Cross-polytope

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Cross-polytopes of dimension 2 to 5
2-orthoplex.svg Octahedron.png
2 dimensions
square
3 dimensions
octahedron
Schlegel wireframe 16-cell.png 5-cube t4.svg
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.

Contents

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 1-norm on Rn, those points x = (x1, x2..., xn) satisfying

An n-orthoplex can be constructed as a bipyramid with an (n−1)-orthoplex base.

The cross-polytope is the dual polytope of the hypercube. The vertex-edge graph of an n-dimensional cross-polytope is the Turán graph T(2n, n) (also known as a cocktail party graph [4] ).

Low-dimensional examples

In 1 dimension the cross-polytope is a line segment, which can be chosen as the interval [1, +1].

In 2 dimensions the cross-polytope is a square. If the vertices are chosen as {(±1, 0), (0, ±1)}, the square's sides are at right angles to the axes; in this orientation a square is often called a diamond.

In 3 dimensions the cross-polytope is a regular octahedron one of the five convex regular polyhedra known as the Platonic solids.

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. [5]

n 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. [6]

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):

[7]

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-faces5-faces6-faces
0 β0 Point
0-orthoplex
.( )CDel node.png
1      
1 β1 Line segment
1-orthoplex
Cross graph 1.svg { }CDel node 1.png
CDel node f1.png
21     
2 β2
111
Square
2-orthoplex
Bicross
2-orthoplex.svg {4}
2{ } = { }+{ }
CDel node 1.pngCDel 4.pngCDel node.png
CDel node f1.pngCDel 2.pngCDel node f1.png
441    
3 β3
011
Octahedron
3-orthoplex
Tricross
3-orthoplex.svg {3,4}
{31,1}
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
61281   
4 β4
111
16-cell
4-orthoplex
Tetracross
4-orthoplex.svg {3,3,4}
{3,31,1}
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
82432161  
5 β5
211
5-orthoplex
Pentacross
5-orthoplex.svg {33,4}
{3,3,31,1}
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
10408080321 
6 β6
311
6-orthoplex
Hexacross
6-orthoplex.svg {34,4}
{33,31,1}
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
1260160240192641
...
nβn
(n−3)11
n-orthoplex
n-cross
{3n  2,4}
{3n  3,31,1}
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
2n0-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. [8]

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 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. [9] 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 = 2p = 3p = 4p = 5p = 6p = 7p = 8
Complex bipartite graph square.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
Complex multipartite graph 16-cell.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

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

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. Bengtsson, Ingemar; Życzkowski, Karol (2017). Geometry of Quantum States: An Introduction to Quantum Entanglement (2nd ed.). Cambridge University Press. p. 162. ISBN   978-1-107-02625-4.
  6. Coxeter 1973, pp. 120–124, §7.2.
  7. Coxeter 1973, p. 121, §7.2.2..
  8. 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 .
  9. Coxeter, Regular Complex Polytopes, p. 108

References

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 polychoron Pentachoron 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 compoundsPolytope operations