Editing Hypercube (section)

== 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 |at=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 16-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-->
|}