View source for 6-orthoplex

{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|6-orthoplex<BR>Hexacross
|-
|bgcolor=#ffffff align=center colspan=2|[[Image:6-cube t5.svg|280px]]<BR>[[Orthogonal projection]]<BR>inside [[Petrie polygon]]
|-
|bgcolor=#e7dcc3|Type||Regular [[6-polytope]]
|-
|bgcolor=#e7dcc3|Family||[[orthoplex]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]s|| {3,3,3,3,4}<BR>{3,3,3,3<sup>1,1</sup>}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node_1|3|node|3|node|3|node|3|node|4|node}}<br>{{CDD|node_1|3|node|3|node|3|node|split1|nodes}} = {{CDD|nodes_10r|3ab|nodes|split5c|nodes}}
|-
|bgcolor=#e7dcc3|5-faces||64 [[5-simplex|{3<sup>4</sup>}]][[Image:5-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|4-faces||192 [[5-cell|{3<sup>3</sup>}]][[Image:4-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Cells||240 [[tetrahedron|{3,3}]][[Image:3-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Faces||160 [[triangle|{3}]][[Image:2-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Edges||60
|-
|bgcolor=#e7dcc3|Vertices||12
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[5-orthoplex]]
|-
|bgcolor=#e7dcc3|[[Petrie polygon]]||[[dodecagon]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||B<sub>6</sub>, [4,3<sup>4</sup>]<BR>D<sub>6</sub>, [3<sup>3,1,1</sup>]
|-
|bgcolor=#e7dcc3|Dual||[[6-cube]]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]], [[Hanner polytope]]
|}
In [[geometry]], a '''6-orthoplex''', or 6-[[cross polytope]], is a regular [[6-polytope]] with 12 [[Vertex (geometry)|vertices]], 60 [[Edge (geometry)|edge]]s, 160 triangle [[Face (geometry)|faces]], 240 tetrahedron [[Cell (mathematics)|cells]], 192 [[5-cell]] ''4-faces'', and 64 ''5-faces''.

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

It is a part of an infinite family of polytopes, called [[cross-polytope]]s or ''orthoplexes''. The [[dual polytope]] is the 6-[[hypercube]], or [[hexeract]].

== Alternate names==
*'''Hexacross''', derived from combining the family name [[cross polytope]] with ''hex'' for six (dimensions) in [[Greek language|Greek]]. 
* '''Hexacontitetrapeton''' as a 64-[[Facet (geometry)|facetted]] [[6-polytope]].

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

== Construction ==

There are three [[Coxeter group]]s associated with the 6-orthoplex, one [[regular polytope|regular]], [[Dual polytope|dual]] of the [[hexeract]] with the C<sub>6</sub> or [4,3,3,3,3] [[Coxeter group]], and a half symmetry with two copies of 5-simplex facets, alternating, with the D<sub>6</sub> or [3<sup>3,1,1</sup>] Coxeter group. A lowest symmetry construction is based on a dual of a 6-[[orthotope]], called a '''6-fusil'''.

{| class="wikitable sortable"
!Name
![[Coxeter diagram|Coxeter]]
![[Schläfli symbol|Schläfli]]
![[Coxeter notation|Symmetry]]
!Order
|- align=center
![[Regular polytope|Regular]] 6-orthoplex
|{{CDD|node_1|3|node|3|node|3|node|3|node|4|node}}
|{3,3,3,3,4}
|[4,3,3,3,3]||46080
|- align=center
![[Quasiregular polytope|Quasiregular]] 6-orthoplex
|{{CDD|node_1|3|node|3|node|3|node|split1|nodes}}
|{3,3,3,3<sup>1,1</sup>}
|[3,3,3,3<sup>1,1</sup>]||23040
|- align=center
!rowspan=10|[[rhombic fusil|6-fusil]]
|{{CDD|node_f1|4|node|3|node|3|node|3|node|2|node_f1}}
||{3,3,3,4}+{}||[4,3,3,3,3]||7680
|- align=center
|{{CDD|node_f1|4|node|3|node|3|node|2|node_f1|4|node}}
||{3,3,4}+{4}||[4,3,3,2,4]||3072
|- align=center
|{{CDD|node_f1|4|node|3|node|2|node_f1|4|node|3|node}}
||2{3,4}||[4,3,2,4,3]||2304
|- align=center
|{{CDD|node_f1|4|node|3|node|3|node|2|node_f1|2|node_f1}}
||{3,3,4}+2{}||[4,3,3,2,2]||1536
|- align=center
|{{CDD|node_f1|4|node|3|node|2|node_f1|4|node|2|node_f1}}
||{3,4}+{4}+{}||[4,3,2,4,2]||768
|- align=center
|{{CDD|node_f1|4|node|2|node_f1|4|node|2|node_f1|4|node}}
||3{4}||[4,2,4,2,4]||512
|- align=center
|{{CDD|node_f1|4|node|3|node|2|node_f1|2|node_f1|2|node_f1}}
||{3,4}+3{}||[4,3,2,2,2]||384
|- align=center
|{{CDD|node_f1|4|node|2|node_f1|4|node|2|node_f1|2|node_f1}}
||2{4}+2{}||[4,2,4,2,2]||256
|- align=center
|{{CDD|node_f1|4|node|2|node_f1|2|node_f1|2|node_f1|2|node_f1}}
||{4}+4{}||[4,2,2,2,2]||128
|- align=center
|{{CDD|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1}}
|6{}
|[2,2,2,2,2]||64
|}

