View source for Hypercube

{{short description|Convex polytope, the n-dimensional analogue of a square and a cube}}
{{other uses}}
{{multiple image
 | footer = In the following [[perspective projection]]s, [[cube (geometry)|cube]] is 3-cube and [[tesseract]] is 4-cube.
 | image1 = Hexahedron.svg
 | image2 = Hypercube.svg
 | total_width = 380px
}}

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 |url-access=subscription }}</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 |url-access=subscription }}</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.

== Construction ==
=== By the number of dimensions ===
[[File:From Point to Tesseract (Looped Version).gif|thumb|An animation showing how to create a tesseract from a point.]]

A hypercube can be defined by increasing the numbers of dimensions of a shape:
:'''0''' – A point is a hypercube of dimension zero.
:'''1''' – If one moves this point one unit length, it will sweep out a line segment, which is a unit hypercube of dimension one.
:'''2''' – If one moves this line segment its length in a [[perpendicular]] direction from itself; it sweeps out a 2-dimensional square.
:'''3''' – If one moves the square one unit length in the direction perpendicular to the plane it lies on, it will generate a 3-dimensional cube.
:'''4''' – If one moves the cube one unit length into the fourth dimension, it generates a 4-dimensional unit hypercube (a unit [[tesseract]]).

This can be generalized to any number of dimensions. This process of sweeping out volumes can be formalized mathematically as a [[Minkowski sum]]: the ''d''-dimensional hypercube is the Minkowski sum of ''d'' mutually perpendicular unit-length line segments, and is therefore an example of a [[zonotope]].

The [[Graph of a polytope|1-skeleton]] of a hypercube is a [[hypercube graph]].

=== Vertex coordinates ===

[[File:8-cell.gif|thumb|Projection of a [[rotation|rotating]] [[tesseract]].]]

A unit hypercube of dimension <math>n</math> is the [[convex hull]] of all the <math>2^n</math> points whose <math>n</math> [[Cartesian coordinate system|Cartesian coordinates]] are each equal to either <math>0</math> or <math>1</math>. These points are its [[vertex (geometry)|vertices]]. The hypercube with these coordinates is also the [[cartesian product]] <math>[0,1]^n</math> of <math>n</math> copies of the unit [[interval (mathematics)|interval]] <math>[0,1]</math>. Another unit hypercube, centered at the origin of the ambient space, can be obtained from this one by a [[translation (geometry)|translation]]. It is the convex hull of the <math>2^n</math> points whose vectors of Cartesian coordinates are

: <math>
\left(\pm \frac{1}{2}, \pm \frac{1}{2}, \cdots, \pm \frac{1}{2}\right)\!.
</math>

Here the symbol <math>\pm</math> means that each coordinate is either equal to <math>1/2</math> or to <math>-1/2</math>. This unit hypercube is also the cartesian product <math>[-1/2,1/2]^n</math>. Any unit hypercube has an edge length of <math>1</math> and an <math>n</math>-dimensional volume of <math>1</math>.

The <math>n</math>-dimensional hypercube obtained as the convex hull of the points with coordinates <math>(\pm 1, \pm 1, \cdots, \pm 1)</math> or, equivalently as the Cartesian product <math>[-1,1]^n</math> is also often considered due to the simpler form of its vertex coordinates. Its edge length is <math>2</math>, and its <math>n</math>-dimensional volume is <math>2^n</math>.

== Faces ==
Every hypercube admits, as its faces, hypercubes of a lower dimension contained in its boundary. A hypercube of dimension <math>n</math> admits <math>2n</math> facets, or faces of dimension <math>n-1</math>: a (<math>1</math>-dimensional) line segment has <math>2</math> endpoints; a (<math>2</math>-dimensional) square has <math>4</math> sides or edges; a <math>3</math>-dimensional cube has <math>6</math> square faces; a (<math>4</math>-dimensional) tesseract has <math>8</math> three-dimensional cubes as its facets. The number of vertices of a hypercube of dimension <math>n</math> is <math>2^n</math> (a usual, <math>3</math>-dimensional cube has <math>2^3=8</math> vertices, for instance).<ref>{{Cite journal |author1=Miroslav Vořechovský |author2=Jan Mašek |author3=Jan Eliáš |title=Distance-based optimal sampling in a hypercube: Analogies to N-body systems |journal=Advances in Engineering Software |volume=137 |date=November 2019 |article-number=102709 |issn=0965-9978 |doi=10.1016/j.advengsoft.2019.102709}}</ref>

