View source for Cross-polytope

{{short description|Regular polytope dual to the hypercube in any number of dimensions}}
{| class=wikitable width=100 align=right
|+ Cross-polytopes of dimension 2 to 5
|-
|style="text-align: center;"|[[Image:2-orthoplex.svg|120px|A 2-dimensional cross-polytope]]
|style="text-align: center;"|[[Image:Octahedron.png|120px|A 3-dimensional cross-polytope]]
|-
|style="text-align: center;"|2 dimensions<BR>[[square (geometry)|square]]
|style="text-align: center;"|3 dimensions<BR>[[octahedron]]
|-
|style="text-align: center;"|[[Image:Schlegel wireframe 16-cell.png|120px|A 4-dimensional cross-polytope]]
|style="text-align: center;"|[[File:5-cube t4.svg|120px|A 5-dimensional cross-polytope]]
|-
|style="text-align: center;"|4 dimensions<BR>[[16-cell]]
|style="text-align: center;"|5 dimensions<BR>[[5-orthoplex]]
|}

In [[geometry]], a '''cross-polytope''',{{sfn|Coxeter|1973|loc=§7.21. illustration Fig 7-2<small>B</small>|pp=121-122}} '''hyperoctahedron''', '''orthoplex''',<ref>{{cite book |first1=J. H. |last1=Conway |first2=N. J. A. |last2=Sloane |chapter=The Cell Structures of Certain Lattices |title=Miscellanea Mathematica |editor-last=Hilton |editor-first=P. |editor2-last=Hirzebruch |editor2-first=F. |editor3-last=Remmert |editor3-first=R. |publisher=Springer |location=Berlin |pages=89–90 |year=1991 |doi=10.1007/978-3-642-76709-8_5 |isbn=978-3-642-76711-1 }}</ref> '''staurotope''',<ref>{{cite book |last=McMullen |first=Peter |author-link=Peter McMullen |date=2020 |title=Geometric Regular Polytopes |url= |location= |publisher=Cambridge University Press |page=92 |isbn=978-1-108-48958-4}}</ref> or '''cocube''' is a [[regular polytope|regular]], [[convex polytope]] that exists in ''n''-[[dimensions|dimensional Euclidean space]]. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular [[octahedron]], and a 4-dimensional cross-polytope is a [[16-cell]]. Its facets are [[simplex]]es of the previous dimension, while the cross-polytope's [[vertex figure]] is another cross-polytope from the previous dimension.

The vertices of a cross-polytope can be chosen as the unit vectors pointing along each co-ordinate axis – i.e. all the permutations of {{nowrap|(±1, 0, 0, ..., 0)}}. The cross-polytope is the [[convex hull]] of its vertices. 
The ''n''-dimensional cross-polytope can also be defined as the closed [[unit ball]] (or, according to some authors, its boundary) in the [[L1-norm|ℓ<sub>1</sub>-norm]] on '''R'''<sup>''n''</sup>:
:<math>\{x\in\mathbb R^n : \|x\|_1 \le 1\}.</math>

In 1 dimension the cross-polytope is simply the [[line segment]] [&minus;1, +1], in 2 dimensions it is a [[Square (geometry)|square]] (or diamond) with vertices {(±1, 0), (0, ±1)}. In 3 dimensions it is an [[octahedron]]&mdash;one of the five convex regular [[polyhedron|polyhedra]] known as the [[Platonic solid]]s. This can be generalised to higher dimensions with an ''n''-orthoplex being constructed as a [[bipyramid#Other dimensions|bipyramid]] with an (''n''−1)-orthoplex base.

The cross-polytope is the [[dual polytope]] of the [[hypercube]]. The 1-[[Skeleton (topology)|skeleton]] of an ''n''-dimensional cross-polytope is the [[Turán graph]] ''T''(2''n'', ''n'') (also known as a ''cocktail party graph'' <ref>{{mathworld|id=CocktailPartyGraph|title=Cocktail Party Graph}}</ref>).

== 4 dimensions ==

The 4-dimensional cross-polytope also goes by the name '''hexadecachoron''' or '''[[16-cell]]'''. It is one of the six [[convex regular 4-polytope]]s. These [[4-polytope]]s were first described by the Swiss mathematician [[Ludwig Schläfli]] in the mid-19th century.

