View source for 8-cube

{{Short description|8-dimensional hypercube}}
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|8-cube<BR>Octeract
|-
|bgcolor=#ffffff align=center colspan=2|[[File:8-cube.svg|285px]]<BR>[[Orthogonal projection]]<BR>inside [[Petrie polygon]]
|-
|bgcolor=#e7dcc3|Type||Regular [[8-polytope]]
|-
|bgcolor=#e7dcc3|Family||[[hypercube]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| {4,3<sup>6</sup>}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node}}<BR>
{{CDD|node_1|2c|node_1|4|node|3|node|3|node|3|node|3|node|}}<BR>
{{CDD|node_1|2c|node_1|2c|node_1|4|node|3|node|3|node|3|node|3|node|}}<BR>
{{CDD|node_1|2c|node_1|2c|node_1|2c|node_1|4|node|3|node|3|node|3|node|}}<BR>
{{CDD|node_1|2c|node_1|2c|node_1|2c|node_1|2c|node_1|4|node|3|node|3|node|}}<BR>
{{CDD|node_1|2c|node_1|2c|node_1|2c|node_1|2c|node_1|2c|node_1|4|node|3|node|}}<BR>
{{CDD|node_1|2c|node_1|2c|node_1|2c|node_1|2c|node_1|2c|node_1|2c|node_1|4|node|}}<BR>
{{CDD|node_1|2c|node_1|2c|node_1|2c|node_1|2c|node_1|2c|node_1|2c|node_1|2c|node_1}}

|-
|bgcolor=#e7dcc3|7-faces||16 [[hepteract|{4,3<sup>5</sup>}]][[Image:7-cube graph.svg|35px]]
|-
|bgcolor=#e7dcc3|6-faces||112 [[hexeract|{4,3<sup>4</sup>}]][[Image:6-cube graph.svg|25px]]
|-
|bgcolor=#e7dcc3|5-faces||448 [[penteract|{4,3<sup>3</sup>}]][[Image:5-cube graph.svg|25px]]
|-
|bgcolor=#e7dcc3|4-faces||1120 [[tesseract|{4,3<sup>2</sup>}]][[Image:4-cube graph.svg|25px]]
|-
|bgcolor=#e7dcc3|Cells||1792 [[cube|{4,3}]][[Image:3-cube.png|25px]]
|-
|bgcolor=#e7dcc3|Faces||1792 [[square (geometry)|{4}]][[File:2-cube.svg|25px]]
|-
|bgcolor=#e7dcc3|Edges||1024
|-
|bgcolor=#e7dcc3|Vertices||256
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[7-simplex]] [[File:7-simplex graph.svg|25px]]
|-
|bgcolor=#e7dcc3|[[Petrie polygon]]||[[hexadecagon]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||C<sub>8</sub>, [3<sup>6</sup>,4]
|-
|bgcolor=#e7dcc3|Dual||[[8-orthoplex]] [[File:8-orthoplex.svg|25px]]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]], [[Hanner polytope]]
|}

In [[geometry]], an '''8-cube''' is an eight-[[dimension]]al [[hypercube]]. It has 256 [[Vertex (geometry)|vertices]], 1024 [[Edge (geometry)|edge]]s, 1792 square [[Face (geometry)|faces]], 1792 cubic [[Cell (mathematics)|cells]], 1120 [[tesseract]] [[4-face]]s, 448 [[5-cube]] [[5-face]]s, 112 [[6-cube]] [[6-face]]s, and 16 [[7-cube]] [[7-face]]s.

It is represented by [[Schläfli symbol]] {4,3<sup>6</sup>}, being composed of 3 [[7-cube]]s around each 6-face. It is called an '''octeract''', a [[portmanteau]] of [[tesseract]] (the ''4-cube'') and ''oct'' for eight (dimensions) in [[Greek language|Greek]]. It can also be called a regular '''hexdeca-8-tope''' or '''hexadecazetton''', being an [[8-polytope|8-dimensional polytope]] constructed from 16 regular [[Facet (geometry)|facet]]s.

