View source for 9-orthoplex

{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Regular 9-orthoplex
Ennecross
|-
|bgcolor=#ffffff align=center colspan=2|[[File:9-orthoplex.svg|280px]]<BR>Orthogonal projection<BR>inside [[Petrie polygon]]
|-
|bgcolor=#e7dcc3|Type||Regular 9-polytope
|-
|bgcolor=#e7dcc3|Family||orthoplex
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| {3<sup>7</sup>,4}<BR>{3<sup>6</sup>,3<sup>1,1</sup>}
|-
|bgcolor=#e7dcc3|Coxeter-Dynkin diagrams||{{CDD|node_1|3|node|3|node|3|node||3|node|3|node|3|node|3|node|4|node}}<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|split1|nodes}}
|-
|bgcolor=#e7dcc3|8-faces||512 {3<sup>7</sup>}[[Image:8-simplex_t0.svg|25px]]
|-
|bgcolor=#e7dcc3|7-faces||2304 [[7-simplex|{3<sup>6</sup>}]][[Image:7-simplex_t0.svg|25px]]
|-
|bgcolor=#e7dcc3|6-faces||4608 [[6-simplex|{3<sup>5</sup>}]][[Image:6-simplex_t0.svg|25px]]
|-
|bgcolor=#e7dcc3|5-faces||5376 [[5-simplex|{3<sup>4</sup>}]][[Image:5-simplex_t0.svg|25px]]
|-
|bgcolor=#e7dcc3|4-faces||4032 [[5-cell|{3<sup>3</sup>}]][[Image:4-simplex_t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Cells||2016 [[Tetrahedron|{3,3}]][[Image:3-simplex_t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Faces||672 [[Triangle|{3}]][[Image:2-simplex_t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Edges||144
|-
|bgcolor=#e7dcc3|Vertices||18
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||Octacross
|-
|bgcolor=#e7dcc3|[[Petrie polygon]]||[[Octadecagon]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||C<sub>9</sub>, [3<sup>7</sup>,4]<BR>D<sub>9</sub>, [3<sup>6,1,1</sup>]
|-
|bgcolor=#e7dcc3|Dual||[[9-cube|9-cube]]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]], [[Hanner polytope]]
|}

In [[Geometry|geometry]], a '''9-orthoplex''' or 9-cross polytope, is a regular 9-polytope with 18 [[Vertex (geometry)|vertices]], 144 [[Edge (geometry)|edge]]s,  672 triangle [[Face (geometry)|faces]], 2016 tetrahedron cells, 4032 [[5-cell|5-cell]]s ''4-faces'', 5376 [[5-simplex|5-simplex]] ''5-faces'', 4608 [[6-simplex|6-simplex]] ''6-faces'', 2304 [[7-simplex|7-simplex]] ''7-faces'', and 512 8-simplex ''8-faces''.

It has two constructed forms, the first being regular with [[Schläfli symbol]] {3<sup>7</sup>,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3<sup>6</sup>,3<sup>1,1</sup>} or Coxeter symbol '''6<sub>11</sub>'''.

It is one of an infinite family of polytopes, called [[Cross-polytope|cross-polytope]]s or ''orthoplexes''. The dual polytope is the 9-[[Hypercube|hypercube]] or enneract.

== Alternate names==
* '''Enneacross''', derived from combining the family name ''cross polytope'' with ''ennea'' for nine (dimensions) in [[Social:Greek language|Greek]]
* '''Pentacosidodecayotton''' as a 512-[[Facet (geometry)|facetted]] 9-polytope (polyyotton)

== Construction ==

There are two [[Coxeter group]]s associated with the 9-orthoplex, one [[Regular polytope|regular]], dual of the enneract with the C<sub>9</sub> or [4,3<sup>7</sup>] symmetry group, and a lower symmetry with two copies of 8-simplex facets, alternating, with the D<sub>9</sub> or [3<sup>6,1,1</sup>] symmetry group.

== Cartesian coordinates ==
Cartesian coordinates for the vertices of a 9-orthoplex, centered at the origin, are
: (±1,0,0,0,0,0,0,0,0), (0,±1,0,0,0,0,0,0,0), (0,0,±1,0,0,0,0,0,0), (0,0,0,±1,0,0,0,0,0), (0,0,0,0,±1,0,0,0,0), (0,0,0,0,0,±1,0,0,0), (0,0,0,0,0,0,±1,0,0), (0,0,0,0,0,0,0,±1,0), (0,0,0,0,0,0,0,0,±1)

Every [[Vertex (geometry)|vertex]] pair is connected by an [[Edge (geometry)|edge]], except opposites.

== Images ==


== References==
* [[Biography:Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]: 
** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973 
** '''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]
* [[Biography:Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. 
* {{KlitzingPolytopes|polyyotta.htm|9D uniform polytopes (polyyotta)|x3o3o3o3o3o3o3o4o - vee}}

== External links ==
*{{GlossaryForHyperspace | anchor=Cross | title=Cross polytope }}
* [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions]
* [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]

{{Polytopes}}

[[Category:9-polytopes]]

{{Sourceattribution|9-orthoplex}}