Editing Hypercube (section)

{{short description|Convex polytope, the n-dimensional analogue of a square and a cube}}
{{other uses}}
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In [[geometry]], a '''hypercube''' is an [[N-dimensional space|''n''-dimensional]] analogue of a [[Square (geometry)|square]] ([[two-dimensional|{{nowrap|1=''n'' = 2}}]]) and a [[cube]] ([[Three-dimensional|{{nowrap|1=''n'' = 3}}]]); the special case for [[Four-dimensional space|{{nowrap|1=''n'' = 4}}]] is known as a ''[[tesseract]]''. It is a [[Closed set|closed]], [[Compact space|compact]], [[Convex polytope|convex]] figure whose 1-[[N-skeleton|skeleton]] consists of groups of opposite [[parallel (geometry)|parallel]] [[line segment]]s aligned in each of the space's [[dimension]]s, [[perpendicular]] to each other and of the same length.  A unit hypercube's longest diagonal in ''n'' dimensions is equal to <math>\sqrt{n}</math>.

An ''n''-dimensional hypercube is more commonly referred to as an '''''n''-cube''' or sometimes as an '''''n''-dimensional cube'''.<ref>{{Cite journal|url=https://dx.doi.org/10.1016/0771-050X%2876%2990005-X|title=An adaptive algorithm for numerical integration over an n-dimensional cube|author1=Paul Dooren|author2=Luc Ridder|journal=Journal of Computational and Applied Mathematics |date=1976 |volume=2 |issue=3 |pages=207–217 |doi=10.1016/0771-050X(76)90005-X }}</ref><ref>{{Cite journal|url=https://www.sciencedirect.com/science/article/pii/S0020025506003173|title=A (4n − 9)/3 diagnosis algorithm on n-dimensional cube network|author1=Xiaofan Yang|author2=Yuan Tang|journal=Information Sciences |date=15 April 2007 |volume=177 |issue=8 |pages=1771–1781 |doi=10.1016/j.ins.2006.10.002 }}</ref> The term '''measure polytope''' (originally from Elte, 1912)<ref>{{cite book|title=The Semiregular Polytopes of the Hyperspaces|last=Elte|first=E. L.|publisher=[[University of Groningen]]|year=1912|location=Netherlands|chapter=IV, Five dimensional semiregular polytope|isbn = 141817968X}}</ref>  is also used, notably in the work of [[Harold Scott MacDonald Coxeter|H. S. M. Coxeter]] who also labels the hypercubes the γ<small>n</small> polytopes.{{Sfn|Coxeter|1973|pp=122-123|loc=§7.2 see illustration Fig 7.2<small>C</small>}}

The hypercube is the special case of a [[hyperrectangle]] (also called an ''n-orthotope'').

A ''unit hypercube'' is a hypercube whose side has length one [[unit (number)|unit]].  Often, the hypercube whose corners (or ''vertices'') are the 2<sup>''n''</sup> points in '''R'''<sup>''n''</sup> with each coordinate equal to 0 or 1 is called ''the'' unit hypercube.