The number of the <math>m</math>-dimensional hypercubes (just referred to as <math>m</math>-cubes from here on) contained in the boundary of an <math>n</math>-cube is

:<math> E_{m,n} = 2^{n-m}{n \choose m} </math>,{{sfn|Coxeter|1973|p=122|loc=§7·25}} &nbsp;&nbsp;&nbsp; where <math>{n \choose m}=\frac{n!}{m!\,(n-m)!}</math> and <math>n!</math> denotes the [[factorial]] of <math>n</math>.

For example, the boundary of a <math>4</math>-cube (<math>n=4</math>) contains <math>8</math> cubes (<math>3</math>-cubes), <math>24</math> squares (<math>2</math>-cubes), <math>32</math> line segments (<math>1</math>-cubes) and <math>16</math> vertices (<math>0</math>-cubes). This identity can be proven by a simple combinatorial argument: for each of the <math>2^n</math> vertices of the hypercube, there are <math>\tbinom n m</math> ways to choose a collection of <math>m</math> edges incident to that vertex. Each of these collections defines one of the <math>m</math>-dimensional faces incident to the considered vertex. Doing this for all the vertices of the hypercube, each of the <math>m</math>-dimensional faces of the hypercube is counted <math>2^m</math> times since it has that many vertices, and we need to divide <math>2^n\tbinom n m</math> by this number.

The number of facets of the hypercube can be used to compute the <math>(n-1)</math>-dimensional volume of its boundary: that volume is <math>2n</math> times the volume of a <math>(n-1)</math>-dimensional hypercube; that is, <math>2ns^{n-1}</math> where <math>s</math> is the length of the edges of the hypercube.

These numbers can also be generated by the linear [[recurrence relation]].

:<math>E_{m,n} = 2E_{m,n-1} + E_{m-1,n-1} \!</math>, &nbsp; &nbsp; with <math>E_{0,0}= 1</math>, and <math>E_{m,n}=0</math> when <math>n < m</math>, <math>n < 0</math>, or <math>m < 0</math>.

For example, extending a square via its 4 vertices adds one extra line segment (edge) per vertex. Adding the opposite square to form a cube provides <math>E_{1,3}=12</math> line segments. 

The extended [[f-vector]] for an ''n''-cube can also be computed by expanding <math>(2x+1)^n</math> (concisely, (2,1)<sup>''n''</sup>), and reading off the coefficients of the resulting [[Polynomial#Multiplication|polynomial]]. For example, the elements of a tesseract is (2,1)<sup>4</sup> = (4,4,1)<sup>2</sup> = (16,32,24,8,1).

