View source for 9-cube

{{Short description|9-dimensional hypercube}}
{{No footnotes|date=September 2017}}
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|9-cube<BR>Enneract
|-
|bgcolor=#ffffff align=center colspan=2|[[File:9-cube.svg|280px]]<BR>[[Orthogonal projection]]<BR>inside [[Petrie polygon]]<BR>Orange vertices are doubled, yellow have 4, and the green center has 8
|-
|bgcolor=#e7dcc3|Type||Regular [[9-polytope]]
|-
|bgcolor=#e7dcc3|Family||[[hypercube]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| {4,3<sup>7</sup>}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
|-
|bgcolor=#e7dcc3|8-faces||18 [[8-cube|{4,3<sup>6</sup>}]][[File:8-cube.svg|25px]]
|-
|bgcolor=#e7dcc3|7-faces||144 [[7-cube|{4,3<sup>5</sup>}]][[Image:7-cube graph.svg|25px]]
|-
|bgcolor=#e7dcc3|6-faces||672 [[6-cube|{4,3<sup>4</sup>}]][[Image:6-cube graph.svg|25px]]
|-
|bgcolor=#e7dcc3|5-faces||2016 [[5-cube|{4,3<sup>3</sup>}]][[Image:5-cube graph.svg|25px]]
|-
|bgcolor=#e7dcc3|4-faces||4032 [[tesseract|{4,3,3}]]
|-
|bgcolor=#e7dcc3|Cells||5376 [[cube|{4,3}]][[Image:3-cube graph.svg|25px]]
|-
|bgcolor=#e7dcc3|Faces||4608 [[square (geometry)|{4}]][[Image:2-cube.svg|25px]]
|-
|bgcolor=#e7dcc3|Edges||2304
|-
|bgcolor=#e7dcc3|Vertices||512
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[8-simplex]] [[Image:8-simplex graph.svg|25px]]
|-
|bgcolor=#e7dcc3|[[Petrie polygon]]||[[octadecagon]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||C<sub>9</sub>, [3<sup>7</sup>,4]
|-
|bgcolor=#e7dcc3|Dual||[[9-orthoplex]] [[Image:9-orthoplex.svg|25px]]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]], [[Hanner polytope]]
|}
In [[geometry]], a '''9-cube''' is a nine-[[dimension]]al [[hypercube]] with 512 [[Vertex (geometry)|vertices]], 2304 [[Edge (geometry)|edge]]s,  4608 [[square]] [[Face (geometry)|faces]], 5376 [[cube|cubic]] [[Cell (mathematics)|cells]], 4032 [[tesseract]] [[4-face]]s, 2016 [[5-cube]] [[5-face]]s, 672 [[6-cube]] [[6-face]]s, 144 [[7-cube]] [[7-face]]s, and 18 [[8-cube]] [[8-face]]s.

It can be named by its [[Schläfli symbol]] {4,3<sup>7</sup>}, being composed of three [[8-cube]]s around each 7-face. It is also called an '''enneract''', a [[portmanteau]] of [[tesseract]] (the ''4-cube'') and ''enne'' for nine (dimensions) in [[Greek language|Greek]]. It can also be called a regular '''octadeca-9-tope''' or '''octadecayotton''', as a [[9-polytope|nine-dimensional polytope]] constructed with 18 regular [[Facet (geometry)|facet]]s.

It is a part of an infinite family of polytopes, called hypercubes. The [[Dual polytope|dual]] of a 9-cube can be called a [[9-orthoplex]], and is a part of the infinite family of [[cross-polytope]]s.

== Cartesian coordinates ==
[[Cartesian coordinates]] for the vertices of a 9-cube centered at the origin and edge length 2 are
: (±1,±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>, ''x''<sub>8</sub>) with −1&nbsp;<&nbsp;''x''<sub>''i''</sub>&nbsp;<&nbsp;1.

== Projections ==

{| class="wikitable" width=320
|- valign=top
|[[Image:9-cube column graph.svg|320px]]<BR>This 9-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 1:9:36:84:126:126:84:36:9:1.
|}

== Images ==

{{B9 Coxeter plane graphs|t0|200}}

== Derived polytopes ==

Applying an ''[[Alternation (geometry)|alternation]]'' operation, deleting alternating vertices of the ''9-cube'', creates another [[uniform polytope]], called a ''[[9-demicube]]'', (part of an infinite family called [[demihypercube]]s), which has 18 [[8-demicube]] and 256 8-simplex facets.

== Notes ==
{{Reflist}}

== 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|polyyotta.htm|9D uniform polytopes (polyyotta)|o3o3o3o3o3o3o3o4x - enne}}

== 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:9-polytopes]]