The vertices of the 4-dimensional hypercube, or [[tesseract]], can be divided into two sets of eight, the convex hull of each set forming a cross-polytope. Moreover, the polytope known as the [[24-cell]] can be constructed by symmetrically arranging three cross-polytopes.<ref>{{cite book|first1=Ingemar |last1=Bengtsson |first2=Karol |last2=Życzkowski |author-link2=Karol Życzkowski |title=Geometry of Quantum States: An Introduction to Quantum Entanglement |title-link=Geometry of Quantum States |edition=2nd |publisher=Cambridge University Press |year=2017 |isbn=978-1-107-02625-4 |page=162}}</ref>

== Higher dimensions ==

The cross-polytope family is one of three [[regular polytope]] families, labeled by [[Coxeter]] as ''β<sub>n</sub>'', the other two being the [[hypercube]] family, labeled as ''γ<sub>n</sub>'', and the [[simplex]] family, labeled as ''α<sub>n</sub>''. A fourth family, the [[hypercubic honeycomb|infinite tessellations of hypercubes]], he labeled as ''δ<sub>n</sub>''.{{Sfn|Coxeter|1973|pp=120-124|loc=§7.2}}

The ''n''-dimensional cross-polytope has 2''n'' vertices, and 2<sup>''n''</sup> facets ((''n''&nbsp;&minus;&nbsp;1)-dimensional components) all of which are (''n''&nbsp;&minus;&nbsp;1)-[[Simplex|simplices]]. The [[vertex figure]]s are all (''n''&nbsp;&minus;&nbsp;1)-cross-polytopes. The [[Schläfli symbol]] of the cross-polytope is {3,3,...,3,4}. 

The [[Dihedral_angle#Dihedral_angles_in_polyhedra|dihedral angle]] of the ''n''-dimensional cross-polytope is <math>\delta_n = \arccos\left(\frac{2-n}{n}\right)</math>. This gives: &delta;<sub>2</sub> = arccos(0/2) = 90°, &delta;<sub>3</sub> = arccos(−1/3) = 109.47°, &delta;<sub>4</sub> = arccos(−2/4) = 120°, &delta;<sub>5</sub> = arccos(−3/5) = 126.87°, ... &delta;<sub>&infin;</sub> = arccos(−1) = 180°.

The hypervolume of the ''n''-dimensional cross-polytope is 
:<math>\frac{2^n}{n!}.</math>

For each pair of non-opposite vertices, there is an edge joining them. More generally, each set of ''k''&nbsp;+&nbsp;1 orthogonal vertices corresponds to a distinct ''k''-dimensional component which contains them. The number of ''k''-dimensional components (vertices, edges, faces, ..., facets) in an ''n''-dimensional cross-polytope is thus given by (see [[binomial coefficient]]):
:<math>2^{k+1}{n \choose {k+1}}</math>{{sfn|Coxeter|1973|p=121|loc=§7.2.2.}}

The extended [[f-vector]] for an ''n''-orthoplex can be computed by ('''1''',2)<sup>''n''</sup>, like the coefficients of [[Polynomial#Multiplication|polynomial products]]. For example a 16-cell is ('''1''',2)<sup>4</sup> = ('''1''',4,4)<sup>2</sup> = ('''1''',8,24,32,16).

There are many possible [[orthographic projection]]s that can show the cross-polytopes as 2-dimensional graphs. [[Petrie polygon]] projections map the points into a regular 2''n''-gon or lower order regular polygons. A second projection takes the 2(''n''−1)-gon petrie polygon of the lower dimension, seen as a [[bipyramid]], projected down the axis, with 2 vertices mapped into the center.