{| class="wikitable"
|+
Number <math>E_{m,n}</math> of <math>m</math>-dimensional faces of a <math>n</math>-dimensional hypercube {{OEIS|A038207}}
|-
! || || || m|| 0|| 1|| 2|| 3|| 4|| 5|| 6|| 7|| 8|| 9|| 10
|-
! [[polytope|''n'']]
! ''n''-cube
! Names
![[Schläfli symbol|Schläfli]]<br>[[Coxeter–Dynkin diagram|Coxeter]]<br>
![[Vertex (geometry)|Vertex]]<br>0-face<br>|| [[Edge (geometry)|Edge]]<br>1-face<br>|| [[Face (geometry)|Face]]<br>2-face<br>|| [[Cell (geometry)|Cell]]<br>3-face<br>|| <br>4-face<br>||<br> 5-face<br>|| <br>6-face<br>|| <br>7-face<br>||<br> 8-face<br>|| <br>9-face<br>||<br>10-face<br>
|-
! [[0-polytope|0]]
! 0-cube
| Point<br>'''Monon'''<br>
| ( )<br>{{CDD|node}}<br>
| 1||  ||rowspan=2| ||rowspan=3| ||rowspan=4| ||rowspan=5| ||rowspan=6| ||rowspan=7| ||rowspan=8| ||rowspan=9| ||rowspan=10|
|-
! [[1-polytope|1]]
! 1-cube
| [[Line segment]]<br>'''Dion'''<ref>Johnson, Norman W.; ''Geometries and Transformations'', Cambridge University Press, 2018, p.224.</ref><br>
|{}<br>{{CDD|node_1}}<br>
| 2|| 1
|-
! [[2-polytope|2]]
! 2-cube
| [[Square (geometry)|Square]]<br>'''Tetragon'''<br>
|{4}<br>{{CDD|node_1|4|node}}<br>
| 4|| 4|| 1
|-
! [[3-polytope|3]]
! 3-cube
| [[Cube]]<br>'''Hexahedron'''<br>
|{4,3}<br>{{CDD|node_1|4|node|3|node}}<br>
| 8|| 12|| 6|| 1
|-
! [[4-polytope|4]]
! 4-cube
| [[Tesseract]]<br>'''Octachoron'''<br>
|{4,3,3}<br>{{CDD|node_1|4|node|3|node|3|node}}<br>
| 16|| 32|| 24|| 8|| 1
|-
! [[5-polytope|5]]
! [[5-cube]]
| Penteract<br>'''Deca-5-tope'''<br>
|{4,3,3,3}<br>{{CDD|node_1|4|node|3|node|3|node|3|node}}<br>
| 32|| 80|| 80|| 40|| 10|| 1
|-
! [[6-polytope|6]]
! [[6-cube]]
| Hexeract<br>'''Dodeca-6-tope'''<br>
|{4,3,3,3,3}<br>{{CDD|node_1|4|node|3|node|3|node|3|node|3|node}}<br>
| 64|| 192|| 240|| 160|| 60|| 12|| 1
|-
! [[7-polytope|7]]
! [[7-cube]]
| Hepteract<br>'''Tetradeca-7-tope'''<br>
|{4,3,3,3,3,3}<br>{{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node}}<br>
| 128|| 448|| 672|| 560|| 280|| 84|| 14|| 1
|-
! [[8-polytope|8]]
! [[8-cube]]
| Octeract<br>'''Hexadeca-8-tope'''<br>
|{4,3,3,3,3,3,3}<br>{{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node}}<br>
| 256|| 1024|| 1792|| 1792|| 1120|| 448|| 112|| 16|| 1
|-
! [[9-polytope|9]]
! [[9-cube]]
| Enneract<br>'''Octadeca-9-tope'''<br>
|{4,3,3,3,3,3,3,3}<br>{{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}<br>
| 512|| 2304|| 4608|| 5376|| 4032|| 2016|| 672|| 144|| 18|| 1
|-
! [[10-polytope|10]]
! [[10-cube]]
| Dekeract<br>'''Icosa-10-tope'''<br>
|{4,3,3,3,3,3,3,3,3}<br>{{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}<br>
|1024||5120||11520||15360||13440||8064||3360||960||180||20||1
|}

=== Graphs ===
An '''''n''-cube''' can be projected inside a regular 2''n''-gonal polygon by a [[Petrie polygon#The hypercube and orthoplex families|skew orthogonal projection]], shown here from the line segment to the 15-cube.

