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In [[geometry]], a '''cross-polytope''',<ref>{{citation | last = [[E. L. Elte|Elte]] | first = E. L. | title = The Semiregular Polytopes of the Hyperspaces | publisher = University of Groningen | location = Groningen | year = 1912}} Chapter IV, five dimensional semiregular polytope [http://www.amazon.com/Semiregular-Polytopes-Hyperspaces-Emanuel-Lodewijk/dp/141817968X]</ref> '''orthoplex''',<ref>[[John Horton Conway|Conway]] calls it an n-'''orthoplex''' for ''[[orthant]] complex''.
</ref> '''hyperoctahedron''', or '''cocube''' is a [[regular polytope|regular]], convex [[polytope]] that exists in any number of dimensions. The vertices of a cross-polytope are all the permutations of (±1, 0, 0, …, 0). The cross-polytope is the [[convex hull]] of its vertices. 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 ''n''-dimensional cross-polytope can also be defined as the closed [[unit ball]] (or, according to some authors, its boundary) in the [[L1-norm|ℓ1-norm]] on '''R'''''n'':
:<math>\{x\in\mathbb R^n : \|x\|_1 \le 1\}.</math>

In 1 dimension the cross-polytope is simply the [[line segment]] [−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]]—one of the five convex regular [[polyhedron|polyhedra]] known as the [[Platonic solid]]s. Higher-dimensional cross-polytopes are generalizations of these.
{| style="margin-left: auto; margin-right: auto"
|style="text-align: center;"|[[Image:2-orthoplex.svg|120px|A 2-dimensional cross-polytope]]
|width="50px"|
|style="text-align: center;"|[[Image:Octahedron.png|120px|A 3-dimensional cross-polytope]]
|width="50px"|
|style="text-align: center;"|[[Image:Schlegel wireframe 16-cell.png|120px|A 4-dimensional cross-polytope]]
|-
|style="text-align: center;"|2 dimensions [[square (geometry)|square]]
|
|style="text-align: center;"|3 dimensions [[octahedron]]
|
|style="text-align: center;"|4 dimensions [[16-cell]]
|}

The cross-polytope is the [[dual polytope]] of the [[hypercube]]. The 1-[[Skeleton (topology)|skeleton]] of a ''n''-dimensional cross-polytope is a [[Turán graph]] ''T''(2''n'',''n'').

== 4 dimensions ==

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

== Higher dimensions ==

The '''cross polytope''' family is one of three [[regular polytope]] families, labeled by [[Coxeter]] as ''βn'', the other two being the [[hypercube]] family, labeled as ''γn'', and the [[simplex|simplices]], labeled as ''αn''. A fourth family, the [[hypercubic honeycomb|infinite tessellations of hypercubes]], he labeled as ''δn''.

The ''n''-dimensional cross-polytope has 2''n'' vertices, and 2''n'' facets (''n''−1 dimensional components) all of which are ''n''−1 [[Simplex|simplices]]. The [[vertex figure]]s are all ''n'' − 1 cross-polytopes. The [[Schläfli symbol]] of the cross-polytope is {3,3,…,3,4}. The [[dihedral angle]] of the ''n''-dimensional cross-polytope is
:<math>\arccos\left(\frac{2-n}{n}\right)</math>.

The number of ''k''-dimensional components (vertices, edges, faces, …, facets) in an ''n''-dimensional cross-polytope is given by (see [[binomial coefficient]]):
:<math>2^{k+1}{n \choose {k+1}}</math>