{| class="wikitable"
|+ 
Cross-polytope elements 
|- 
! [[polytope|''n'']]
!β<sub>''n''</sub><BR>''k''<sub>11</sub>
! Name(s)<BR>[[Graph (discrete mathematics)|Graph]]
!Graph<BR>2''n''-gon
![[Schläfli symbol|Schläfli]]
![[Coxeter-Dynkin diagram|Coxeter-Dynkin<BR>diagrams]]
! [[Vertex (geometry)|Vertices]]
! [[Edge (geometry)|Edges]]
! [[Face (geometry)|Faces]]
! [[Cell (geometry)|Cells]]
! 4-faces
! 5-faces
! 6-faces
! 7-faces
! 8-faces
! 9-faces
! 10-faces
|- 
| [[0-polytope|0]]
!β<sub>0</sub>
|[[Vertex (geometry)|Point]]<BR>0-orthoplex
|.
|( )
|{{CDD|node}}<BR>
|align=right| 1
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
|-
| [[1-polytope|1]]
!β<sub>1</sub>
|[[Edge (geometry)|Line segment]]<BR>1-orthoplex
|[[Image:Cross graph 1.svg|50px]]
|{ }
|{{CDD|node_1}}<BR>{{CDD|node_f1}}
|align=right| 2
|align=right| 1
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
|- 
| [[2-polytope|2]]
!β<sub>2</sub><BR> &minus;1<sub>11</sub>
|[[Square (geometry)|Square]]<BR>2-orthoplex<BR>'''Bicross'''
|[[Image:Cross graph 2.png|50px]]
|{4}<BR>2{ } = { }+{ }
|{{CDD|node_1|4|node}}<BR>{{CDD|node_f1|2|node_f1}}
|align=right| 4
|align=right| 4
|align=right| 1
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
|- 
| [[3-polytope|3]]
!β<sub>3</sub><BR>0<sub>11</sub>
| [[Octahedron]]<BR>3-orthoplex<BR>'''Tricross'''
|[[Image:3-orthoplex.svg|50px]]
|{3,4}<BR>{3<sup>1,1</sup>}<BR>3{ }
|{{CDD|node_1|3|node|4|node}}<BR>{{CDD|node_1|split1|nodes}}<BR>{{CDD|node_f1|2|node_f1|2|node_f1}}
|align=right| 6
|align=right| 12
|align=right| 8
|align=right| 1
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
|- 
| [[4-polytope|4]]
!β<sub>4</sub><BR>1<sub>11</sub>
|[[16-cell]]<BR>4-orthoplex<BR>'''Tetracross'''
|[[Image:4-orthoplex.svg|50px]]
|{3,3,4}<BR>{3,3<sup>1,1</sup>}<BR>4{ }
|{{CDD|node_1|3|node|3|node|4|node}}<BR>{{CDD|node_1|3|node|split1|nodes}}<BR>{{CDD|node_f1|2|node_f1|2|node_f1|2|node_f1}}
|align=right| 8
|align=right| 24
|align=right| 32
|align=right| 16
|align=right| 1
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
|- 
| [[5-polytope|5]]
!β<sub>5</sub><BR>2<sub>11</sub>
|[[5-orthoplex]]<BR>'''Pentacross'''
|[[Image:5-orthoplex.svg|50px]]
|{3<sup>3</sup>,4}<BR>{3,3,3<sup>1,1</sup>}<BR>5{ }
|{{CDD|node_1|3|node|3|node|3|node|4|node}}<BR>{{CDD|node_1|3|node|3|node|split1|nodes}}<BR>{{CDD|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1}}
|align=right| 10
|align=right| 40
|align=right| 80
|align=right| 80
|align=right| 32
|align=right| 1
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
|- 
| [[6-polytope|6]]
!β<sub>6</sub><BR>3<sub>11</sub>
| [[6-orthoplex]]<BR>'''Hexacross'''
|[[Image:6-orthoplex.svg|50px]]
|{3<sup>4</sup>,4}<BR>{3<sup>3</sup>,3<sup>1,1</sup>}<BR>6{ }
|{{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}}<BR>{{CDD|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1}}
