Cross-polytope
From Wikipedia, the free encyclopedia
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:
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.
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).
Contents |
[edit] 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.
[edit] Higher dimensions
The cross polytope family is the first 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 tessellation of hypercubes he labeled as δn.
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):
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.
n | βn | 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 | Line segment 1-cross-polytope |
{} |
2 | ||||||||||||
2 | β2 | -111 | Bicross square 2-cross-polytope |
{4} = {}x{} |
4 | 4 | ||||||||||
3 | β3 | 011 | Tricross octahedron 3-cross-polytope |
{3,4} = t1{3,3} |
6 | 12 | 8 | |||||||||
4 | β4 | 111 | Tetracross 16-cell hexadecachoron 4-cross-polytope |
{3,3,4} = {31,1,1} |
8 | 24 | 32 | 16 | ||||||||
5 | β5 | 211 | Pentacross triacontakaidi-5-tope 5-cross-polytope |
{33,4} = {32,1,1} |
10 | 40 | 80 | 80 | 32 | |||||||
6 | β6 | 311 | Hexacross hexacontatetra-6-tope 6-cross-polytope |
{34,4} = {33,1,1} |
12 | 60 | 160 | 240 | 192 | 64 | ||||||
7 | β7 | 411 | Heptacross hecticosiocta-7-tope 7-cross-polytope |
{35,4} = {34,1,1} |
14 | 84 | 280 | 560 | 672 | 448 | 128 | |||||
8 | β8 | 511 | Octacross dihectapentacontahexa-8-tope 8-cross-polytope |
{36,4} = {35,1,1} |
16 | 112 | 448 | 1120 | 1792 | 1792 | 1024 | 256 | ||||
9 | β9 | 611 | Enneacross pentahectadodeca-9-tope 9-cross-polytope |
{37,4} = {36,1,1} |
18 | 144 | 672 | 2016 | 4032 | 5376 | 4608 | 2304 | 512 | |||
10 | β10 | 711 | Decacross 10-cross-polytope |
{38,4} = {37,1,1} |
20 | 180 | 960 | 3360 | 8064 | 13440 | 15360 | 11520 | 5120 | 1024 | ||
... | ||||||||||||||||
n | βn | (n-3)11 | n-cross n-orthoplex n-cross-polytope |
{3n-2,4} = {3n-3,1,1} ... ... |
2n |
[edit] See also
[edit] 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)
[edit] External links
- Eric W. Weisstein, Cross polytope at MathWorld.
- Polytope Viewer (Click <polytopes...> to select cross polytope.)
- Olshevsky, George, Cross polytope at Glossary for Hyperspace.