71.202.143.72: /* Examples */ improved parentheses

2014-10-17T03:53:25Z

Examples: improved parentheses

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en>David Eppstein: /* Other results */ nontrivial

2014-02-26T23:32:20Z

Other results: nontrivial

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203.10.91.11: /* Definition */

2013-10-30T20:41:08Z

Definition

← Older revision		Revision as of 22:41, 30 October 2013
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	~~Irwin Butts~~ is ~~what my spouse loves to call me although I don~~'~~t really like being~~ called ~~like that~~. ~~Doing ceramics~~ is ~~what adore performing~~. ~~In her professional life she~~ is ~~a payroll clerk but she'~~s ~~usually wanted her own company. For years he'~~s ~~been residing in North Dakota~~ and ~~his family loves~~ it.<br><br>~~Also visit my web site ::~~ [~~http:~~//~~carnavalsite~~.~~com~~/~~demo-page-1~~/~~solid-advice-~~in-~~relation~~-to-~~yeast-infection~~/ ~~std home test~~]		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]].

			==Automorphism group==
			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]]
			:<math>(\mathbb Z_2)^n \rtimes S_n</math>
			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]]).

			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.

			==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 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.

			==Coarse geometry==
			In [[coarse structure\|coarse geometry]], the integer lattice is coarsely equivalent to [[Euclidean space]].

			==See also==
			*[[Regular grid]]

			==References==
			{{refimprove\|date=August 2013}}
			* {{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}}

			{{DEFAULTSORT:Integer Lattice}}
			[[Category:Euclidean geometry]]
			[[Category:Lattice points]]
			[[Category:Diophantine geometry]]

en>Nosuchforever: /* Examples */

2012-03-25T14:13:56Z

Examples

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Irwin Butts is what my spouse loves to call me although I don't really like being called like that. Doing ceramics is what adore performing. In her professional life she is a payroll clerk but she's usually wanted her own company. For years he's been residing in North Dakota and his family loves it.<br><br>Also visit my web site :: [http://carnavalsite.com/demo-page-1/solid-advice-in-relation-to-yeast-infection/ std home test]

Invariant basis number - Revision history

71.202.143.72: /* Examples */ improved parentheses

en>David Eppstein: /* Other results */ nontrivial

203.10.91.11: /* Definition */

en>Nosuchforever: /* Examples */