cross-polytope

{{short description|Regular polytope dual to the hypercube in any number of dimensions}}

class="wikitable floatright"

|+ Cross-polytopes of dimension 2 to 5

style="text-align: center;"|Image:2-orthoplex.svg

|style="text-align: center;"|Image:Octahedron.png

style="text-align: center;"|2 dimensions
square

|style="text-align: center;"|3 dimensions
octahedron

style="text-align: center;"|Image:Schlegel wireframe 16-cell.png

|style="text-align: center;"|File:5-cube t4.svg

style="text-align: center;"|4 dimensions
16-cell

|style="text-align: center;"|5 dimensions
5-orthoplex

In geometry, a cross-polytope,{{sfn|Coxeter|1973|loc=§7.21. illustration Fig 7-2B|pp=121-122}} hyperoctahedron, orthoplex,{{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 }} staurotope,{{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}} or cocube is a regular, convex polytope that exists in n-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 simplexes 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 on Rⁿ:

: $\{x\in\mathbb R^n : \|x\|_1 \le 1\}.$

In 1 dimension the cross-polytope is simply the line segment [−1, +1], in 2 dimensions it is a square (or diamond) with vertices {(±1, 0), (0, ±1)}. In 3 dimensions it is an octahedron—one of the five convex regular polyhedra known as the Platonic solids. This can be generalised to higher dimensions with an n-orthoplex being constructed as a bipyramid with an (n−1)-orthoplex base.

The cross-polytope is the dual polytope of the hypercube. The 1-skeleton of an n-dimensional cross-polytope is the Turán graph T(2n, n) (also known as a cocktail party graph {{mathworld|id=CocktailPartyGraph|title=Cocktail Party Graph}}).

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-polytopes. These 4-polytopes 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.{{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}}

Higher dimensions

The cross-polytope family is one of three regular polytope families, labeled by Coxeter as β_n, the other two being the hypercube family, labeled as γ_n, and the simplex family, labeled as α_n. A fourth family, the infinite tessellations of hypercubes, he labeled as δ_n.{{Sfn|Coxeter|1973|pp=120-124|loc=§7.2}}

The n-dimensional cross-polytope has 2n vertices, and 2ⁿ facets ((n − 1)-dimensional components) all of which are (n − 1)-simplices. The vertex figures 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 $\delta_n = \arccos\left(\frac{2-n}{n}\right)$ . This gives: δ₂ = arccos(0/2) = 90°, δ₃ = arccos(−1/3) = 109.47°, δ₄ = arccos(−2/4) = 120°, δ₅ = arccos(−3/5) = 126.87°, ... δ_∞ = arccos(−1) = 180°.

The hypervolume of the n-dimensional cross-polytope is

: $\frac{2^n}{n!}.$

For each pair of non-opposite vertices, there is an edge joining them. More generally, each set of k + 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):

: $2^{k+1}{n \choose {k+1}}$ {{sfn|Coxeter|1973|p=121|loc=§7.2.2.}}

The extended f-vector for an n-orthoplex can be computed by (1,2)ⁿ, like the coefficients of polynomial products. For example a 16-cell is (1,2)⁴ = (1,4,4)² = (1,8,24,32,16).