|align=right| 12
|align=right| 60
|align=right| 160
|align=right| 240
|align=right| 192
|align=right| 64
|align=right| 1
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
|- 
| [[7-polytope|7]]
!β<sub>7</sub><BR>4<sub>11</sub>
|[[7-orthoplex]]<BR>'''Heptacross'''
|[[Image:7-orthoplex.svg|50px]]
|{3<sup>5</sup>,4}<BR>{3<sup>4</sup>,3<sup>1,1</sup>}<BR>7{ }
|{{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}}<BR>{{CDD|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1}}
|align=right| 14
|align=right| 84
|align=right| 280
|align=right| 560
|align=right| 672
|align=right| 448
|align=right| 128
|align=right| 1
| &nbsp;
| &nbsp;
| &nbsp;
|- 
| [[8-polytope|8]]
!β<sub>8</sub><BR>5<sub>11</sub>
|[[8-orthoplex]]<BR>'''Octacross'''
|[[Image:8-orthoplex.svg|50px]]
|{3<sup>6</sup>,4}<BR>{3<sup>5</sup>,3<sup>1,1</sup>}<BR>8{ }
|{{CDD|node_1|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|split1|nodes}}<BR>{{CDD|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1}}
|align=right| 16
|align=right| 112
|align=right| 448
|align=right| 1120
|align=right| 1792
|align=right| 1792
|align=right| 1024
|align=right| 256
|align=right| 1
| &nbsp;
| &nbsp;
|- 
| [[9-polytope|9]]
!β<sub>9</sub><BR>6<sub>11</sub>
|[[9-orthoplex]]<BR>'''Enneacross'''
|[[Image:9-orthoplex.svg|50px]]
|{3<sup>7</sup>,4}<BR>{3<sup>6</sup>,3<sup>1,1</sup>}<BR>9{ }
|{{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}}<BR>{{CDD|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1}}
|align=right| 18
|align=right| 144
|align=right| 672
|align=right| 2016
|align=right| 4032
|align=right| 5376
|align=right| 4608
|align=right| 2304
|align=right| 512
|align=right| 1
| &nbsp;
|- 
! [[10-polytope|10]]
! β<sub>10</sub><BR>7<sub>11</sub>
|[[10-orthoplex]]<BR>'''Decacross'''
|[[Image:10-orthoplex.svg|50px]]
|{3<sup>8</sup>,4}<BR>{3<sup>7</sup>,3<sup>1,1</sup>}<BR>10{ }
|{{CDD|node_1|3|node|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|3|node|split1|nodes}}<BR>{{CDD|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1}}
|align=right|20
|align=right|180
|align=right|960
|align=right|3360
|align=right|8064
|align=right|13440
|align=right|15360
|align=right|11520
|align=right|5120
|align=right|1024
|align=right|1
|- 
| colspan=17 style="text-align:center" |...
|- 
! '''''n'''''
! β<sub>''n''</sub><BR>''k''<sub>11</sub>
|''n''-orthoplex<BR>''n''-cross
|
|{3<sup>''n''&nbsp;−&nbsp;2</sup>,4}<BR>{3<sup>''n''&nbsp;−&nbsp;3</sup>,3<sup>1,1</sup>}<BR>''n''{}
|{{CDD|node_1|3|node|3|node}}...{{CDD|3|node|3|node|4|node}}<BR>{{CDD|node_1|3|node|3}}...{{CDD|node|3|node|split1|nodes}}<BR>{{CDD|node_f1|2|node_f1|2|node_f1|2|node_f1|2}}...{{CDD|2|node_f1}}
| colspan=11 style="text-align:center" | 2''n'' '''0-faces''', ... <math>2^{k+1}{n\choose k+1}</math> '''''k''-faces''' ..., 2<sup>''n''</sup> '''(''n''−1)-faces'''
|}

