Cross-polytope (original) (raw)

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Regular polytope dual to the hypercube in any number of dimensions

Cross-polytopes of dimension 2 to 5

A 2-dimensional cross-polytope A 3-dimensional cross-polytope
2 dimensionssquare 3 dimensionsoctahedron
A 4-dimensional cross-polytope A 5-dimensional cross-polytope
4 dimensions16-cell 5 dimensions5-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:

{ x ∈ R n : ‖ x ‖ 1 ≤ 1 } . {\displaystyle \{x\in \mathbb {R} ^{n}:\|x\|_{1}\leq 1\}.} {\displaystyle \{x\in \mathbb {R} ^{n}:\|x\|_{1}\leq 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(2_n, n) (also known as a cocktail party graph [4]).

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 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 2_n vertices, and 2_n_ 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 δ n = arccos ⁡ ( 2 − n n ) {\displaystyle \delta _{n}=\arccos \left({\frac {2-n}{n}}\right)} {\displaystyle \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

2 n n ! . {\displaystyle {\frac {2^{n}}{n!}}.} {\displaystyle {\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 k + 1 ) {\displaystyle 2^{k+1}{n \choose {k+1}}} {\displaystyle 2^{k+1}{n \choose {k+1}}}[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 2_n_-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 β_n_ _k_11 Name(s)Graph Graph2_n_-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 Point0-orthoplex . ( ) 1
1 β1 Line segment1-orthoplex { } 2 1
2 β2 −111 Square2-orthoplexBicross {4}2{ } = { }+{ } 4 4 1
3 β3011 Octahedron3-orthoplexTricross {3,4}{31,1}3{ } 6 12 8 1
4 β4111 16-cell4-orthoplexTetracross {3,3,4}{3,31,1}4{ } 8 24 32 16 1
5 β5211 5-orthoplexPentacross {33,4}{3,3,31,1}5{ } 10 40 80 80 32 1
6 β6311 6-orthoplexHexacross {34,4}{33,31,1}6{ } 12 60 160 240 192 64 1
7 β7411 7-orthoplexHeptacross {35,4}{34,31,1}7{ } 14 84 280 560 672 448 128 1
8 β8511 8-orthoplexOctacross {36,4}{35,31,1}8{ } 16 112 448 1120 1792 1792 1024 256 1
9 β9611 9-orthoplexEnneacross {37,4}{36,31,1}9{ } 18 144 672 2016 4032 5376 4608 2304 512 1
10 β10711 10-orthoplexDecacross {38,4}{37,31,1}10{ } 20 180 960 3360 8064 13440 15360 11520 5120 1024 1
...
n β_n_ _k_11 _n_-orthoplex_n_-cross {3_n_ − 2,4}{3_n_ − 3,31,1}n{} ......... 2_n_ 0-faces, ... 2 k + 1 ( n k + 1 ) {\displaystyle 2^{k+1}{n \choose k+1}} {\displaystyle 2^{k+1}{n \choose k+1}} _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 Manhattan distance (L1 norm). Kusner's conjecture states that this set of 2_d_ points is the largest possible equidistant set for this distance.[7]

Generalized orthoplex

[edit]

Regular complex polytopes can be defined in complex Hilbert space called generalized orthoplexes (or cross polytopes), β_p_
n = 2{3}2{3}...2{4}p, or ... Real solutions exist with p = 2, i.e. β2
n = β_n_ = 2{3}2{3}...2{4}2 = {3,3,..,4}. For p > 2, they exist in C n {\displaystyle \mathbb {\mathbb {C} } ^{n}} {\displaystyle \mathbb {\mathbb {C} } ^{n}}. A p_-generalized n_-orthoplex has pn vertices. Generalized orthoplexes have regular simplexes (real) as facets.[8] Generalized orthoplexes make complete multipartite graphs, β_p
2 make K_p,p for complete bipartite graph, β_p_
3 make K_p_,p,p for complete tripartite graphs. β_p_
n creates K_p_ n. 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
R 2 {\displaystyle \mathbb {R} ^{2}} {\displaystyle \mathbb {R} ^{2}} 2{4}2 = {4} = K2,2 C 2 {\displaystyle \mathbb {\mathbb {C} } ^{2}} {\displaystyle \mathbb {\mathbb {C} } ^{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
R 3 {\displaystyle \mathbb {R} ^{3}} {\displaystyle \mathbb {R} ^{3}} 2{3}2{4}2 = {3,4} = K2,2,2 C 3 {\displaystyle \mathbb {\mathbb {C} } ^{3}} {\displaystyle \mathbb {\mathbb {C} } ^{3}} 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
R 4 {\displaystyle \mathbb {R} ^{4}} {\displaystyle \mathbb {R} ^{4}} 2{3}2{3}2{3,3,4} = K2,2,2,2 C 4 {\displaystyle \mathbb {\mathbb {C} } ^{4}} {\displaystyle \mathbb {\mathbb {C} } ^{4}} 2{3}2{3}2{4}3K3,3,3,3 2{3}2{3}2{4}4K4,4,4,4 2{3}2{3}2{4}5K5,5,5,5 2{3}2{3}2{4}6K6,6,6,6 2{3}2{3}2{4}7K7,7,7,7 2{3}2{3}2{4}8K8,8,8,8
R 5 {\displaystyle \mathbb {R} ^{5}} {\displaystyle \mathbb {R} ^{5}} 2{3}2{3}2{3}2{4}2{3,3,3,4} = K2,2,2,2,2 C 5 {\displaystyle \mathbb {\mathbb {C} } ^{5}} {\displaystyle \mathbb {\mathbb {C} } ^{5}} 2{3}2{3}2{3}2{4}3K3,3,3,3,3 2{3}2{3}2{3}2{4}4K4,4,4,4,4 2{3}2{3}2{3}2{4}5K5,5,5,5,5 2{3}2{3}2{3}2{4}6K6,6,6,6,6 2{3}2{3}2{3}2{4}7K7,7,7,7,7 2{3}2{3}2{3}2{4}8K8,8,8,8,8
R 6 {\displaystyle \mathbb {R} ^{6}} {\displaystyle \mathbb {R} ^{6}} 2{3}2{3}2{3}2{3}2{4}2{3,3,3,3,4} = K2,2,2,2,2,2 C 6 {\displaystyle \mathbb {\mathbb {C} } ^{6}} {\displaystyle \mathbb {\mathbb {C} } ^{6}} 2{3}2{3}2{3}2{3}2{4}3K3,3,3,3,3,3 2{3}2{3}2{3}2{3}2{4}4K4,4,4,4,4,4 2{3}2{3}2{3}2{3}2{4}5K5,5,5,5,5,5 2{3}2{3}2{3}2{3}2{4}6K6,6,6,6,6,6 2{3}2{3}2{3}2{3}2{4}7K7,7,7,7,7,7 2{3}2{3}2{3}2{3}2{4}8K8,8,8,8,8,8

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

  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
vteFundamental convex regular and uniform polytopes in dimensions 2–10
Family A n B n _I_2(p) / D n _E_6 / _E_7 / _E_8 / _F_4 / _G_2 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