There are many possible orthographic projections 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
n !β_n k₁₁ ! Name(s) Graph !Graph 2n-gon !Schläfli !Coxeter-Dynkin diagrams ! Vertices ! Edges ! Faces ! Cells ! 4-faces ! 5-faces ! 6-faces ! 7-faces ! 8-faces ! 9-faces ! 10-faces
0 !β₀ \|Point 0-orthoplex \|. \|( ) \|{{CDD\|node}} \|align=right\| 1 \| \| \| \| \| \| \| \| \| \|
1 !β₁ \|Line segment 1-orthoplex \|50px \|{ } \|{{CDD\|node_1}} {{CDD\|node_f1}} \|align=right\| 2 \|align=right\| 1 \| \| \| \| \| \| \| \| \|
2 !β₂ −1₁₁ \|Square 2-orthoplex Bicross \|50px \|{4} 2{ } = { }+{ } \|{{CDD\|node_1\|4\|node}} {{CDD\|node_f1\|2\|node_f1}} \|align=right\| 4 \|align=right\| 4 \|align=right\| 1 \| \| \| \| \| \| \| \|
3 !β₃ 0₁₁ \| Octahedron 3-orthoplex Tricross \|50px \|{3,4} {3^1,1} 3{ } \|{{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 \|align=right\| 1 \| \| \| \| \| \| \|
4 !β₄ 1₁₁ \|16-cell 4-orthoplex Tetracross \|50px \|{3,3,4} {3,3^1,1} 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 \|align=right\| 1 \| \| \| \| \| \|
5 !β₅ 2₁₁ \|5-orthoplex Pentacross \|50px \|{3³,4} {3,3,3^1,1} 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 \|align=right\| 1 \| \| \| \| \|
6 !β₆ 3₁₁ \| 6-orthoplex Hexacross \|50px \|{3⁴,4} {3³,3^1,1} 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 \|align=right\| 1 \| \| \| \|
7 !β₇ 4₁₁ \|7-orthoplex Heptacross \|50px \|{3⁵,4} {3⁴,3^1,1} 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 \|align=right\| 1 \| \| \|
8 !β₈ 5₁₁ \|8-orthoplex Octacross \|50px \|{3⁶,4} {3⁵,3^1,1} 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 \|align=right\| 1 \| \|
9 !β₉ 6₁₁ \|9-orthoplex Enneacross \|50px \|{3⁷,4} {3⁶,3^1,1} 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 \|align=right\| 1 \|
10 ! β₁₀ 7₁₁ \|10-orthoplex Decacross \|50px \|{3⁸,4} {3⁷,3^1,1} 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 \|align=right\|1
colspan=17 style="text-align:center" \|...
n ! β_n k₁₁ \|n-orthoplex n-cross \| \|{3^n − 2,4} {3^n − 3,3^1,1} 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" \| 2n 0-faces, ... $2^{k+1}{n\choose k+1}$ k-faces ..., 2ⁿ (n−1)-faces

class="wikitable"

Cross-polytope elements

!β_n
k₁₁

! Name(s)
Graph

!Graph
2n-gon

!Schläfli

!Coxeter-Dynkin
diagrams

! Vertices

! Edges

! Faces

! Cells

! 4-faces

! 5-faces

! 6-faces

! 7-faces

! 8-faces

! 9-faces

! 10-faces

!β₀

|Point
0-orthoplex

|( )

|{{CDD|node}}

|align=right| 1

!β₁

|Line segment
1-orthoplex

|50px

|{ }

|{{CDD|node_1}}
{{CDD|node_f1}}

|align=right| 2

|align=right| 1

!β₂
−1₁₁

|Square
2-orthoplex
Bicross

|50px

|{4}
2{ } = { }+{ }

|{{CDD|node_1|4|node}}
{{CDD|node_f1|2|node_f1}}

|align=right| 4

|align=right| 1

!β₃
0₁₁

| Octahedron
3-orthoplex
Tricross

|50px

|{3,4}
{3^1,1}
3{ }

|{{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

|align=right| 1

!β₄
1₁₁

|16-cell
4-orthoplex
Tetracross

|50px

|{3,3,4}
{3,3^1,1}
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

|align=right| 1

!β₅
2₁₁

|5-orthoplex
Pentacross

|50px

|{3³,4}
{3,3,3^1,1}
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| 32

|align=right| 1

!β₆
3₁₁

| 6-orthoplex
Hexacross

|50px

|{3⁴,4}
{3³,3^1,1}
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

|align=right| 1

!β₇
4₁₁

|7-orthoplex
Heptacross

|50px

|{3⁵,4}
{3⁴,3^1,1}
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

|align=right| 1

!β₈
5₁₁

|8-orthoplex
Octacross

|50px

|{3⁶,4}
{3⁵,3^1,1}
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| 1024

|align=right| 256

|align=right| 1

!β₉
6₁₁

|9-orthoplex
Enneacross

|50px

|{3⁷,4}
{3⁶,3^1,1}
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

|align=right| 1

! β₁₀
7₁₁

|10-orthoplex
Decacross

|50px

|{3⁸,4}
{3⁷,3^1,1}
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

|align=right|1

colspan=17 style="text-align:center" |...