{| class="wikitable skin-invert-image"
|+ [[Petrie polygon]] [[Orthographic projection]]s
|- align=center valign=bottom
|[[File:1-simplex t0.svg|160px]]<br />[[Line segment]]
|[[File:2-cube.svg|160px]]<br />[[Square (geometry)|Square]]
|[[File:3-cube graph.svg|160px]]<br />[[Cube]]
|[[File:4-cube graph.svg|160px]]<br />[[Tesseract]]
|- align=center
|[[File:5-cube graph.svg|160px]]<br />[[5-cube]]
|[[File:6-cube graph.svg|160px]]<br />[[6-cube]]
|[[File:7-cube graph.svg|160px]]<br />[[7-cube]]
|[[File:8-cube.svg|160px]]<br />[[8-cube]]
|- align=center
|[[File:9-cube.svg|160px]]<br />[[9-cube]]
|[[File:10-cube.svg|160px]]<br />[[10-cube]]
|[[File:11-cube.svg|160px]]<br />11-cube
|[[File:12-cube.svg|160px]]<br />12-cube
|- align=center
|[[File:13-cube.svg|160px]]<br />13-cube
|[[File:14-cube.svg|160px]]<br />14-cube
|[[File:15-cube.svg|160px]]<br />15-cube
|<!--[[File:16-cube t0 A15.svg|160px]]<br />[[16-cube]] - this is not in the B16 Coxeter plane-->
|}

== Related families of polytopes ==
The hypercubes are one of the few families of [[regular polytope]]s that are represented in any number of dimensions.<ref>{{cite journal|title=Transmitting in the n-dimensional cube|author1=Noga Alon|journal=Discrete Applied Mathematics |date=1992 |volume=37-38 |pages=9–11 |doi=10.1016/0166-218X(92)90121-P |doi-access=free }}</ref>

The '''hypercube''' family is one of three [[regular polytope]] families, labeled by [[Coxeter]] as ''γ<sub>n</sub>''. The other two are the hypercube dual family, the '''[[cross-polytope]]s''', labeled as ''β<sub>n,</sub>'' and the '''[[simplex|simplices]]''', labeled as ''α<sub>n</sub>''. A fourth family, the [[hypercubic honeycomb|infinite tessellations of hypercubes]], is labeled as ''δ<sub>n</sub>''.

Another related family of semiregular and [[uniform polytope]]s is the '''[[demihypercube]]s''', which are constructed from hypercubes with alternate vertices deleted and [[simplex]] facets added in the gaps, labeled as ''hγ<sub>n</sub>''.

''n''-cubes can be combined with their duals (the [[cross-polytope]]s) to form compound polytopes:

* In two dimensions, we obtain the [[octagram]]mic star figure {8/2},
* In three dimensions we obtain the [[compound of cube and octahedron]],
* In four dimensions we obtain the compound of tesseract and 16-cell.

== {{anchor|Relation to n-simplices}}Relation to (''n''−1)-simplices ==
The graph of the ''n''-hypercube's edges is [[isomorphism|isomorphic]] to the [[Hasse diagram]] of the (''n''−1)-[[simplex]]'s [[Convex polytope#The face lattice|face lattice]]. This can be seen by orienting the ''n''-hypercube so that two opposite vertices lie vertically, corresponding to the (''n''−1)-simplex itself and the null polytope, respectively. Each vertex connected to the top vertex then uniquely maps to one of the (''n''−1)-simplex's facets (''n''−2 faces), and each vertex connected to those vertices maps to one of the simplex's ''n''−3 faces, and so forth, and the vertices connected to the bottom vertex map to the simplex's vertices.

This relation may be used to generate the face lattice of an (''n''−1)-simplex efficiently, since face lattice enumeration algorithms applicable to general polytopes are more computationally expensive.

== Generalized hypercubes ==
Regular [[Complex polytope#Regular complex polytopes|complex polytopes]] can be defined in [[Complex number|complex]] [[Hilbert space]] called ''generalized hypercubes'', &gamma;{{supsub|''p''|''n''}} = <sub>''p''</sub>{4}<sub>2</sub>{3}...<sub>2</sub>{3}<sub>2</sub>, or {{CDD|pnode_1|4|node|3}}..{{CDD|3|node|3|node}}. Real solutions exist with ''p'' = 2, i.e. &gamma;{{supsub|2|''n''}} = &gamma;<sub>''n''</sub> = <sub>2</sub>{4}<sub>2</sub>{3}...<sub>2</sub>{3}<sub>2</sub> = {4,3,..,3}. For ''p'' > 2, they exist in <math>\mathbb{C}^n</math>. The facets are generalized (''n''−1)-cube and the [[vertex figure]] are regular [[simplex]]es.