The vertices of an axis-aligned cross polytope are all at equal distance from each other in the [[taxicab geometry|Manhattan distance]] ([[Lp space|L<sup>1</sup> norm]]). [[Kusner's conjecture]] states that this set of 2''d'' points is the largest possible [[equidistant set]] for this distance.<ref>{{citation
 | last = Guy | first = Richard K.
 | issue = 3
 | journal = American Mathematical Monthly
 | pages = 196–200
 | title = An olla-podrida of open problems, often oddly posed
 | jstor = 2975549
 | volume = 90
 | year = 1983
 | doi = 10.2307/2975549 }}.</ref>

== Generalized orthoplex ==
Regular [[Complex polytope#Regular_complex_polytopes|complex polytopes]] can be defined in [[Complex number|complex]] [[Hilbert space]] called ''generalized orthoplexes'' (or cross polytopes), β{{supsub|''p''|''n''}} = <sub>2</sub>{3}<sub>2</sub>{3}...<sub>2</sub>{4}<sub>''p''</sub>, or {{CDD|node_1|3|node|3}}..{{CDD|3|node|4|pnode}}. Real solutions exist with ''p'' = 2, i.e. β{{supsub|2|''n''}} = β<sub>''n''</sub> = <sub>2</sub>{3}<sub>2</sub>{3}...<sub>2</sub>{4}<sub>2</sub> = {3,3,..,4}. For ''p'' > 2, they exist in <math>\mathbb{\Complex}^n</math>. A ''p''-generalized ''n''-orthoplex has ''pn'' vertices. ''Generalized orthoplexes'' have regular [[simplex]]es (real) as [[Facet (geometry)|facets]].<ref>Coxeter, Regular Complex Polytopes, p. 108</ref> Generalized orthoplexes make [[complete multipartite graph]]s, β{{supsub|''p''|2}} make K<sub>''p'',''p''</sub> for [[complete bipartite graph]], β{{supsub|''p''|3}} make K<sub>''p'',''p'',''p''</sub> for complete tripartite graphs. β{{supsub|''p''|''n''}} creates K<sub>''p''<sup>''n''</sup></sub>. An [[orthogonal projection]] can be defined that maps all the vertices equally-spaced on a circle, with all pairs of vertices connected, except multiples of ''n''. The [[regular polygon]] perimeter in these orthogonal projections is called a [[petrie polygon]].

