View source for 7-orthoplex

{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Regular 7-orthoplex<BR>(heptacross)
|-
|bgcolor=#ffffff align=center colspan=2|[[Image:7-orthoplex.svg|280px]]<BR>Orthogonal projection<BR>inside [[Petrie polygon]]
|-
|bgcolor=#e7dcc3|Type||Regular 7-polytope
|-
|bgcolor=#e7dcc3|Family||orthoplex
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| {3<sup>5</sup>,4}<BR>{3,3,3,3,3<sup>1,1</sup>}
|-
|bgcolor=#e7dcc3|Coxeter-Dynkin diagrams||{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|4|node}}<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|split1|nodes}}
|-
|bgcolor=#e7dcc3|6-faces||128 [[6-simplex|{3<sup>5</sup>}]][[Image:6-simplex_t0.svg|25px]]
|-
|bgcolor=#e7dcc3|5-faces||448 [[5-simplex|{3<sup>4</sup>}]][[Image:5-simplex_t0.svg|25px]]
|-
|bgcolor=#e7dcc3|4-faces||672 [[5-cell|{3<sup>3</sup>}]][[Image:4-simplex_t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Cells||560 [[Tetrahedron|{3,3}]][[Image:3-simplex_t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Faces||280 [[Triangle|{3}]][[Image:2-simplex_t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Edges||84
|-
|bgcolor=#e7dcc3|Vertices||14
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[6-orthoplex|6-orthoplex]]
|-
|bgcolor=#e7dcc3|[[Petrie polygon]]||[[Tetradecagon|tetradecagon]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||C<sub>7</sub>, [3,3,3,3,3,4]<BR>D<sub>7</sub>, [3<sup>4,1,1</sup>]
|-
|bgcolor=#e7dcc3|Dual||[[7-cube|7-cube]]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]], [[Hanner polytope]]
|}
In [[Geometry|geometry]], a '''7-orthoplex''', or 7-cross polytope, is a regular 7-polytope with 14 [[Vertex (geometry)|vertices]], 84 [[Edge (geometry)|edge]]s,  280 triangle [[Face (geometry)|faces]], 560 tetrahedron cells, 672 [[5-cell|5-cell]]s ''4-faces'', 448  ''5-faces'', and 128 ''6-faces''.

It has two constructed forms, the first being regular with [[Schläfli symbol]] {3<sup>5</sup>,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,3,3,3<sup>1,1</sup>} or Coxeter symbol '''4<sub>11</sub>'''.

It is a part of an infinite family of polytopes, called [[Cross-polytope|cross-polytope]]s or ''orthoplexes''. The dual polytope is the 7-[[Hypercube|hypercube]], or hepteract.

== Alternate names==
* '''Heptacross''', derived from combining the family name ''cross polytope'' with ''hept'' for seven (dimensions) in [[Social:Greek language|Greek]]. 
* '''Hecatonicosoctaexon''' as a 128-[[Facet (geometry)|facetted]] 7-polytope (polyexon).

== As a configuration==
This [[Regular 4-polytope#As configurations|configuration matrix]] represents the 7-orthoplex. 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-orthoplex. 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}
14 & 12 & 60 & 160 & 240 & 192 & 64 
\\ 2 & 84 & 10 & 40 & 80 & 80 & 32 
\\ 3 & 3 & 280 & 8 & 24 & 32 & 16 
\\ 4 & 6 & 4 & 560 & 6 & 12 & 8 
\\ 5 & 10 & 10 & 5 & 672 & 4 & 4 
\\ 6 & 15 & 20 & 15 & 6 & 448 & 2 
\\ 7 & 21 & 35 & 35 & 21 & 7 & 128 
\end{matrix}\end{bmatrix}</math>

== Images ==

{{7-cube Coxeter plane graphs|t6|150}}

== Construction ==

There are two [[Coxeter group]]s associated with the 7-orthoplex, one [[Regular polytope|regular]], dual of the hepteract with the C<sub>7</sub> or [4,3,3,3,3,3] symmetry group, and a half symmetry with two copies of 6-simplex facets, alternating, with the D<sub>7</sub> or [3<sup>4,1,1</sup>] symmetry group. A lowest symmetry construction is based on a dual of a 7-orthotope, called a '''7-fusil'''.

{| class=wikitable
!Name
!Coxeter diagram
![[Schläfli symbol]]
!Symmetry
!Order
![[Vertex figure]]
|- align=center
!regular 7-orthoplex
|{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|4|node}}
|{3,3,3,3,3,4}
|[3,3,3,3,3,4]||645120 
|{{CDD|node_1|3|node|3|node|3|node|3|node|4|node}}
|- align=center
!Quasiregular 7-orthoplex
|{{CDD|node_1|3|node|3|node|3|node|split1|nodes}}
|{3,3,3,3,3<sup>1,1</sup>}
|[3,3,3,3,3<sup>1,1</sup>]||322560
|{{CDD|node_1|3|node|3|node|3|node|split1|nodes}}
|- align=center
!7-fusil
|{{CDD|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1}}
|7{}
|[2<sup>6</sup>]||128
|{{CDD|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1}}
|}

== Cartesian coordinates ==
Cartesian coordinates for the vertices of a 7-orthoplex, centered at the origin are
: (±1,0,0,0,0,0,0), (0,±1,0,0,0,0,0), (0,0,±1,0,0,0,0), (0,0,0,±1,0,0,0), (0,0,0,0,±1,0,0), (0,0,0,0,0,±1,0), (0,0,0,0,0,0,±1)

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

== See also ==
* Rectified 7-orthoplex
* Truncated 7-orthoplex

== References ==
<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. (1966)
* {{KlitzingPolytopes|polyexa.htm|7D uniform polytopes (polyexa)|x3o3o3o3o3o4o - zee}}

== 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:7-polytopes]]

{{Sourceattribution|7-orthoplex}}