Cross-polytope

From Wikipedia, the free encyclopedia

(Redirected from Cross polytope)

Categories: Polytopes | 4-dimensional geometry

In geometry, a cross-polytope, or orthoplex, or hyperoctahedron, is a regular, convex polytope that exists in any number of dimensions. The vertices of a cross-polytope consist of all permutations of (±1, 0, 0, …, 0). The cross-polytope is the convex hull of its vertices. (Note: some authors define a cross-polytope only as the boundary of this region.)

The n-dimensional cross-polytope can also be defined as the closed unit ball in the ℓ₁-norm on Rⁿ:

Failed to parse (Missing texvc executable; please see math/README to configure.): \{x\in\mathbb R^n : \|x\|_1 \le 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 regular polyhedra known as the Platonic solids. Higher-dimensional cross-polytopes are generalizations of these.

Image:Square diamond (shape).svg		Image:Octahedron.svg		Image:Cell16-4dpolytope.svg
2 dimensions square		3 dimensions octahedron		4 dimensions 16-cell

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).

1 4 dimensions
2 Higher dimensions
3 See also
4 References
5 External links

4 dimensions

The 4-dimensional cross-polytope also goes by the name hexadecachoron or 16-cell. It is one of six convex regular 4-polytope. These polychora were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century.

Higher dimensions

The cross polytope family is the first of three regular polytope families, labeled by Coxeter as β_n, the other two being the hypercube family, labeled as γ_n, and the simplices, labeled as α_n. A fourth family, the infinite tessellation of hypercubes he labeled as δ_n.

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 number of k-dimensional components (vertices, edges, faces, …, facets) in an n-dimensional cross-polytope is given by (see binomial coefficient):

Failed to parse (Missing texvc executable; please see math/README to configure.): 2^{k+1}{n \choose {k+1}}

For n≠1, a two dimensional graph of the edges of the n-dimensional cross-polytope can be constructed by drawing 2n vertices on a circle and connecting all pairs of vertices except for vertices exactly on opposite sides of the circle. (These unattached pairs represent the vertex pairs on opposite directions of one coordinate axis of the polytope.) To put this more abstractly, the graph is the complement of a matching of n edges.