The [[regular polygon]] perimeter seen in these orthogonal projections is called a [[Petrie polygon]]. The generalized squares (''n'' = 2) are shown with edges outlined as red and blue alternating color ''p''-edges, while the higher ''n''-cubes are drawn with black outlined ''p''-edges.

The number of ''m''-face elements in a ''p''-generalized ''n''-cube are: <math>p^{n-m}{n \choose m}</math>. This is ''p''<sup>''n''</sup> vertices and ''pn'' facets.<ref>{{citation
 | last = Coxeter | first = H. S. M.
 | mr = 0370328
 | page = 180
 | publisher = [[Cambridge University Press]] | location = London & New York
 | title = Regular complex polytopes
 | year = 1974}}.</ref>

{| class=wikitable
|+ Generalized hypercubes
! || ''p''=2 || ||''p''=3 ||''p''=4||''p''=5||''p''=6||''p''=7||''p''=8
|- align=center valign=bottom
!valign=middle|<math>\mathbb{R}^2</math>
|[[File:2-generalized-2-cube.svg|class=skin-invert-image|100px]]<BR>&gamma;{{supsub|2|2}} = [[square|{4}]] = {{CDD|node_1|4|node}}<BR>4 vertices
!valign=middle|<math>\mathbb{C}^2</math>
|[[File:3-generalized-2-cube skew.svg|class=skin-invert-image|100px]]<BR>&gamma;{{supsub|3|2}} = {{CDD|3node_1|4|node}}<BR>9 vertices
|[[File:4-generalized-2-cube.svg|class=skin-invert-image|100px]]<BR>&gamma;{{supsub|4|2}} = {{CDD|4node_1|4|node}}<BR>16 vertices
|[[File:5-generalized-2-cube skew.svg|class=skin-invert-image|100px]]<BR>&gamma;{{supsub|5|2}} = {{CDD|5node_1|4|node}}<BR>25 vertices
|[[File:6-generalized-2-cube.svg|class=skin-invert-image|100px]]<BR>&gamma;{{supsub|6|2}} = {{CDD|6node_1|4|node}}<BR>36 vertices
|[[File:7-generalized-2-cube skew.svg|class=skin-invert-image|100px]]<BR>&gamma;{{supsub|7|2}} = {{CDD|7node_1|4|node}}<BR>49 vertices
|[[File:8-generalized-2-cube.svg|class=skin-invert-image|100px]]<BR>&gamma;{{supsub|8|2}} = {{CDD|8node_1|4|node}}<BR>64 vertices
|- align=center valign=bottom
!valign=middle|<math>\mathbb{R}^3</math>
|[[File:2-generalized-3-cube.svg|class=skin-invert-image|100px]]<BR>&gamma;{{supsub|2|3}} = [[cube|{4,3}]] = {{CDD|node_1|4|node|3|node}}<BR>8 vertices
!valign=middle|<math>\mathbb{C}^3</math>
|[[File:3-generalized-3-cube.svg|class=skin-invert-image|100px]]<BR>&gamma;{{supsub|3|3}} = {{CDD|3node_1|4|node|3|node}}<BR>27 vertices
|[[File:4-generalized-3-cube.svg|class=skin-invert-image|100px]]<BR>&gamma;{{supsub|4|3}} = {{CDD|4node_1|4|node|3|node}}<BR>64 vertices
|[[File:5-generalized-3-cube.svg|class=skin-invert-image|100px]]<BR>&gamma;{{supsub|5|3}} = {{CDD|5node_1|4|node|3|node}}<BR>125 vertices
|[[File:6-generalized-3-cube.svg|class=skin-invert-image|100px]]<BR>&gamma;{{supsub|6|3}} = {{CDD|6node_1|4|node|3|node}}<BR>216 vertices
|[[File:7-generalized-3-cube.svg|class=skin-invert-image|100px]]<BR>&gamma;{{supsub|7|3}} = {{CDD|7node_1|4|node|3|node}}<BR>343 vertices
