Cross-polytope


Cross-polytope

In geometry, a cross-polytope,[1] orthoplex,[2] hyperoctahedron, or cocube is a regular, convex polytope that exists in any number of dimensions. The vertices of a cross-polytope are all the permutations of (±1, 0, 0, …, 0). The cross-polytope is the convex hull of its vertices. Its facets are simplexes of the previous dimension, while the cross-polytope's vertex figure is another cross-polytope from the previous dimension.

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:

\{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 convex regular polyhedra known as the Platonic solids. Higher-dimensional cross-polytopes are generalizations of these.

A 2-dimensional cross-polytope A 3-dimensional cross-polytope A 4-dimensional cross-polytope
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

4 dimensions

The 4-dimensional cross-polytope also goes by the name hexadecachoron or 16-cell. It is one of six convex regular 4-polytopes. These polychora 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.

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):

2^{k+1}{n \choose {k+1}}

The volume of the n-dimensional cross-polytope is

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

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
Graph
2(n-1)-gon
Schläfli Coxeter-Dynkin
diagrams
Vertices Edges Faces Cells 4-faces 5-faces 6-faces 7-faces 8-faces 9-faces
1 β1 Line segment
1-orthoplex
Cross graph 1.svg {} CDel node 1.png 2                  
2 β2
−111
Bicross
square
2-orthoplex
Cross graph 2.png 2-orthoplex B1.svg {4}
{} x {}
CDel node 1.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 2.pngCDel node 1.png
4 4                
3 β3
011
Tricross
octahedron
3-orthoplex
3-orthoplex.svg 3-orthoplex B2.svg {3,4}
{30,1,1}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel split1.pngCDel nodes.png
6 12 8              
4 β4
111
Tetracross
16-cell
4-orthoplex
4-orthoplex.svg 4-orthoplex B3.svg {3,3,4}
{31,1,1}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
8 24 32 16            
5 β5
211
Pentacross
5-orthoplex
5-orthoplex.svg 5-orthoplex B4.svg {33,4}
{32,1,1}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
10 40 80 80 32          
6 β6
311
Hexacross
6-orthoplex
6-orthoplex.svg 6-orthoplex B5.svg {34,4}
{33,1,1}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
12 60 160 240 192 64        
7 β7
411
Heptacross
7-orthoplex
7-orthoplex.svg 7-orthoplex B6.svg {35,4}
{34,1,1}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
14 84 280 560 672 448 128      
8 β8
511
Octacross
8-orthoplex
8-orthoplex.svg 8-orthoplex B7.svg {36,4}
{35,1,1}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
16 112 448 1120 1792 1792 1024 256    
9 β9
611
Enneacross
9-orthoplex
9-orthoplex.svg 9-orthoplex B8.svg {37,4}
{36,1,1}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
18 144 672 2016 4032 5376 4608 2304 512  
10 β10
711
Decacross
10-orthoplex
10-orthoplex.svg 10-orthoplex B9.svg {38,4}
{37,1,1}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
20 180 960 3360 8064 13440 15360 11520 5120 1024
...
n βn
k11
n-cross
n-orthoplex
{3n − 2,4}
{3n − 3,1,1}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png...CDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.png...CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
2n 0-faces, ... 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.[3]

See also

Notes

  1. ^ Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces, Groningen: University of Groningen  Chapter IV, five dimensional semiregular polytope [1]
  2. ^ Conway calls it an n-orthoplex for orthant complex.
  3. ^ Guy, Richard K. (1983), "An olla-podrida of open problems, often oddly posed", American Mathematical Monthly 90 (3): 196–200, http://www.jstor.org/stable/2975549 .

References

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

External links


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