Cross-polytope elements
n	β_n	k₁₁	Graph	Name(s)	Schläfli symbol and Coxeter-Dynkin diagrams	Vertices	Edges	Faces	Cells	4-faces	5-faces	6-faces	7-faces	8-faces	9-faces
1	β₁		Image:Cross graph 1.svg	Line segment 1-cross-polytope	{} Image:CDW ring.png	2
2	β₂	-1₁₁	Image:Cross graph 2.svg	Bicross square 2-cross-polytope	{4} = {}x{} Image:CDW ring.png Image:CDW 4.png Image:CDW dot.png Image:CD ring.png Image:CD 2.png Image:CD ring.png	4	4
3	β₃	0₁₁	Image:Cross graph 3.svg	Tricross octahedron 3-cross-polytope	{3,4} = t₁{3,3} Image:CDW ring.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 4.png Image:CDW dot.png Image:CD downbranch-10.png Image:CD 3b.png Image:CD dot.png	6	12	8
4	β₄	1₁₁	Image:Cross graph 4.svg	Tetracross 16-cell hexadecachoron 4-cross-polytope	{3,3,4} = {3^1,1,1} Image:CDW ring.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 4.png Image:CDW dot.png Image:CD ring.png Image:CD 3b.png Image:CD downbranch-00.png Image:CD 3b.png Image:CD dot.png	8	24	32	16
5	β₅	2₁₁	Image:Cross graph 5.svg	Pentacross triacontakaidi-5-tope 5-cross-polytope	{3³,4} = {3^2,1,1} Image:CDW ring.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 4.png Image:CDW dot.png Image:CD ring.png Image:CD 3b.png Image:CD dot.png Image:CD 3b.png Image:CD downbranch-00.png Image:CD 3b.png Image:CD dot.png	10	40	80	80	32
6	β₆	3₁₁	Image:Cross graph 6 Nodes highlighted.svg	Hexacross hexacontatetra-6-tope 6-cross-polytope	{3⁴,4} = {3^3,1,1} Image:CDW ring.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 4.png Image:CDW dot.png Image:CD ring.png Image:CD 3b.png Image:CD dot.png Image:CD 3b.png Image:CD dot.png Image:CD 3b.png Image:CD downbranch-00.png Image:CD 3b.png Image:CD dot.png	12	60	160	240	192	64
7	β₇	4₁₁	Image:Cross graph 7 Nodes highlighted.svg	Heptacross hecticosiocta-7-tope 7-cross-polytope	{3⁵,4} = {3^4,1,1} Image:CDW ring.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 4.png Image:CDW dot.png Image:CD ring.png Image:CD 3b.png Image:CD dot.png Image:CD 3b.png Image:CD dot.png Image:CD 3b.png Image:CD dot.png Image:CD 3b.png Image:CD downbranch-00.png Image:CD 3b.png Image:CD dot.png	14	84	280	560	672	448	128
8	β₈	5₁₁	Image:Cross graph 8 Nodes highlighted.svg	Octacross dihectapentacontahexa-8-tope 8-cross-polytope	{3⁶,4} = {3^5,1,1} Image:CDW ring.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 4.png Image:CDW dot.png Image:CD ring.png Image:CD 3b.png Image:CD dot.png Image:CD 3b.png Image:CD dot.png Image:CD 3b.png Image:CD dot.png Image:CD 3b.png Image:CD dot.png Image:CD 3b.png Image:CD downbranch-00.png Image:CD 3b.png Image:CD dot.png	16	112	448	1120	1792	1792	1024	256
9	β₉	6₁₁	Image:Cross graph 9 Nodes highlighted.svg	Enneacross pentahectadodeca-9-tope 9-cross-polytope	{3⁷,4} = {3^6,1,1} Image:CDW ring.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 4.png Image:CDW dot.png Image:CD ring.png Image:CD 3b.png Image:CD dot.png Image:CD 3b.png Image:CD dot.png Image:CD 3b.png Image:CD dot.png Image:CD 3b.png Image:CD dot.png Image:CD 3b.png Image:CD dot.png Image:CD 3b.png Image:CD downbranch-00.png Image:CD 3b.png Image:CD dot.png	18	144	672	2016	4032	5376	4608	2304	512
10	β₁₀	7₁₁		Decacross 10-cross-polytope	{3⁸,4} = {3^7,1,1} Image:CDW ring.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 4.png Image:CDW dot.png Image:CD ring.png Image:CD 3b.png Image:CD dot.png Image:CD 3b.png Image:CD dot.png Image:CD 3b.png Image:CD dot.png Image:CD 3b.png Image:CD dot.png Image:CD 3b.png Image:CD dot.png Image:CD 3b.png Image:CD dot.png Image:CD 3b.png Image:CD downbranch-00.png Image:CD 3b.png Image:CD dot.png	20	180	960	3360	8064	13440	15360	11520	5120	1024
...
n	β_n	(n-3)₁₁		n-cross n-orthoplex n-cross-polytope	{3^n-2,4} = {3^n-3,1,1} Image:CDW ring.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 3b.png Image:CDW dot.png...Image:CDW 3b.png Image:CDW dot.png Image:CDW 3b.png Image:CDW dot.png Image:CDW 4.png Image:CDW dot.png Image:CD ring.png Image:CD 3b.png Image:CD dot.png Image:CD 3b.png...Image:CD dot.png Image:CD 3b.png Image:CD downbranch-00.png Image:CD 3b.png Image:CD dot.png	2n	...

References

Coxeter, H. S. M. (1973). Regular Polytopes, 3rd ed., New York: Dover Publications, 121–122. ISBN 0-486-61480-8. p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n>=5)