! β_n
k₁₁

|n-orthoplex
n-cross

|{3^n − 2,4}
{3^n − 3,3^1,1}
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" | 2n 0-faces, ... $2^{k+1}{n\choose k+1}$ k-faces ..., 2ⁿ (n−1)-faces

The vertices of an axis-aligned cross polytope are all at equal distance from each other in the Manhattan distance (L¹ norm). Kusner's conjecture states that this set of 2d points is the largest possible equidistant set for this distance.{{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 }}.

Generalized orthoplex

Regular complex polytopes can be defined in complex Hilbert space called generalized orthoplexes (or cross polytopes), β{{supsub|p|n}} = ₂{3}₂{3}...₂{4}_p, 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_n = ₂{3}₂{3}...₂{4}₂ = {3,3,..,4}. For p > 2, they exist in $\mathbb{\Complex}^n$ . A p-generalized n-orthoplex has pn vertices. Generalized orthoplexes have regular simplexes (real) as facets.Coxeter, Regular Complex Polytopes, p. 108 Generalized orthoplexes make complete multipartite graphs, β{{supsub|p|2}} make K_p,p for complete bipartite graph, β{{supsub|p|3}} make K_p,p,p for complete tripartite graphs. β{{supsub|p|n}} creates K_pⁿ. 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 ! $\mathbb{R}^2$ \|100px ₂{4}₂ = {4} = {{CDD\|node_1\|4\|node}} K_2,2 ! $\mathbb{\Complex}^2$ \|100px ₂{4}₃ = {{CDD\|node_1\|4\|3node}} K_3,3 \|100px ₂{4}₄ = {{CDD\|node_1\|4\|4node}} K_4,4 \|100px ₂{4}₅ = {{CDD\|node_1\|4\|5node}} K_5,5 \|100px ₂{4}₆ = {{CDD\|node_1\|4\|6node}} K_6,6 \|100px ₂{4}₇ = {{CDD\|node_1\|4\|7node}} K_7,7 \|100px ₂{4}₈ = {{CDD\|node_1\|4\|8node}} K_8,8
align=center ! $\mathbb{R}^3$ \|100px ₂{3}₂{4}₂ = {3,4} = {{CDD\|node_1\|3\|node\|4\|node}} K_2,2,2 ! $\mathbb{\Complex}^3$ \|100px ₂{3}₂{4}₃ = {{CDD\|node_1\|3\|node\|4\|3node}} K_3,3,3 \|100px ₂{3}₂{4}₄ = {{CDD\|node_1\|3\|node\|4\|4node}} K_4,4,4 \|100px ₂{3}₂{4}₅ = {{CDD\|node_1\|3\|node\|4\|5node}} K_5,5,5 \|100px ₂{3}₂{4}₆ = {{CDD\|node_1\|3\|node\|4\|6node}} K_6,6,6 \|100px ₂{3}₂{4}₇ = {{CDD\|node_1\|3\|node\|4\|7node}} K_7,7,7 \|100px ₂{3}₂{4}₈ = {{CDD\|node_1\|3\|node\|4\|8node}} K_8,8,8
align=center ! $\mathbb{R}^4$ \|100px ₂{3}₂{3}₂ {3,3,4} = {{CDD\|node_1\|3\|node\|3\|node\|4\|node}} K_2,2,2,2 ! $\mathbb{\Complex}^4$ \|100px ₂{3}₂{3}₂{4}₃ {{CDD\|node_1\|3\|node\|3\|node\|4\|3node}} K_3,3,3,3 \|100px ₂{3}₂{3}₂{4}₄ {{CDD\|node_1\|3\|node\|3\|node\|4\|4node}} K_4,4,4,4 \|100px ₂{3}₂{3}₂{4}₅ {{CDD\|node_1\|3\|node\|3\|node\|4\|5node}} K_5,5,5,5 \|100px ₂{3}₂{3}₂{4}₆ {{CDD\|node_1\|3\|node\|3\|node\|4\|6node}} K_6,6,6,6 \|100px ₂{3}₂{3}₂{4}₇ {{CDD\|node_1\|3\|node\|3\|node\|4\|7node}} K_7,7,7,7 \|100px ₂{3}₂{3}₂{4}₈ {{CDD\|node_1\|3\|node\|3\|node\|4\|8node}} K_8,8,8,8
