View source for 5-orthoplex

{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Regular 5-orthoplex<BR>(pentacross)
|-
|bgcolor=#ffffff align=center colspan=2|[[Image:5-cube t4.svg|281px]]<BR>[[Orthogonal projection]]<BR>inside [[Petrie polygon]]
|-
|bgcolor=#e7dcc3|Type||Regular [[5-polytope]]
|-
|bgcolor=#e7dcc3|Family||[[orthoplex]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| {3,3,3,4}<BR>{3,3,3<sup>1,1</sup>}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node_1|3|node|3|node|3|node|4|node}}<BR>{{CDD|node_1|3|node|3|node|split1|nodes}}
|-
|bgcolor=#e7dcc3|4-faces||32 [[5-cell|{3<sup>3</sup>}]][[Image:Cross graph 4.png|31px]]
|-
|bgcolor=#e7dcc3|Cells||80 [[tetrahedron|{3,3}]][[Image:Cross graph 3.png|31px]]
|-
|bgcolor=#e7dcc3|Faces||80 [[triangle|{3}]][[Image:Cross graph 2.png|31px]]
|-
|bgcolor=#e7dcc3|Edges||40
|-
|bgcolor=#e7dcc3|Vertices||10
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Pentacross verf.png|60px]]<BR>[[16-cell]]
|-
|bgcolor=#e7dcc3|[[Petrie polygon]]||[[decagon]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||BC<sub>5</sub>, [3,3,3,4]<BR>D<sub>5</sub>, [3<sup>2,1,1</sup>]
|-
|bgcolor=#e7dcc3|Dual||[[5-cube]]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]], [[Hanner polytope]]
|}
In [[Five-dimensional space|five-dimensional]] [[geometry]], a '''5-orthoplex''', or 5-[[cross polytope]], is a five-dimensional polytope with 10 [[Vertex (geometry)|vertices]], 40 [[Edge (geometry)|edge]]s,  80 triangle [[Face (geometry)|faces]], 80 tetrahedron [[Cell (mathematics)|cells]], 32 [[5-cell]] [[4-face]]s.

It has two constructed forms, the first being regular with [[Schläfli symbol]] {3<sup>3</sup>,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,3<sup>1,1</sup>} or [[Coxeter symbol]] '''2<sub>11</sub>'''.

It is a part of an infinite family of polytopes, called [[cross-polytope]]s or ''orthoplexes''. The [[dual polytope]] is the 5-[[hypercube]] or [[5-cube]].

== Alternate names==
* '''pentacross''', derived from combining the family name ''cross polytope'' with ''pente'' for five (dimensions) in [[Greek language|Greek]].
* '''Triacontaditeron''' (or ''triacontakaiditeron'') - as a 32-[[Facet (geometry)|facetted]] [[5-polytope]] (polyteron).

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

== Cartesian coordinates ==
[[Cartesian coordinates]] for the vertices of a 5-orthoplex, centered at the origin are
: (±1,0,0,0,0), (0,±1,0,0,0), (0,0,±1,0,0), (0,0,0,±1,0), (0,0,0,0,±1)

== Construction ==

There are three [[Coxeter group]]s associated with the 5-orthoplex, one [[regular polytope|regular]], [[Dual polytope|dual]] of the [[penteract]] with the C<sub>5</sub> or [4,3,3,3] [[Coxeter group]], and a lower symmetry with two copies of ''5-cell'' facets, alternating, with the D<sub>5</sub> or [3<sup>2,1,1</sup>] Coxeter group, and the final one as a dual 5-[[orthotope]], called a '''5-fusil''' which can have a variety of subsymmetries.

{| class=wikitable
!Name
![[Coxeter diagram]]
![[Schläfli symbol]]
![[Coxeter notation|Symmetry]]
!Order
![[Vertex figure]](s)
|- align=center
!regular 5-orthoplex
|{{CDD|node_1|3|node|3|node|3|node|4|node}}
|{3,3,3,4}
|[3,3,3,4]||3840
|{{CDD|node_1|3|node|3|node|4|node}}
|- align=center
![[Quasiregular polytope|Quasiregular]] 5-orthoplex
|{{CDD|node_1|3|node|3|node|split1|nodes}}
|{3,3,3<sup>1,1</sup>}
|[3,3,3<sup>1,1</sup>]||1920
|{{CDD|node_1|3|node|split1|nodes}}
|- align=center
!rowspan=8|[[rhombic fusil|5-fusil]]
|- align=center
|{{CDD|node_f1|4|node|3|node|3|node|3|node}}
||{3,3,3,4}||[4,3,3,3]||3840||{{CDD|node_f1|4|node|3|node|3|node}}
|- align=center
|{{CDD|node_f1|4|node|3|node|3|node|2|node_f1}}
||{3,3,4}+{}||[4,3,3,2]||768||{{CDD|node_f1|4|node|3|node|2|node_f1}}
|- align=center
|{{CDD|node_f1|4|node|3|node|2|node_f1|4|node}}
||{3,4}+{4}||[4,3,2,4]||384||{{CDD|node_f1|4|node|3|node|2|node_f1}}<BR>{{CDD|node_f1|4|node|2|node_f1|4|node}}
|- align=center
|{{CDD|node_f1|4|node|3|node|2|node_f1|2|node_f1}}
||{3,4}+2{}||[4,3,2,2]||192||{{CDD|node_f1|4|node|3|node|2|node_f1}}<BR>{{CDD|node_f1|4|node|2|node_f1|2|node_f1}}
|- align=center
|{{CDD|node_f1|4|node|2|node_f1|4|node|2|node_f1}}
||2{4}+{}||[4,2,4,2]||128||{{CDD|node_f1|4|node|2|node_f1|4|node}}
|- align=center
|{{CDD|node_f1|4|node|2|node_f1|2|node_f1|2|node_f1}}
||{4}+3{}||[4,2,2,2]||64||{{CDD|node_f1|4|node|2|node_f1|2|node_f1}}<BR>{{CDD|node_f1|2|node_f1|2|node_f1|2|node_f1}}
|- align=center
|{{CDD|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1}}
|5{}
|[2,2,2,2]||32
|{{CDD|node_f1|2|node_f1|2|node_f1|2|node_f1}}
|}

== Other images ==

{{5-cube Coxeter plane graphs|t4|150}}

{| class=wikitable width=300px]
|align=center|[[Image:Pentacross wire.png|220px]]<BR>The [[perspective projection]] (3D to 2D) of a [[stereographic projection]] (4D to 3D) of the [[Schlegel diagram]] (5D to 4D) of the 5-orthoplex. 10 sets of 4 edges form 10 circles in the 4D Schlegel diagram: two of these circles are straight lines in the stereographic projection because they contain the center of projection.
|}

== Related polytopes and honeycombs ==

{{2 k1 polytopes}}

This polytope is one of 31 [[uniform 5-polytope]]s generated from the B<sub>5</sub> [[Coxeter plane]], including the regular [[5-cube]] and 5-orthoplex.

{{Penteract family}}

== References ==
<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|polytera.htm|5D uniform polytopes (polytera)|x3o3o3o4o - tac}}

== 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:5-polytopes]]