{| class=wikitable
|+ Generalized orthoplexes
! || ''p'' = 2 || ||''p'' = 3 ||''p'' = 4||''p'' = 5||''p'' = 6||''p'' = 7||''p'' = 8
|- align=center
!<math>\mathbb{R}^2</math>
|[[File:Complex_bipartite_graph_square.svg|100px]]<BR><sub>2</sub>{4}<sub>2</sub> = [[square|{4}]] = {{CDD|node_1|4|node}}<BR>K<sub>2,2</sub>
!<math>\mathbb{\Complex}^2</math>
|[[File:Complex polygon 2-4-3-bipartite graph.png|100px]]<BR>[[3-3_duoprism#Related_complex_polygon|<sub>2</sub>{4}<sub>3</sub>]] = {{CDD|node_1|4|3node}}<BR>K<sub>3,3</sub>
|[[File:Complex polygon 2-4-4 bipartite graph.png|100px]]<BR>[[16-cell#Related_complex_polygons|<sub>2</sub>{4}<sub>4</sub>]] = {{CDD|node_1|4|4node}}<BR>K<sub>4,4</sub>
|[[File:Complex_polygon_2-4-5-bipartite_graph.png|100px]]<BR>[[5-5_duoprism#Related_complex_polygon|<sub>2</sub>{4}<sub>5</sub>]] = {{CDD|node_1|4|5node}}<BR>K<sub>5,5</sub>
|[[File:6-generalized-2-orthoplex.svg|100px]]<BR>[[6-6_duoprism#Related_complex_polygon|<sub>2</sub>{4}<sub>6</sub>]] =  {{CDD|node_1|4|6node}}<BR>K<sub>6,6</sub>
|[[File:7-generalized-2-orthoplex.svg|100px]]<BR><sub>2</sub>{4}<sub>7</sub> = {{CDD|node_1|4|7node}}<BR>K<sub>7,7</sub>
|[[File:8-generalized-2-orthoplex.svg|100px]]<BR>[[8-8_duoprism#Related_complex_polygon|<sub>2</sub>{4}<sub>8</sub>]] = {{CDD|node_1|4|8node}}<BR>K<sub>8,8</sub>
|- align=center
!<math>\mathbb{R}^3</math>
|[[File:Complex tripartite graph octahedron.svg|100px]]<BR><sub>2</sub>{3}<sub>2</sub>{4}<sub>2</sub> = [[octahedron|{3,4}]] = {{CDD|node_1|3|node|4|node}}<BR>K<sub>2,2,2</sub>
!<math>\mathbb{\Complex}^3</math>
|[[File:3-generalized-3-orthoplex-tripartite.svg|100px]]<BR><sub>2</sub>{3}<sub>2</sub>{4}<sub>3</sub> = {{CDD|node_1|3|node|4|3node}}<BR>K<sub>3,3,3</sub>
|[[File:4-generalized-3-orthoplex.svg|100px]]<BR><sub>2</sub>{3}<sub>2</sub>{4}<sub>4</sub> = {{CDD|node_1|3|node|4|4node}}<BR>K<sub>4,4,4</sub>
|[[File:5-generalized-3-orthoplex.svg|100px]]<BR><sub>2</sub>{3}<sub>2</sub>{4}<sub>5</sub> = {{CDD|node_1|3|node|4|5node}}<BR>K<sub>5,5,5</sub>
|[[File:6-generalized-3-orthoplex.svg|100px]]<BR><sub>2</sub>{3}<sub>2</sub>{4}<sub>6</sub> = {{CDD|node_1|3|node|4|6node}}<BR>K<sub>6,6,6</sub>
|[[File:7-generalized-3-orthoplex.svg|100px]]<BR><sub>2</sub>{3}<sub>2</sub>{4}<sub>7</sub> = {{CDD|node_1|3|node|4|7node}}<BR>K<sub>7,7,7</sub>
|[[File:8-generalized-3-orthoplex.svg|100px]]<BR><sub>2</sub>{3}<sub>2</sub>{4}<sub>8</sub> = {{CDD|node_1|3|node|4|8node}}<BR>K<sub>8,8,8</sub>
|- align=center
!<math>\mathbb{R}^4</math>
|[[File:Complex_multipartite_graph_16-cell.svg|100px]]<BR><sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub><BR>[[16-cell|{3,3,4}]] = {{CDD|node_1|3|node|3|node|4|node}}<BR>K<sub>2,2,2,2</sub>
!<math>\mathbb{\Complex}^4</math>
|[[File:3-generalized-4-orthoplex.svg|100px]]<BR><sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{4}<sub>3</sub><BR>{{CDD|node_1|3|node|3|node|4|3node}}<BR>K<sub>3,3,3,3</sub>
|[[File:4-generalized-4-orthoplex.svg|100px]]<BR><sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{4}<sub>4</sub><BR>{{CDD|node_1|3|node|3|node|4|4node}}<BR>K<sub>4,4,4,4</sub>
|[[File:5-generalized-4-orthoplex.svg|100px]]<BR><sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{4}<sub>5</sub><BR>{{CDD|node_1|3|node|3|node|4|5node}}<BR>K<sub>5,5,5,5</sub>
|[[File:6-generalized-4-orthoplex.svg|100px]]<BR><sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{4}<sub>6</sub><BR>{{CDD|node_1|3|node|3|node|4|6node}}<BR>K<sub>6,6,6,6</sub>
|[[File:7-generalized-4-orthoplex.svg|100px]]<BR><sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{4}<sub>7</sub><BR>{{CDD|node_1|3|node|3|node|4|7node}}<BR>K<sub>7,7,7,7</sub>
|[[File:8-generalized-4-orthoplex.svg|100px]]<BR><sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{4}<sub>8</sub><BR>{{CDD|node_1|3|node|3|node|4|8node}}<BR>K<sub>8,8,8,8</sub>
|- align=center
!<math>\mathbb{R}^5</math>