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

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 ''2n''-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]]
!βn k11
! Name(s) [[Graph (mathematics)|Graph]]
!Graph 2n-gon
!Graph 2(n-1)-gon
![[Schläfli symbol|Schläfli]]
![[Coxeter-Dynkin diagram|Coxeter-Dynkin 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
|-
| [[1-polytope|1]]
!β1
|[[Edge (geometry)|Line segment]] 1-orthoplex
|[[Image:Cross graph 1.svg|50px]]
|
|{}
|{{CDD|node_1}} {{CDD|node_f1}}
|align=right| 2
|  
|  
|  
|  
|  
|  
|  
|  
|  
|-
| [[2-polytope|2]]
!β2 −111
|[[square (geometry)|square]] 2-orthoplex '''Bicross'''
|[[Image:Cross graph 2.png|50px]]
|[[Image:2-orthoplex B1.svg|50px]]
|{4} {}+{}
|{{CDD|node_1|4|node}} {{CDD|node_f1|2|node_f1}}
|align=right| 4
|align=right| 4
|  
|  
|  
|  
|  
|  
|  
|  
|-
| [[3-polytope|3]]
!β3 011
| [[octahedron]] 3-orthoplex '''Tricross'''
|[[Image:3-orthoplex.svg|50px]]
|[[Image:3-orthoplex B2.svg|50px]]
|{3,4} {30,1,1} {}+{}+{}
|{{CDD|node_1|3|node|4|node}} {{CDD|node_1|split1|nodes}} {{CDD|node_f1|2|node_f1|2|node_f1}}
|align=right| 6
|align=right| 12
|align=right| 8
|  
|  
|  
|  
|  
|  
|  
|-
| [[4-polytope|4]]
!β4 111
|[[16-cell]] 4-orthoplex '''Tetracross'''
|[[Image:4-orthoplex.svg|50px]]
|[[Image:4-orthoplex B3.svg|50px]]
|{3,3,4} {31,1,1} 4{}
|{{CDD|node_1|3|node|3|node|4|node}} {{CDD|node_1|3|node|split1|nodes}} {{CDD|node_f1|2|node_f1|2|node_f1|2|node_f1}}
|align=right| 8
|align=right| 24
|align=right| 32
|align=right| 16
|  
|  
|  
|  
|  
|  
|-
| [[5-polytope|5]]
!β5 211
|[[5-orthoplex]] '''Pentacross'''
|[[Image:5-orthoplex.svg|50px]]
|[[Image:5-orthoplex B4.svg|50px]]
|{33,4} {32,1,1} 5{}
|{{CDD|node_1|3|node|3|node|3|node|4|node}} {{CDD|node_1|3|node|3|node|split1|nodes}} {{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
|  
|  
|  
|  
|  
|-
| [[6-polytope|6]]
!β6 311
| [[6-orthoplex]] '''Hexacross'''
|[[Image:6-orthoplex.svg|50px]]
|[[Image:6-orthoplex B5.svg|50px]]
|{34,4} {33,1,1} 6{}
|{{CDD|node_1|3|node|3|node|3|node|3|node|4|node}} {{CDD|node_1|3|node|3|node|3|node|split1|nodes}} {{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
|  
|  
|  
|  
|-
| [[7-polytope|7]]
!β7 411
|[[7-orthoplex]] '''Heptacross'''
|[[Image:7-orthoplex.svg|50px]]
|[[Image:7-orthoplex B6.svg|50px]]
|{35,4} {34,1,1} 7{}
|{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|4|node}} {{CDD|node_1|3|node|3|node|3|node|3|node|split1|nodes}} {{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
|  
|  
|  
|-
| [[8-polytope|8]]
!β8 511
|[[8-orthoplex]] '''Octacross'''
|[[Image:8-orthoplex.svg|50px]]
|[[Image:8-orthoplex B7.svg|50px]]
|{36,4} {35,1,1} 8{}
|{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|4|node}} {{CDD|node_1|3|node|3|node|3|node|3|node|3|node|split1|nodes}} {{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
|  
|  
|-
| [[9-polytope|9]]
!β9 611
|[[9-orthoplex]] '''Enneacross'''
|[[Image:9-orthoplex.svg|50px]]
|[[Image:9-orthoplex B8.svg|50px]]
|{37,4} {36,1,1} 9{}
|{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node}} {{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|split1|nodes}} {{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
|  
|-
! [[10-polytope|10]]
! β10 711
|[[10-orthoplex]] '''Decacross'''
|[[Image:10-orthoplex.svg|50px]]
|[[Image:10-orthoplex B9.svg|50px]]
|{38,4} {37,1,1} 10{}
|{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node}} {{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|split1|nodes}} {{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
|-
| colspan=17 style="text-align:center" |...
|-
! '''n'''
! βn ''k''11
|''n''-orthoplex ''n''-cross
|
|
|{3''n'' − 2,4} {3''n'' − 3,1,1} n{}
|{{CDD|node_1|3|node|3|node}}...{{CDD|3|node|3|node|4|node}} {{CDD|node_1|3|node|3}}...{{CDD|node|3|node|split1|nodes}} {{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''n'' '''(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|L1 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}}.</ref>

== See also ==

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

==Notes==
{{reflist}}

==References==
*{{cite book | first = H. S. M. | last = Coxeter | authorlink = H. S. M. Coxeter | year = 1973 | title = [[Regular Polytopes (book)|Regular Polytopes]] | edition = 3rd ed. | publisher = Dover Publications | location = New York | pages = 121–122 | isbn = 0-486-61480-8}} p. 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}}
* [http://www.geocities.com/shapirojon34/tesseract/TesseractApplet.html Polytope Viewer]{{dead link|date=November 2010|bot=AnomieBOT}} (Click <polytopes...> to select cross polytope.)
*{{GlossaryForHyperspace | anchor=Cross | title=Cross polytope }}

{{Dimension topics}}
{{Polytopes}}

{{DEFAULTSORT:Cross-Polytope}}
[[Category:Polytopes]]
[[Category:Multi-dimensional geometry]]
[[Category:Four-dimensional geometry]]

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