In geometry , a cross-polytope , hyperoctahedron , orthoplex ,[ 2] staurotope ,[ 3] 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 R n
{ x ∈ ℝ n : ‖ x ‖ 1 ≤ 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 [ 4]
4 dimensions The 4-dimensional cross-polytope also goes by the name hexadecachoron or 16-cell 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 .
The n -dimensional cross-polytope has 2n vertices, and 2n 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 ) 2 = 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 ! . 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 ) The extended f-vector for an n -orthoplex can be computed by (1 ,2)n 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 k 11
Name(s)Graph
Graphn -gon
Schläfli
Coxeter-Dynkin
Vertices
Edges
Faces
Cells
4-faces
5-faces
6-faces
7-faces
8-faces
9-faces
10-faces
0
β0
Point .
( )
File:CDel node.png 1
1
β1
Line segment File:Cross graph 1.svg { }
File:CDel node 1.png File:CDel node f1.png 2
1
2
β2 11
Square Bicross
File:Cross graph 2.png {4}
File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png 4
4
1
3
β3 11
Octahedron Tricross
File:3-orthoplex.svg {3,4}1,1 }
File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel node 1.png File:CDel split1.png File:CDel nodes.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png 6
12
8
1
4
β4 11
16-cell Tetracross
File:4-orthoplex.svg {3,3,4}1,1 }
File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel split1.png File:CDel nodes.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png 8
24
32
16
1
5
β5 11
5-orthoplex Pentacross
File:5-orthoplex.svg {33 ,4}1,1 }
File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel split1.png File:CDel nodes.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png 10
40
80
80
32
1
6
β6 11
6-orthoplex Hexacross
File:6-orthoplex.svg {34 ,4}3 ,31,1 }
File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel split1.png File:CDel nodes.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png 12
60
160
240
192
64
1
7
β7 11
7-orthoplex Heptacross
File:7-orthoplex.svg {35 ,4}4 ,31,1 }
File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel split1.png File:CDel nodes.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png 14
84
280
560
672
448
128
1
8
β8 11
8-orthoplex Octacross
File:8-orthoplex.svg {36 ,4}5 ,31,1 }
File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel split1.png File:CDel nodes.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png 16
112
448
1120
1792
1792
1024
256
1
9
β9 11
9-orthoplex Enneacross
File:9-orthoplex.svg {37 ,4}6 ,31,1 }
File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel split1.png File:CDel nodes.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png 18
144
672
2016
4032
5376
4608
2304
512
1
10
β10 11
10-orthoplex Decacross
File:10-orthoplex.svg {38 ,4}7 ,31,1 }
File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel split1.png File:CDel nodes.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png 20
180
960
3360
8064
13440
15360
11520
5120
1024
1
...
n βn k 11
n -orthoplexn -cross
{3n − 2n − 31,1 }n {}
File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel split1.png File:CDel nodes.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel node f1.png File:CDel 2.png File:CDel 2.png File:CDel node f1.png 2n 0-faces , ... 2 k + 1 ( n k + 1 ) k -facesn (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 File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel pnode.png p = 2, i.e. β2 n n 2 {3}2 {3}...2 {4}2 = {3,3,..,4}. For p > 2, they exist in ℂ n p -generalized n -orthoplex has pn vertices. Generalized orthoplexes have regular simplexes (real) as facets .[ 8] complete multipartite graphs , βp 2 p ,p complete bipartite graph , βp 3 p ,p ,p p n p n 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
ℝ 2 File:Complex bipartite graph square.svg 2 {4}2 = {4} = File:CDel node 1.png File:CDel 4.png File:CDel node.png 2,2
ℂ 2 File:Complex polygon 2-4-3-bipartite graph.png 2 {4}3 File:CDel node 1.png File:CDel 4.png File:CDel 3node.png 3,3
File:Complex polygon 2-4-4 bipartite graph.png 2 {4}4 File:CDel node 1.png File:CDel 4.png File:CDel 4node.png 4,4
File:Complex polygon 2-4-5-bipartite graph.png 2 {4}5 File:CDel node 1.png File:CDel 4.png File:CDel 5node.png 5,5
File:6-generalized-2-orthoplex.svg 2 {4}6 File:CDel node 1.png File:CDel 4.png File:CDel 6node.png 6,6
File:7-generalized-2-orthoplex.svg 2 {4}7 = File:CDel node 1.png File:CDel 4.png File:CDel 7node.png 7,7