|[[File:2-generalized-5-orthoplex.svg|100px]]<BR><sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{4}<sub>2</sub><BR>[[5-orthoplex|{3,3,3,4}]] = {{CDD|node_1|3|node|3|node|3|node|4|node}}<BR>K<sub>2,2,2,2,2</sub>
!<math>\mathbb{\Complex}^5</math>
|[[File:3-generalized-5-orthoplex.svg|100px]]<BR><sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{4}<sub>3</sub><BR>{{CDD|node_1|3|node|3|node|3|node|4|3node}}<BR>K<sub>3,3,3,3,3</sub>
|[[File:4-generalized-5-orthoplex.svg|100px]]<BR><sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{4}<sub>4</sub><BR>{{CDD|node_1|3|node|3|node|3|node|4|4node}}<BR>K<sub>4,4,4,4,4</sub>
|[[File:5-generalized-5-orthoplex.svg|100px]]<BR><sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{4}<sub>5</sub><BR>{{CDD|node_1|3|node|3|node|3|node|4|5node}}<BR>K<sub>5,5,5,5,5</sub>
|[[File:6-generalized-5-orthoplex.svg|100px]]<BR><sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{4}<sub>6</sub><BR>{{CDD|node_1|3|node|3|node|3|node|4|6node}}<BR>K<sub>6,6,6,6,6</sub>
|[[File:7-generalized-5-orthoplex.svg|100px]]<BR><sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{4}<sub>7</sub><BR>{{CDD|node_1|3|node|3|node|3|node|4|7node}}<BR>K<sub>7,7,7,7,7</sub>
|[[File:8-generalized-5-orthoplex.svg|100px]]<BR><sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{4}<sub>8</sub><BR>{{CDD|node_1|3|node|3|node|3|node|4|8node}}<BR>K<sub>8,8,8,8,8</sub>
|- align=center
!<math>\mathbb{R}^6</math>
|[[File:2-generalized-6-orthoplex.svg|100px]]<BR><sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{4}<sub>2</sub><BR>[[6-orthoplex|{3,3,3,3,4}]] = {{CDD|node_1|3|node|3|node|3|node|3|node|4|node}}<BR>K<sub>2,2,2,2,2,2</sub>
!<math>\mathbb{\Complex}^6</math>
|[[File:3-generalized-6-orthoplex.svg|100px]]<BR><sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{4}<sub>3</sub><BR>{{CDD|node_1|3|node|3|node|3|node|3|node|4|3node}}<BR>K<sub>3,3,3,3,3,3</sub>
|[[File:4-generalized-6-orthoplex.svg|100px]]<BR><sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{4}<sub>4</sub><BR>{{CDD|node_1|3|node|3|node|3|node|3|node|4|4node}}<BR>K<sub>4,4,4,4,4,4</sub>
|[[File:5-generalized-6-orthoplex.svg|100px]]<BR><sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{4}<sub>5</sub><BR>{{CDD|node_1|3|node|3|node|3|node|3|node|4|5node}}<BR>K<sub>5,5,5,5,5,5</sub>
|[[File:6-generalized-6-orthoplex.svg|100px]]<BR><sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{4}<sub>6</sub><BR>{{CDD|node_1|3|node|3|node|3|node|3|node|4|6node}}<BR>K<sub>6,6,6,6,6,6</sub>
|[[File:7-generalized-6-orthoplex.svg|100px]]<BR><sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{4}<sub>7</sub><BR>{{CDD|node_1|3|node|3|node|3|node|3|node|4|7node}}<BR>K<sub>7,7,7,7,7,7</sub>
|[[File:8-generalized-6-orthoplex.svg|100px]]<BR><sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{3}<sub>2</sub>{4}<sub>8</sub><BR>{{CDD|node_1|3|node|3|node|3|node|3|node|4|8node}}<BR>K<sub>8,8,8,8,8,8</sub>
|}

== Related polytope families ==

Cross-polytopes can be combined with their dual cubes to form compound polytopes:

*In two dimensions, we obtain the [[octagram]]mic star figure {{mset|{{sfrac|8|2}}}},
*In three dimensions we obtain the [[compound of cube and octahedron]],
*In four dimensions we obtain the compound of tesseract and 16-cell.

== See also ==

* [[List of regular polytopes]]
* [[Hyperoctahedral group]], the symmetry group of the cross-polytope

==Citations==
{{reflist}}

==References==
* {{Cite book | last=Coxeter | first=H.S.M. | author-link=Harold Scott MacDonald Coxeter | year=1973 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=Regular Polytopes (book) }}
** pp.&nbsp;121-122, §7.21. see illustration Fig 7.2<small>B</small>
** p.&nbsp;296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)

== External links ==
{{commons category|Cross-polytope graphs}}
* {{mathworld | urlname = CrossPolytope | title = Cross polytope}}

{{Dimension topics}}
{{Polytopes}}

{{DEFAULTSORT:Cross-Polytope}}
[[Category:Regular polytopes]]
[[Category:Multi-dimensional geometry]]