View source for 7-cube

{{Short description|7-dimensional hypercube}}
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|7-cube<BR>Hepteract
|-
|bgcolor=#ffffff align=center colspan=2|[[File:7-cube t0.svg|280px]]<BR>[[Orthogonal projection]]<BR>inside [[Petrie polygon]]<BR>The central orange vertex is doubled
|-
|bgcolor=#e7dcc3|Type||Regular [[7-polytope]]
|-
|bgcolor=#e7dcc3|Family||[[hypercube]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| {4,3<sup>5</sup>}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node_1|4|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|}}<BR>
{{CDD|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|4|node|3|node|}}<BR>
{{CDD|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}}
|-
|bgcolor=#e7dcc3|6-faces||14 [[hexeract|{4,3<sup>4</sup>}]] [[File:6-cube graph.svg|25px]]
|-
|bgcolor=#e7dcc3|5-faces||84 [[penteract|{4,3<sup>3</sup>}]] [[File:5-cube graph.svg|25px]]
|-
|bgcolor=#e7dcc3|4-faces||280 [[tesseract|{4,3,3}]] [[File:4-cube graph.svg|25px]]
|-
|bgcolor=#e7dcc3|Cells||560 [[cube|{4,3}]] [[File:3-cube graph.svg|25px]]
|-
|bgcolor=#e7dcc3|Faces||672 [[square (geometry)|{4}]] [[File:2-cube.svg|25px]]
|-
|bgcolor=#e7dcc3|Edges||448
|-
|bgcolor=#e7dcc3|Vertices||128
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[6-simplex]] [[File:6-simplex graph.svg|25px]]
|-
|bgcolor=#e7dcc3|[[Petrie polygon]]||[[tetradecagon]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||C<sub>7</sub>, [3<sup>5</sup>,4]
|-
|bgcolor=#e7dcc3|Dual||[[7-orthoplex]]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]], [[Hanner polytope]]
|}

In [[geometry]], a '''7-cube''' is a [[seven-dimensional space|seven-dimensional]] [[hypercube]] with 128 [[Vertex (geometry)|vertices]], 448 [[Edge (geometry)|edge]]s,  672 square [[Face (geometry)|faces]], 560 cubic [[Cell (mathematics)|cells]], 280 [[tesseract]] [[4-face]]s, 84 [[penteract]] [[5-face]]s, and 14 [[hexeract]] [[6-face]]s.

It can be named by its [[Schläfli symbol]] {4,3<sup>5</sup>}, being composed of 3 [[6-cube]]s around each 5-face.  It can be called a '''hepteract''',  a [[portmanteau]] of [[tesseract]] (the ''4-cube'') and ''hepta'' for seven (dimensions) in [[Greek language|Greek]]. It can also be called a regular '''tetradeca-7-tope''' or '''tetradecaexon''', being a [[7-polytope|7 dimensional polytope]] constructed from 14 regular [[Facet (geometry)|facet]]s.

== Related polytopes==
The ''7-cube'' is 7th in a series of [[hypercube]]:
{{Hypercube polytopes}}

The [[Dual polytope|dual]] of a 7-cube is called a [[7-orthoplex]], and is a part of the infinite family of [[cross-polytope]]s.

Applying an ''[[Alternation (geometry)|alternation]]'' operation, deleting alternating vertices of the hepteract, creates another [[uniform polytope]], called a [[demihepteract]], (part of an infinite family called [[demihypercube]]s), which has 14 [[demihexeract]]ic and 64 [[6-simplex]] 6-faces.

== As a configuration==
This [[Regular 4-polytope#As configurations|configuration matrix]] represents the 7-cube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces and 6-faces. The diagonal numbers say how many of each element occur in the whole 7-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}
128 & 7 & 21 & 35 & 35 & 21 & 7 
\\ 2 & 448 & 6 & 15 & 20 & 15 & 6 
\\ 4 & 4 & 672 & 5 & 10 & 10 & 5 
\\ 8 & 12 & 6 & 560 & 4 & 6 & 4 
\\ 16 & 32 & 24 & 8 & 280 & 3 & 3 
\\ 32 & 80 & 80 & 40 & 10 & 84 & 2 
\\ 64 & 192 & 240 & 160 & 60 & 12 & 14 
\end{matrix}\end{bmatrix}</math>

== Cartesian coordinates ==
[[Cartesian coordinates]] for the vertices of a hepteract centered at the origin and edge length 2 are
: (±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>) with -1 < x<sub>i</sub> < 1.
{{-}}

== Projections ==
{| class="wikitable" width=240 align=right
|- valign=top
|[[Image:7-cube column graph.svg|240px]]<BR>This hypercube 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 1:7:21:35:35:21:7:1.
|}


{{7-cube Coxeter plane graphs|t0|150}}

== References ==
<references />
* [[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.&nbsp;296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973, p.&nbsp;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|polyexa.htm|7D uniform polytopes (polyexa)|o3o3o3o3o3o4x - hept}}

== External links ==
* {{MathWorld|title=Hypercube|urlname=Hypercube}}
* {{MathWorld|title=Hypercube graph|urlname=HypercubeGraph}}
*{{GlossaryForHyperspace | anchor=Measure | title=Measure polytope }}
* [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary: hypercube] Garrett Jones
* [http://www.4d-screen.de/related-space/4d-space/7d-cube.htm Rotation of 7D-Cube] www.4d-screen.de

{{Polytopes}}

[[Category:7-polytopes]]
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