It is a part of an infinite family of polytopes, called [[hypercube]]s. The [[Dual polytope|dual]] of an 8-cube can be called an [[8-orthoplex]] and is a part of the infinite family of [[cross-polytope]]s.

== Cartesian coordinates ==
[[Cartesian coordinates]] for the vertices of an 8-cube centered at the origin and edge length 2 are
: (±1,±1,±1,±1,±1,±1,±1,±1)
while the interior of the same consists of all points (x<sub>0</sub>, x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>, x<sub>4</sub>, x<sub>5</sub>, x<sub>6</sub>, x<sub>7</sub>) with -1 < x<sub>i</sub> < 1.

== As a configuration==
This [[Regular 4-polytope#As configurations|configuration matrix]] represents the 8-cube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces, 6-faces, and 7-faces. The diagonal numbers say how many of each element occur in the whole 8-cube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.<ref>Coxeter, Regular Polytopes, sec 1.8 Configurations</ref><ref>Coxeter, Complex Regular Polytopes, p.117</ref>

<math>\begin{bmatrix}\begin{matrix}
256 & 8 & 28 & 56 & 70 & 56 & 28 & 8
\\ 2 & 1024 & 7 & 21 & 35 & 35 & 21 & 7
\\ 4 & 4 &  1792 & 6 & 15 & 20 & 15 & 6
\\ 8 & 12 & 6 &  1792 & 5 & 10 & 10 & 5
\\ 16 & 32 & 24 & 8 & 1120 & 4 & 6 & 4
\\ 32 & 80 & 80 & 40 & 10 & 448 & 3 & 3
\\ 64 & 192 & 240 & 160 & 60 & 12 &  112 & 2
\\ 128 & 448 & 672 & 560 & 280 & 84 & 14 & 16
\end{matrix}\end{bmatrix}</math>

The diagonal [[f-vector]] numbers are derived through the [[Wythoff construction]], dividing the full group order of a subgroup order by removing one mirror at a time.<ref>{{KlitzingPolytopes|../incmats/octo.htm| o3o3o3o3o3o3o4x - octo}}</ref>