|[[File:8-generalized-3-cube.svg|class=skin-invert-image|100px]]<BR>&gamma;{{supsub|8|3}} = {{CDD|8node_1|4|node|3|node}}<BR>512 vertices
|- align=center valign=bottom
!valign=middle|<math>\mathbb{R}^4</math>
|[[File:2-generalized-4-cube.svg|class=skin-invert-image|100px]]<BR>&gamma;{{supsub|2|4}} = [[tesseract|{4,3,3}]]<BR>= {{CDD|node_1|4|node|3|node|3|node}}<BR>16 vertices
!valign=middle|<math>\mathbb{C}^4</math>
|[[File:3-generalized-4-cube.svg|class=skin-invert-image|100px]]<BR>&gamma;{{supsub|3|4}} = {{CDD|3node_1|4|node|3|node|3|node}}<BR>81 vertices
|[[File:4-generalized-4-cube.svg|class=skin-invert-image|100px]]<BR>&gamma;{{supsub|4|4}} = {{CDD|4node_1|4|node|3|node|3|node}}<BR>256 vertices
|[[File:5-generalized-4-cube.svg|class=skin-invert-image|100px]]<BR>&gamma;{{supsub|5|4}} = {{CDD|5node_1|4|node|3|node|3|node}}<BR>625 vertices
|[[File:6-generalized-4-cube.svg|class=skin-invert-image|100px]]<BR>&gamma;{{supsub|6|4}} = {{CDD|6node_1|4|node|3|node|3|node}}<BR>1296 vertices
|[[File:7-generalized-4-cube.svg|class=skin-invert-image|100px]]<BR>&gamma;{{supsub|7|4}} = {{CDD|7node_1|4|node|3|node|3|node}}<BR>2401 vertices
|[[File:8-generalized-4-cube.svg|class=skin-invert-image|100px]]<BR>&gamma;{{supsub|8|4}} = {{CDD|8node_1|4|node|3|node|3|node}}<BR>4096 vertices
|- align=center valign=bottom
!valign=middle|<math>\mathbb{R}^5</math>
|[[File:2-generalized-5-cube.svg|class=skin-invert-image|100px]]<BR>&gamma;{{supsub|2|5}} = [[5-cube|{4,3,3,3}]]<BR>= {{CDD|node_1|4|node|3|node|3|node|3|node}}<BR>32 vertices
!valign=middle|<math>\mathbb{C}^5</math>
|[[File:3-generalized-5-cube.svg|class=skin-invert-image|100px]]<BR>&gamma;{{supsub|3|5}} = {{CDD|3node_1|4|node|3|node|3|node|3|node}}<BR>243 vertices
|[[File:4-generalized-5-cube.svg|class=skin-invert-image|100px]]<BR>&gamma;{{supsub|4|5}} = {{CDD|4node_1|4|node|3|node|3|node|3|node}}<BR>1024 vertices
|[[File:5-generalized-5-cube.svg|class=skin-invert-image|100px]]<BR>&gamma;{{supsub|5|5}} = {{CDD|5node_1|4|node|3|node|3|node|3|node}}<BR>3125 vertices
|[[File:6-generalized-5-cube.svg|class=skin-invert-image|100px]]<BR>&gamma;{{supsub|6|5}} = {{CDD|6node_1|4|node|3|node|3|node|3|node}}<BR>7776 vertices
|&gamma;{{supsub|7|5}} = {{CDD|7node_1|4|node|3|node|3|node|3|node}}<BR>16,807 vertices
|&gamma;{{supsub|8|5}} = {{CDD|8node_1|4|node|3|node|3|node|3|node}}<BR>32,768 vertices
|- align=center valign=bottom
!valign=middle|<math>\mathbb{R}^6</math>
|[[File:2-generalized-6-cube.svg|class=skin-invert-image|100px]]<BR>&gamma;{{supsub|2|6}} = [[6-cube|{4,3,3,3,3}]]<BR>= {{CDD|node_1|4|node|3|node|3|node|3|node|3|node}}<BR>64 vertices
!valign=middle|<math>\mathbb{C}^6</math>
