Cross-polytope

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] hyperoctahedron, orthoplex,[2] staurotope,[3] 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 (±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:

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 [4]).

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.

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 family, labeled as αn. A fourth family, the infinite tessellations of hypercubes, he labeled as δn.[5]

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 hypervolume of the n-dimensional cross-polytope is

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):

[6]

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.

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 10-faces
0 β0 Point
0-orthoplex 		. ( )
1                    
1 β1 Line segment
1-orthoplex 		{ }
2 1                  
2 β2
−111 		Square
2-orthoplex
Bicross 		{4}
2{ } = { }+{ }
4 4 1                
3 β3
011 		Octahedron
3-orthoplex
Tricross 		{3,4}
{31,1}
3{ }

6 12 8 1              
4 β4
111 		16-cell
4-orthoplex
Tetracross 		{3,3,4}
{3,31,1}
4{ }

8 24 32 16 1            
5 β5
211 		5-orthoplex
Pentacross 		{33,4}
{3,3,31,1}
5{ }

10 40 80 80 32 1          
6 β6
311 		6-orthoplex
Hexacross 		{34,4}
{33,31,1}
6{ }

12 60 160 240 192 64 1        
7 β7
411 		7-orthoplex
Heptacross 		{35,4}
{34,31,1}
7{ }

14 84 280 560 672 448 128 1      
8 β8
511 		8-orthoplex
Octacross 		{36,4}
{35,31,1}
8{ }

16 112 448 1120 1792 1792 1024 256 1    
9 β9
611 		9-orthoplex
Enneacross 		{37,4}
{36,31,1}
9{ }

18 144 672 2016 4032 5376 4608 2304 512 1  
10 β10
711 		10-orthoplex
Decacross 		{38,4}
{37,31,1}
10{ }

20 180 960 3360 8064 13440 15360 11520 5120 1024 1
...
n βn
k11 		n-orthoplex
n-cross 		{3n − 2,4}
{3n − 3,31,1}
n{} 		...
...
... 		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.[7]

Generalized orthoplex

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 ... 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.[8] 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{4}2 = {4} =
K2,2
2{4}3 =
K3,3
2{4}4 =
K4,4
2{4}5 =
K5,5
2{4}6 =
K6,6
2{4}7 =
K7,7
2{4}8 =
K8,8

2{3}2{4}2 = {3,4} =
K2,2,2
2{3}2{4}3 =
K3,3,3
2{3}2{4}4 =
K4,4,4
2{3}2{4}5 =
K5,5,5
2{3}2{4}6 =
K6,6,6
2{3}2{4}7 =
K7,7,7
2{3}2{4}8 =
K8,8,8

2{3}2{3}2
{3,3,4} =
K2,2,2,2
2{3}2{3}2{4}3

K3,3,3,3
2{3}2{3}2{4}4

K4,4,4,4
2{3}2{3}2{4}5

K5,5,5,5
2{3}2{3}2{4}6

K6,6,6,6
2{3}2{3}2{4}7

K7,7,7,7
2{3}2{3}2{4}8

K8,8,8,8

2{3}2{3}2{3}2{4}2
{3,3,3,4} =
K2,2,2,2,2
2{3}2{3}2{3}2{4}3

K3,3,3,3,3
2{3}2{3}2{3}2{4}4

K4,4,4,4,4
2{3}2{3}2{3}2{4}5

K5,5,5,5,5
2{3}2{3}2{3}2{4}6

K6,6,6,6,6
2{3}2{3}2{3}2{4}7

K7,7,7,7,7
2{3}2{3}2{3}2{4}8

K8,8,8,8,8

2{3}2{3}2{3}2{3}2{4}2
{3,3,3,3,4} =
K2,2,2,2,2,2
2{3}2{3}2{3}2{3}2{4}3

K3,3,3,3,3,3
2{3}2{3}2{3}2{3}2{4}4

K4,4,4,4,4,4
2{3}2{3}2{3}2{3}2{4}5

K5,5,5,5,5,5
2{3}2{3}2{3}2{3}2{4}6

K6,6,6,6,6,6
2{3}2{3}2{3}2{3}2{4}7

K7,7,7,7,7,7
2{3}2{3}2{3}2{3}2{4}8

K8,8,8,8,8,8

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

  • In two dimensions, we obtain the octagrammic star figure {⁠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

Citations

  1. ^ Coxeter 1973, pp. 121–122, §7.21. illustration Fig 7-2B.
  2. ^ Conway, J. H.; Sloane, N. J. A. (1991). "The Cell Structures of Certain Lattices". In Hilton, P.; Hirzebruch, F.; Remmert, R. (eds.). Miscellanea Mathematica. Berlin: Springer. pp. 89–90. doi:10.1007/978-3-642-76709-8_5. ISBN 978-3-642-76711-1.
  3. ^ McMullen, Peter (2020). Geometric Regular Polytopes. Cambridge University Press. p. 92. ISBN 978-1-108-48958-4.
  4. ^ Weisstein, Eric W. "Cocktail Party Graph". MathWorld.
  5. ^ Coxeter 1973, pp. 120–124, §7.2.
  6. ^ Coxeter 1973, p. 121, §7.2.2..
  7. ^ 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.
  8. ^ Coxeter, Regular Complex Polytopes, p. 108

References

  • Coxeter, H.S.M. (1973). Regular Polytopes (3rd ed.). New York: Dover.
    • 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)
Wikimedia Commons has media related to Cross-polytope graphs.
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 Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

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