|publisher=Mathematical Association of America|year=2000|isbn=0-88385-643-3|url=http://books.google.com/books?id=Bycut_duHr8C&printsec=frontcover}}
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| {{DEFAULTSORT:Integer Lattice}}
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| [[Category:Euclidean geometry]]
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| [[Category:Lattice points]]
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| [[Category:Diophantine geometry]]
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