== Cartesian coordinates ==
[[Cartesian coordinates]] for the vertices of a 6-orthoplex, centered at the origin are
: (±1,0,0,0,0,0), (0,±1,0,0,0,0), (0,0,±1,0,0,0), (0,0,0,±1,0,0), (0,0,0,0,±1,0), (0,0,0,0,0,±1)

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

==Images==
{{6-cube Coxeter plane graphs|t5|150}}

== Related polytopes==
The 6-orthoplex can be projected down to 3-dimensions into the vertices of a [[regular icosahedron]].<ref>''[[Quasicrystals and Geometry]]'', Marjorie Senechal, 1996, Cambridge University Press, p64. 2.7.1 ''The I<sub>6</sub> crystal''</ref>
{| class=wikitable width=640
!colspan=2|2D
!colspan=2|3D
|- align=center valign=top
|[[File:Icosahedron H3 projection.svg|160px]]<BR>Icosahedron<BR>{3,5} = {{CDD|node_1|3|node|5|node}}<BR>H<sub>3</sub> Coxeter plane
|[[File:6-cube t5 B5.svg|155px]]<BR>6-orthoplex<BR>{3,3,3,3<sup>1,1</sup>} = {{CDD|node_1|3|node|3|node|3|node|split1|nodes}}<BR>D<sub>6</sub> Coxeter plane
|[[File:Icosahedron_frame.png|155px]]<BR>Icosahedron
|[[File:Hexacross.png|160px]]<BR>6-orthoplex
|-
|colspan=4|This construction can be geometrically seen as the 12 vertices of the 6-orthoplex projected to 3 dimensions as the vertices of a [[regular icosahedron]]. This represents a [[Coxeter–Dynkin diagram#Geometric folding|geometric folding]] of the D<sub>6</sub> to H<sub>3</sub> [[Coxeter group]]s: [[File:Geometric folding Coxeter graph D6 H3.png]]: {{CDD|nodes_10r|3ab|nodes|split5a|nodes}} to {{CDD|node_1|3|node|5|node}}. On the left, seen by these 2D [[Coxeter plane]] orthogonal projections, the two overlapping central vertices define the third axis in this mapping. Every pair of vertices of the 6-orthoplex are connected, except opposite ones: 30 edges are shared with the icosahedron, while 30 more edges from the 6-orthoplex project to the interior of the icosahedron.
|}

It is in a dimensional series of uniform polytopes and honeycombs, expressed by [[Coxeter]] as 3<sub>k1</sub> series. (A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral [[hosohedron]].)
{{3_k1_polytopes}}

This polytope is one of 63 [[uniform 6-polytope]]s generated from the B<sub>6</sub> [[Coxeter plane]], including the regular [[6-cube]] or 6-orthoplex.

{{Hexeract family}}

== References==
* [[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]
* [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. 1966
* {{KlitzingPolytopes|polypeta.htm|6D uniform polytopes (polypeta)| x3o3o3o3o4o - gee}}

;Specific
<references />

== 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:6-polytopes]]