Cross-polytope

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Cross-polytopes of dimension 2 to 5
A 2-dimensional cross-polytope A 3-dimensional cross-polytope
2 dimensions
square
3 dimensions
octahedron
A 4-dimensional cross-polytope A 5-dimensional cross-polytope
4 dimensions
16-cell
5 dimensions
5-orthoplex

In geometry, a cross-polytope,[1] orthoplex,[2] hyperoctahedron, or cocube is a regular, convex polytope that exists in n-dimensions. A 2-orthoplex is a square, a 3-orthoplex is a regular octahedron, and a 4-orthoplex 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 are all the permutations of (±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 1-norm on Rn:

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. Higher-dimensional cross-polytopes are generalizations of these.

The cross-polytope is the dual polytope of the hypercube. The 1-skeleton of a n-dimensional cross-polytope is a Turán graph T(2n,n).

4 dimensions[edit]

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

Higher dimensions[edit]

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 simplices, labeled as αn. A fourth family, the infinite tessellations of hypercubes, he labeled as δn.

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 . 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 volume of the n-dimensional cross-polytope is

The number of k-dimensional components (vertices, edges, faces, …, facets) in an n-dimensional cross-polytope is given by (see binomial coefficient):

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.

Cross-polytope elements
n βn
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
1 β1 Line segment
1-orthoplex
Cross graph 1.svg { } CDel node 1.png
CDel node f1.png
2                  
2 β2
−111
square
2-orthoplex
Bicross
Cross graph 2.png {4}
2{ } = { }+{ }
CDel node 1.pngCDel 4.pngCDel node.png
CDel node f1.pngCDel 2.pngCDel node f1.png
4 4                
3 β3
011
octahedron
3-orthoplex
Tricross
3-orthoplex.svg {3,4}
{30,1,1}
3{ }
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel split1.pngCDel nodes.png
CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.png
6 12 8              
4 β4
111
16-cell
4-orthoplex
Tetracross
4-orthoplex.svg {3,3,4}
{31,1,1}
4{ }
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.png
8 24 32 16            
5 β5
211
5-orthoplex
Pentacross
5-orthoplex.svg {33,4}
{32,1,1}
5{ }
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.png
10 40 80 80 32          
6 β6
311
6-orthoplex
Hexacross
6-orthoplex.svg {34,4}
{33,1,1}
6{ }
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.png
12 60 160 240 192 64        
7 β7
411
7-orthoplex
Heptacross
7-orthoplex.svg {35,4}
{34,1,1}
7{ }
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.png
14 84 280 560 672 448 128      
8 β8
511
8-orthoplex
Octacross
8-orthoplex.svg {36,4}
{35,1,1}
8{ }
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.png
16 112 448 1120 1792 1792 1024 256    
9 β9
611
9-orthoplex
Enneacross
9-orthoplex.svg {37,4}
{36,1,1}
9{ }
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.png
18 144 672 2016 4032 5376 4608 2304 512  
10 β10
711
10-orthoplex
Decacross
10-orthoplex.svg {38,4}
{37,1,1}
10{ }
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.png
20 180 960 3360 8064 13440 15360 11520 5120 1024
...
n βn
k11
n-orthoplex
n-cross
{3n − 2,4}
{3n − 3,1,1}
n{}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png...CDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.png...CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.png...CDel 2.pngCDel node f1.png
2n 0-faces, ... 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.[3]

Generalized orthoplex[edit]

Regular complex polytopes can be defined in complex Hilbert space called generalized orthoplexes (or cross polytopes), βp
n
= 2{3}2{3}...2{4}p, or CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.png..CDel 3.pngCDel node.pngCDel 4.pngCDel pnode.png. Real solutions exist with p=2, i.e. β2
n
= βn = 2{3}2{3}...2{4}2 = {3,3,..,4}. For p>2, they exist in . A p-generalized n-orthoplex has pn vertices. Generalized orthoplexes have regular simplexes (real) as facets.[4] Generalized orthoplexes make complete multipartite graphs, βp
2
make Kp,p for complete bipartite graph, βp
3
make Kp,p,p for complete tripartite graphs. βp
n
creates Kpn. 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.