|[[File:3-generalized-6-cube.svg|class=skin-invert-image|100px]]<BR>&gamma;{{supsub|3|6}} = {{CDD|3node_1|4|node|3|node|3|node|3|node|3|node}}<BR>729 vertices
|[[File:4-generalized-6-cube.svg|class=skin-invert-image|100px]]<BR>&gamma;{{supsub|4|6}} = {{CDD|4node_1|4|node|3|node|3|node|3|node|3|node}}<BR>4096 vertices
|[[File:5-generalized-6-cube.svg|class=skin-invert-image|100px]]<BR>&gamma;{{supsub|5|6}} = {{CDD|5node_1|4|node|3|node|3|node|3|node|3|node}}<BR>15,625 vertices
|&gamma;{{supsub|6|6}} = {{CDD|6node_1|4|node|3|node|3|node|3|node|3|node}}<BR>46,656 vertices
|&gamma;{{supsub|7|6}} = {{CDD|7node_1|4|node|3|node|3|node|3|node|3|node}}<BR>117,649 vertices
|&gamma;{{supsub|8|6}} = {{CDD|8node_1|4|node|3|node|3|node|3|node|3|node}}<BR>262,144 vertices
|- align=center valign=bottom
!valign=middle|<math>\mathbb{R}^7</math>
|[[File:2-generalized-7-cube.svg|class=skin-invert-image|100px]]<BR>&gamma;{{supsub|2|7}} = [[7-cube|{4,3,3,3,3,3}]]<BR>= {{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node}}<br>128 vertices
!valign=middle|<math>\mathbb{C}^7</math>
|[[File:3-generalized-7-cube.svg|class=skin-invert-image|100px]]<BR>&gamma;{{supsub|3|7}} = {{CDD|3node_1|4|node|3|node|3|node|3|node|3|node|3|node}}<BR>2187 vertices
|&gamma;{{supsub|4|7}} = {{CDD|4node_1|4|node|3|node|3|node|3|node|3|node|3|node}}<BR>16,384 vertices
|&gamma;{{supsub|5|7}} = {{CDD|5node_1|4|node|3|node|3|node|3|node|3|node|3|node}}<BR>78,125 vertices
|&gamma;{{supsub|6|7}} = {{CDD|6node_1|4|node|3|node|3|node|3|node|3|node|3|node}}<BR>279,936 vertices
|&gamma;{{supsub|7|7}} = {{CDD|7node_1|4|node|3|node|3|node|3|node|3|node|3|node}}<BR>823,543 vertices
|&gamma;{{supsub|8|7}} = {{CDD|8node_1|4|node|3|node|3|node|3|node|3|node|3|node}}<BR>2,097,152 vertices
|- align=center valign=bottom
!valign=middle|<math>\mathbb{R}^8</math>
|[[File:2-generalized-8-cube.svg|class=skin-invert-image|100px]]<BR>&gamma;{{supsub|2|8}} = [[8-cube|{4,3,3,3,3,3,3}]]<BR>= {{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node}}<BR>256 vertices
!valign=middle|<math>\mathbb{C}^8</math>
|[[File:3-generalized-8-cube.svg|class=skin-invert-image|100px]]<BR>&gamma;{{supsub|3|8}} = {{CDD|3node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node}}<BR>6561 vertices
|&gamma;{{supsub|4|8}} = {{CDD|4node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node}}<BR>65,536 vertices
|&gamma;{{supsub|5|8}} = {{CDD|5node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node}}<BR>390,625 vertices
|&gamma;{{supsub|6|8}} = {{CDD|6node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node}}<BR>1,679,616 vertices
|&gamma;{{supsub|7|8}} = {{CDD|7node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node}}<BR>5,764,801 vertices
|&gamma;{{supsub|8|8}} = {{CDD|8node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node}}<BR>16,777,216 vertices
|}

== Relation to exponentiation ==
Any positive integer raised to another positive integer power will yield a third integer, with this third integer being a specific type of [[figurate number]] corresponding to an ''n''-cube with a number of dimensions corresponding to the exponential. For example, the exponent 2 will yield a [[square number]] or "perfect square", which can be arranged into a square shape with a side length corresponding to that of the base. Similarly, the exponent 3 will yield [[Cube (algebra)#In integers|a perfect cube]], an integer which can be arranged into a cube shape with a side length of the base. As a result, the act of raising a number to 2 or 3 is more commonly referred to as "[[Square (algebra)|squaring]]" and "cubing", respectively. However, the names of higher-order hypercubes do not appear to be in common use for higher powers.

== See also ==
{{Portal|Mathematics}}
* [[Multiple instruction, multiple data#Hypercube interconnection network|Hypercube interconnection network]] of computer architecture
* [[Hyperoctahedral group]], the symmetry group of the hypercube
* [[Hypersphere]]
* [[Simplex]]
* [[Parallelohedron#Parallelotope|Parallelotope]]
* ''[[Crucifixion (Corpus Hypercubus)]]'', a painting by Salvador Dalí featuring an unfolded 4-cube

== Notes ==
{{reflist}}

== References ==
* {{cite journal|author-link=Jonathan Bowen |last=Bowen |first=J. P. | title=Hypercube | journal=[[Practical Computing]] | volume=5 | issue=4 | pages=97–99 | date=April 1982 |url=http://www.jpbowen.com/publications/ndcubes.html |archive-url=https://web.archive.org/web/20080630081518/http://www.jpbowen.com/publications/ndcubes.html |url-status=dead |archive-date=2008-06-30 | access-date=June 30, 2008 }}
* {{cite book
|author-link = Harold Scott MacDonald Coxeter
|last = Coxeter
|first = H. S. M.
|title = [[Regular Polytopes (book)|Regular Polytopes]]
|edition = 3rd
|publisher = [[Dover Publications|Dover]]
|year = 1973
|pages = [https://archive.org/details/regularpolytopes0000coxe/page/122 122-123]
|chapter= §7.2. see illustration Fig. 7-2c
|isbn = 0-486-61480-8
}} p.&nbsp;296, Table I (iii): Regular Polytopes, three regular polytopes in ''n'' dimensions (''n''&nbsp;≥&nbsp;5)
* {{cite book
|first = Frederick J.
|last = Hill
|author2 = Gerald R. Peterson
|title = Introduction to Switching Theory and Logical Design: Second Edition
|year = 1974
|publisher = [[John Wiley & Sons]]
|place = New York
|isbn = 0-471-39882-9
}} Cf Chapter 7.1 "Cubical Representation of Boolean Functions" wherein the notion of "hypercube" is introduced as a means of demonstrating a distance-1 code ([[Gray code]]) as the vertices of a hypercube, and then the hypercube with its vertices so labelled is squashed into two dimensions to form either a [[Veitch diagram]] or [[Karnaugh map]].

== External links ==
{{Commons category|Hypercubes}}
* {{MathWorld|title=Hypercube|urlname=Hypercube}}
* {{MathWorld|title=Hypercube graphs|urlname=HypercubeGraph}}
* ''[http://demonstrations.wolfram.com/RotatingAHypercube/  Rotating a Hypercube]'' by Enrique Zeleny, [[Wolfram Demonstrations Project]].
* [https://web.archive.org/web/20130326090312/http://www.cs.sjsu.edu/~rucker/hypercube.htm Rudy Rucker and Farideh Dormishian's Hypercube Downloads]
* [https://oeis.org/A001787 A001787&nbsp;&nbsp;&nbsp;&nbsp;Number of edges in an n-dimensional hypercube.] at [[OEIS]]

{{Dimension topics}}
{{Polytopes}}


[[Category:Regular polytopes]]
[[Category:Cubes]]