align=center ! $\mathbb{R}^5$ \|100px ₂{3}₂{3}₂{3}₂{4}₂ {3,3,3,4} = {{CDD\|node_1\|3\|node\|3\|node\|3\|node\|4\|node}} K_2,2,2,2,2 ! $\mathbb{\Complex}^5$ \|100px ₂{3}₂{3}₂{3}₂{4}₃ {{CDD\|node_1\|3\|node\|3\|node\|3\|node\|4\|3node}} K_3,3,3,3,3 \|100px ₂{3}₂{3}₂{3}₂{4}₄ {{CDD\|node_1\|3\|node\|3\|node\|3\|node\|4\|4node}} K_4,4,4,4,4 \|100px ₂{3}₂{3}₂{3}₂{4}₅ {{CDD\|node_1\|3\|node\|3\|node\|3\|node\|4\|5node}} K_5,5,5,5,5 \|100px ₂{3}₂{3}₂{3}₂{4}₆ {{CDD\|node_1\|3\|node\|3\|node\|3\|node\|4\|6node}} K_6,6,6,6,6 \|100px ₂{3}₂{3}₂{3}₂{4}₇ {{CDD\|node_1\|3\|node\|3\|node\|3\|node\|4\|7node}} K_7,7,7,7,7 \|100px ₂{3}₂{3}₂{3}₂{4}₈ {{CDD\|node_1\|3\|node\|3\|node\|3\|node\|4\|8node}} K_8,8,8,8,8
align=center ! $\mathbb{R}^6$ \|100px ₂{3}₂{3}₂{3}₂{3}₂{4}₂ {3,3,3,3,4} = {{CDD\|node_1\|3\|node\|3\|node\|3\|node\|3\|node\|4\|node}} K_2,2,2,2,2,2 ! $\mathbb{\Complex}^6$ \|100px ₂{3}₂{3}₂{3}₂{3}₂{4}₃ {{CDD\|node_1\|3\|node\|3\|node\|3\|node\|3\|node\|4\|3node}} K_3,3,3,3,3,3 \|100px ₂{3}₂{3}₂{3}₂{3}₂{4}₄ {{CDD\|node_1\|3\|node\|3\|node\|3\|node\|3\|node\|4\|4node}} K_4,4,4,4,4,4 \|100px ₂{3}₂{3}₂{3}₂{3}₂{4}₅ {{CDD\|node_1\|3\|node\|3\|node\|3\|node\|3\|node\|4\|5node}} K_5,5,5,5,5,5 \|100px ₂{3}₂{3}₂{3}₂{3}₂{4}₆ {{CDD\|node_1\|3\|node\|3\|node\|3\|node\|3\|node\|4\|6node}} K_6,6,6,6,6,6 \|100px ₂{3}₂{3}₂{3}₂{3}₂{4}₇ {{CDD\|node_1\|3\|node\|3\|node\|3\|node\|3\|node\|4\|7node}} K_7,7,7,7,7,7 \|100px ₂{3}₂{3}₂{3}₂{3}₂{4}₈ {{CDD\|node_1\|3\|node\|3\|node\|3\|node\|3\|node\|4\|8node}} K_8,8,8,8,8,8

Related polytope families

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

In two dimensions, we obtain the octagrammic 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.

Citations

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. 121-122, §7.21. see illustration Fig 7.2B
p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)

External links

{{mathworld | urlname = CrossPolytope | title = Cross polytope}}

Category:Regular polytopes

Category:Multi-dimensional geometry

cross-polytope

4 dimensions

Higher dimensions

Generalized orthoplex

Related polytope families

See also

Citations

References

External links