Generalized orthoplexes
p=2 p=3 p=4 p=5 p=6 p=7 p=8
2-generalized-2-orthoplex.svg
2{4}2 = {4} = CDel node 1.pngCDel 4.pngCDel node.png
K2,2
Complex polygon 2-4-3-bipartite graph.png
2{4}3 = CDel node 1.pngCDel 4.pngCDel 3node.png
K3,3
Complex polygon 2-4-4 bipartite graph.png
2{4}4 = CDel node 1.pngCDel 4.pngCDel 4node.png
K4,4
Complex polygon 2-4-5-bipartite graph.png
2{4}5 = CDel node 1.pngCDel 4.pngCDel 5node.png
K5,5
6-generalized-2-orthoplex.svg
2{4}6 = CDel node 1.pngCDel 4.pngCDel 6node.png
K6,6
7-generalized-2-orthoplex.svg
2{4}7 = CDel node 1.pngCDel 4.pngCDel 7node.png
K7,7
8-generalized-2-orthoplex.svg
2{4}8 = CDel node 1.pngCDel 4.pngCDel 8node.png
K8,8
Complex tripartite graph octahedron.svg
2{3}2{4}2 = {3,4} = CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
K2,2,2
3-generalized-3-orthoplex-tripartite.svg
2{3}2{4}3 = CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 3node.png
K3,3,3
4-generalized-3-orthoplex.svg
2{3}2{4}4 = CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 4node.png
K4,4,4
5-generalized-3-orthoplex.svg
2{3}2{4}5 = CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 5node.png
K5,5,5
6-generalized-3-orthoplex.svg
2{3}2{4}6 = CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 6node.png
K6,6,6
7-generalized-3-orthoplex.svg
2{3}2{4}7 = CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 7node.png
K7,7,7
8-generalized-3-orthoplex.svg
2{3}2{4}8 = CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 8node.png
K8,8,8
2-generalized-4-orthoplex.svg
2{3}2{3}2
{3,3,4} = CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
K2,2,2,2
3-generalized-4-orthoplex.svg
2{3}2{3}2{4}3
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 3node.png
K3,3,3,3
4-generalized-4-orthoplex.svg
2{3}2{3}2{4}4
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 4node.png
K4,4,4,4
5-generalized-4-orthoplex.svg
2{3}2{3}2{4}5
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 5node.png
K5,5,5,5
6-generalized-4-orthoplex.svg
2{3}2{3}2{4}6
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 6node.png
K6,6,6,6
7-generalized-4-orthoplex.svg
2{3}2{3}2{4}7
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 7node.png
K7,7,7,7
8-generalized-4-orthoplex.svg
2{3}2{3}2{4}8
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 8node.png
K8,8,8,8
2-generalized-5-orthoplex.svg
2{3}2{3}2{3}2{4}2
{3,3,3,4} = CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
K2,2,2,2,2
3-generalized-5-orthoplex.svg
2{3}2{3}2{3}2{4}3
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 3node.png
K3,3,3,3,3
4-generalized-5-orthoplex.svg
2{3}2{3}2{3}2{4}4
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 4node.png
K4,4,4,4,4
5-generalized-5-orthoplex.svg
2{3}2{3}2{3}2{4}5
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 5node.png
K5,5,5,5,5
6-generalized-5-orthoplex.svg
2{3}2{3}2{3}2{4}6
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 6node.png
K6,6,6,6,6
7-generalized-5-orthoplex.svg
2{3}2{3}2{3}2{4}7
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 7node.png
K7,7,7,7,7
8-generalized-5-orthoplex.svg
2{3}2{3}2{3}2{4}8
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 8node.png
K8,8,8,8,8
2-generalized-6-orthoplex.svg
2{3}2{3}2{3}2{3}2{4}2
{3,3,3,3,4} = CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
K2,2,2,2,2,2
3-generalized-6-orthoplex.svg
2{3}2{3}2{3}2{3}2{4}3
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 3node.png
K3,3,3,3,3,3
4-generalized-6-orthoplex.svg
2{3}2{3}2{3}2{3}2{4}4
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 4node.png
K4,4,4,4,4,4
5-generalized-6-orthoplex.svg
2{3}2{3}2{3}2{3}2{4}5
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 5node.png
K5,5,5,5,5,5
6-generalized-6-orthoplex.svg
2{3}2{3}2{3}2{3}2{4}6
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 6node.png
K6,6,6,6,6,6
7-generalized-6-orthoplex.svg
2{3}2{3}2{3}2{3}2{4}7
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 7node.png
K7,7,7,7,7,7
8-generalized-6-orthoplex.svg
2{3}2{3}2{3}2{3}2{4}8
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 8node.png
K8,8,8,8,8,8

See also[edit]

Notes[edit]

  1. ^ Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces, Groningen: University of Groningen  Chapter IV, five dimensional semiregular polytope [1]
  2. ^ Conway calls it an n-orthoplex for orthant complex.
  3. ^ Guy, Richard K. (1983), "An olla-podrida of open problems, often oddly posed", American Mathematical Monthly, 90 (3): 196–200, doi:10.2307/2975549, JSTOR 2975549 .
  4. ^ Coxeter, Regular Complex Polytopes, p. 108

References[edit]

External links[edit]

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds