Cross-polytope

 2 dimensionssquare 3 dimensionsoctahedron 4 dimensions16-cell 5 dimensions5-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 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:

${\displaystyle \{x\in \mathbb {R} ^{n}:\|x\|_{1}\leq 1\}.}$

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 a n-dimensional cross-polytope is a Turán graph T(2n,n).

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 simplices, labeled as αn. A fourth family, the infinite tessellations of hypercubes, he labeled as δn.[3]

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 ${\displaystyle \delta _{n}=\arccos \left({\frac {2-n}{n}}\right)}$. 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 volume of the n-dimensional cross-polytope is

${\displaystyle {\frac {2^{n}}{n!}}.}$

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):

${\displaystyle 2^{k+1}{n \choose {k+1}}}$[4]

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, ... ${\displaystyle 2^{k+1}{n \choose k+1}}$ 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.[5]

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 ${\displaystyle \mathbb {C} ^{n}}$. A p-generalized n-orthoplex has pn vertices. Generalized orthoplexes have regular simplexes (real) as facets.[6] 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
${\displaystyle \mathbb {R} ^{2}}$
2{4}2 = {4} =
K2,2
${\displaystyle \mathbb {C} ^{2}}$
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
${\displaystyle \mathbb {R} ^{3}}$
2{3}2{4}2 = {3,4} =
K2,2,2
${\displaystyle \mathbb {C} ^{3}}$
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
${\displaystyle \mathbb {R} ^{4}}$
2{3}2{3}2
{3,3,4} =
K2,2,2,2
${\displaystyle \mathbb {C} ^{4}}$
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
${\displaystyle \mathbb {R} ^{5}}$
2{3}2{3}2{3}2{4}2
{3,3,3,4} =
K2,2,2,2,2
${\displaystyle \mathbb {C} ^{5}}$
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
${\displaystyle \mathbb {R} ^{6}}$
2{3}2{3}2{3}2{3}2{4}2
{3,3,3,3,4} =
K2,2,2,2,2,2
${\displaystyle \mathbb {C} ^{6}}$
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

Related polytope families

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

Citations

1. ^ Coxeter 1973, pp. 121-122, §7.21. illustration Fig 7-2B.
2. ^ Conway calls it an n-orthoplex for orthant complex.
3. ^ Coxeter 1973, pp. 120-124, §7.2.
4. ^ Coxeter 1973, p. 121, §7.2.2..
5. ^ 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.
6. ^ 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)