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):
The volume of the n-dimensional cross-polytope is
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.
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 |
{} | 2 | |||||||||||||
2 | β2 −111 |
Bicross square 2-orthoplex |
{4} {} x {} |
4 | 4 | ||||||||||||
3 | β3 011 |
Tricross octahedron 3-orthoplex |
{3,4} {30,1,1} |
6 | 12 | 8 | |||||||||||
4 | β4 111 |
Tetracross 16-cell 4-orthoplex |
{3,3,4} {31,1,1} |
8 | 24 | 32 | 16 | ||||||||||
5 | β5 211 |
Pentacross 5-orthoplex |
{33,4} {32,1,1} |
10 | 40 | 80 | 80 | 32 | |||||||||
6 | β6 311 |
Hexacross 6-orthoplex |
{34,4} {33,1,1} |
12 | 60 | 160 | 240 | 192 | 64 | ||||||||
7 | β7 411 |
Heptacross 7-orthoplex |
{35,4} {34,1,1} |
14 | 84 | 280 | 560 | 672 | 448 | 128 | |||||||
8 | β8 511 |
Octacross 8-orthoplex |
{36,4} {35,1,1} |
16 | 112 | 448 | 1120 | 1792 | 1792 | 1024 | 256 | ||||||
9 | β9 611 |
Enneacross 9-orthoplex |
{37,4} {36,1,1} |
18 | 144 | 672 | 2016 | 4032 | 5376 | 4608 | 2304 | 512 | |||||
10 | β10 711 |
Decacross 10-orthoplex |
{38,4} {37,1,1} |
20 | 180 | 960 | 3360 | 8064 | 13440 | 15360 | 11520 | 5120 | 1024 | ||||
... | |||||||||||||||||
n | βn k11 |
n-cross n-orthoplex |
{3n − 2,4} {3n − 3,1,1} |
... ... |
2n 0-faces, ... 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.
Read more about this topic: Cross-polytope
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