File:8-generalized-2-orthoplex.svg 2 {4}8 File:CDel node 1.png File:CDel 4.png File:CDel 8node.png 8,8
ℝ 3 File:Complex tripartite graph octahedron.svg 2 {3}2 {4}2 = {3,4} = File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png 2,2,2
ℂ 3 File:3-generalized-3-orthoplex-tripartite.svg 2 {3}2 {4}3 = File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel 3node.png 3,3,3
File:4-generalized-3-orthoplex.svg 2 {3}2 {4}4 = File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel 4node.png 4,4,4
File:5-generalized-3-orthoplex.svg 2 {3}2 {4}5 = File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel 5node.png 5,5,5
File:6-generalized-3-orthoplex.svg 2 {3}2 {4}6 = File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel 6node.png 6,6,6
File:7-generalized-3-orthoplex.svg 2 {3}2 {4}7 = File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel 7node.png 7,7,7
File:8-generalized-3-orthoplex.svg 2 {3}2 {4}8 = File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel 8node.png 8,8,8
ℝ 4 File:Complex multipartite graph 16-cell.svg 2 {3}2 {3}2 {3,3,4} = File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png 2,2,2,2
ℂ 4 File:3-generalized-4-orthoplex.svg 2 {3}2 {3}2 {4}3 File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel 3node.png 3,3,3,3
File:4-generalized-4-orthoplex.svg 2 {3}2 {3}2 {4}4 File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel 4node.png 4,4,4,4
File:5-generalized-4-orthoplex.svg 2 {3}2 {3}2 {4}5 File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel 5node.png 5,5,5,5
File:6-generalized-4-orthoplex.svg 2 {3}2 {3}2 {4}6 File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel 6node.png 6,6,6,6
File:7-generalized-4-orthoplex.svg 2 {3}2 {3}2 {4}7 File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel 7node.png 7,7,7,7
File:8-generalized-4-orthoplex.svg 2 {3}2 {3}2 {4}8 File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel 8node.png 8,8,8,8
ℝ 5 File:2-generalized-5-orthoplex.svg 2 {3}2 {3}2 {3}2 {4}2 {3,3,3,4} = File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png 2,2,2,2,2
ℂ 5 File:3-generalized-5-orthoplex.svg 2 {3}2 {3}2 {3}2 {4}3 File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel 3node.png 3,3,3,3,3
File:4-generalized-5-orthoplex.svg 2 {3}2 {3}2 {3}2 {4}4 File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel 4node.png 4,4,4,4,4
File:5-generalized-5-orthoplex.svg 2 {3}2 {3}2 {3}2 {4}5 File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel 5node.png 5,5,5,5,5
File:6-generalized-5-orthoplex.svg 2 {3}2 {3}2 {3}2 {4}6 File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel 6node.png 6,6,6,6,6
File:7-generalized-5-orthoplex.svg 2 {3}2 {3}2 {3}2 {4}7 File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel 7node.png 7,7,7,7,7
File:8-generalized-5-orthoplex.svg 2 {3}2 {3}2 {3}2 {4}8 File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel 8node.png 8,8,8,8,8
ℝ 6 File:2-generalized-6-orthoplex.svg 2 {3}2 {3}2 {3}2 {3}2 {4}2 {3,3,3,3,4} = File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png 2,2,2,2,2,2
ℂ 6 File:3-generalized-6-orthoplex.svg 2 {3}2 {3}2 {3}2 {3}2 {4}3 File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel 3node.png 3,3,3,3,3,3
File:4-generalized-6-orthoplex.svg 2 {3}2 {3}2 {3}2 {3}2 {4}4 File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel 4node.png 4,4,4,4,4,4
File:5-generalized-6-orthoplex.svg 2 {3}2 {3}2 {3}2 {3}2 {4}5 File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel 5node.png 5,5,5,5,5,5
File:6-generalized-6-orthoplex.svg 2 {3}2 {3}2 {3}2 {3}2 {4}6 File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel 6node.png 6,6,6,6,6,6
File:7-generalized-6-orthoplex.svg 2 {3}2 {3}2 {3}2 {3}2 {4}7 File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel 7node.png 7,7,7,7,7,7
File:8-generalized-6-orthoplex.svg 2 {3}2 {3}2 {3}2 {3}2 {4}8 File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel 8node.png 8,8,8,8,8,8
Related polytope families Cross-polytopes can be combined with their dual cubes to form compound polytopes:
See also Citations
↑ 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 ↑ McMullen, Peter (2020). Geometric Regular Polytopes . Cambridge University Press. p. 92. ISBN 978-1-108-48958-4 ↑ Weisstein, Eric W. "Cocktail Party Graph" . MathWorld ↑ 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 ↑ Coxeter, Regular Complex Polytopes, p. 108
References Coxeter, H.S.M. (1973). Regular Polytopes 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
Dimensional spaces Other dimensions Polytopes and shapes Number systems Dimensions by number See also