{| class=wikitable
!B<sub>8</sub>||{{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node_1}}||k-face|| f<sub>k</sub> || f<sub>0</sub> || f<sub>1</sub>||f<sub>2</sub>||f<sub>3</sub>||f<sub>4</sub>||f<sub>5</sub>||f<sub>6</sub>||f<sub>7</sub>||[[vertex figure|''k''-figure]]|| notes
|- align=right
||A<sub>7</sub>||{{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node|2|node_x}}||( )
!f<sub>0</sub>
|BGCOLOR="#ffe0e0" |'''256'''||8||28||56||70||56||28||8||[[7-simplex|{3,3,3,3,3,3}]]||B<sub>8</sub>/A<sub>7</sub> = 2^8*8!/8! = 256
|- align=right
||A<sub>6</sub>A<sub>1</sub>||{{CDD|node|3|node|3|node|3|node|3|node|3|node|2|node_x|2|node_1}}||{ }
!f<sub>1</sub>
||2|| BGCOLOR="#ffffe0" |'''1024'''||7||21||35||35||21||7||[[6-simplex|{3,3,3,3,3}]]||B<sub>8</sub>/A<sub>6</sub>A<sub>1</sub> = 2^8*8!/7!/2 = 1024
|- align=right
||A<sub>5</sub>B<sub>2</sub>||{{CDD|node|3|node|3|node|3|node|3|node|2|node_x|2|node|4|node_1}}||[[Square|{4}]]
!f<sub>2</sub>
||4||4|| BGCOLOR="#e0ffe0" |'''1792'''||6||15||20||15||6||[[5-simplex|{3,3,3,3}]]||B<sub>8</sub>/A<sub>5</sub>B<sub>2</sub> = 2^8*8!/6!/4/2 = 1792
|- align=right
||A<sub>4</sub>B<sub>3</sub>||{{CDD|node|3|node|3|node|3|node|2|node_x|2|node|3|node|4|node_1}}||[[Cube|{4,3}]]
!f<sub>3</sub>
||8||12||6|| BGCOLOR="#e0ffff" |'''1792'''||5||10||10||5||[[5-cell|{3,3,3}]]||B<sub>8</sub>/A<sub>4</sub>B<sub>3</sub> = 2^8*8!/5!/8/3! = 1792
|- align=right
||A<sub>3</sub>B<sub>4</sub>||{{CDD|node|3|node|3|node|2|node_x|2|node|3|node|3|node|4|node_1}}||[[Tesseract|{4,3,3}]]
!f<sub>4</sub>
||16||32||24||8|| BGCOLOR="#e0e0ff" |'''1120'''||4||6||4||[[tetrahedron|{3,3}]]||B<sub>8</sub>/A<sub>3</sub>B<sub>4</sub> = 2^8*8!/4!/2^4/4! = 1120
|- align=right
||A<sub>2</sub>B<sub>5</sub>||{{CDD|node|3|node|2|node_x|2|node|3|node|3|node|3|node|4|node_1}}||[[5-cube|{4,3,3,3}]]
!f<sub>5</sub>
||32||80||80||40||10|| BGCOLOR="#ffe0ff" |'''448'''||3||3||[[triangle|{3}]]||B<sub>8</sub>/A<sub>2</sub>B<sub>5</sub> = 2^8*8!/3!/2^5/5! = 448
|- align=right
||A<sub>1</sub>B<sub>6</sub>||{{CDD|node|2|node_x|2|node|3|node|3|node|3|node|3|node|4|node_1}}||[[6-cube|{4,3,3,3,3}]]
!f<sub>6</sub>
||64||192||240||160||60||12|| BGCOLOR="#ffe0e0" |'''112'''||2||{ }||B<sub>8</sub>/A<sub>1</sub>B<sub>6</sub> = 2^8*8!/2/2^6/6!= 112
|- align=right
||B<sub>7</sub>||{{CDD|node_x||2|node|3|node|3|node|3|node|3|node|3|node|4|node_1}}||[[7-cube|{4,3,3,3,3,3}]]
!f<sub>7</sub>
||128||448||672||560||280||84||14|| BGCOLOR="#ffffe0" |'''16'''||( )||B<sub>8</sub>/B<sub>7</sub> = 2^8*8!/2^7/7! = 16
|}

{{-}}

== Projections ==
{| class="wikitable" width=240 align=right
|- valign=top
|[[Image:8-cube column graph.svg|240px]]<BR>This 8-cube graph is an [[orthogonal projection]]. This orientation shows columns of vertices positioned a vertex-edge-vertex distance from one vertex on the left to one vertex on the right, and edges attaching adjacent columns of vertices. The number of vertices in each column represents rows in [[Pascal's triangle]], being 1:8:28:56:70:56:28:8:1.
|}

{{8-cube Coxeter plane graphs|t0|200}}

== Derived polytopes ==

Applying an ''[[Alternation (geometry)|alternation]]'' operation, deleting alternating vertices of the octeract, creates another [[uniform polytope]], called a ''[[8-demicube]]'', (part of an infinite family called [[demihypercube]]s), which has 16 [[demihepteract]]ic and 128 8-simplex facets.

== Related polytopes==
The ''8-cube'' is 8th in an infinite series of [[hypercube]]:
{{Hypercube polytopes}}

== References ==
{{reflist}}
* [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]: 
** Coxeter, ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, {{ISBN|0-486-61480-8}}, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
** '''Kaleidoscopes: Selected Writings of H.S.M. Coxeter''', edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, {{ISBN|978-0-471-01003-6}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
*** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10]
*** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591]
*** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
* [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. (1966)
* {{KlitzingPolytopes|polyzetta.htm|8D uniform polytopes (polyzetta)| o3o3o3o3o3o3o4x - octo}}

== External links ==
* {{MathWorld|title=Hypercube|urlname=Hypercube}}
*{{GlossaryForHyperspace | anchor=Measure | title=Measure polytope }}
* [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary: hypercube] Garrett Jones

{{Polytopes}}

[[Category:8-polytopes]]