In geometry, a cross-polytope, hyperoctahedron, orthoplex,[2] or cocube is a regular, convex polytope that exists in n-dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahedron, and a 4-dimensional cross-polytope 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 ℓ1-norm on Rn:
![{\displaystyle \{x\in \mathbb {R} ^{n}:\|x\|_{1}\leq 1\}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/75adeabc1f154d6ecc42c62508cce91b7f34d6ab)
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 an n-dimensional cross-polytope is the Turán graph T(2n, n) (also known as a cocktail party graph [3]).
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 simplex family, 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 dihedral angle of the n-dimensional cross-polytope is
. 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 hypervolume of the n-dimensional cross-polytope is
![{\displaystyle {\frac {2^{n}}{n!}}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/46461c896c17260a514ead32605a9d109129dd27)
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}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/f7c32d2c92a1e4f90614d8a29d7dd066cccdede2)
The extended f-vector for an n-orthoplex can be computed by (1,2)n, like the coefficients of polynomial products. For example a 16-cell is (1,2)4 = (1,4,4)2 = (1,8,24,32,16).
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.
More information , ...
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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 |
. |
( ) |
![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png)
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1 |
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1 |
β1 |
Line segment 1-orthoplex |
![](//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Cross_graph_1.svg/50px-Cross_graph_1.svg.png) |
{ } |
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png)
![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) |
2 |
1 |
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2 |
β2 −111 |
square 2-orthoplex Bicross |
![](//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Cross_graph_2.png/50px-Cross_graph_2.png) |
{4} 2{ } = { }+{ } |
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png)
![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) |
4 |
4 |
1 |
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3 |
β3 011 |
octahedron 3-orthoplex Tricross |
![](//upload.wikimedia.org/wikipedia/commons/thumb/2/25/3-orthoplex.svg/50px-3-orthoplex.svg.png) |
{3,4} {31,1} 3{ } |
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png)
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/a/a1/CDel_split1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/1f/CDel_nodes.png)
![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) |
6 |
12 |
8 |
1 |
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4 |
β4 111 |
16-cell 4-orthoplex Tetracross |
![](//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/4-orthoplex.svg/50px-4-orthoplex.svg.png) |
{3,3,4} {3,31,1} 4{ } |
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png)
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/a/a1/CDel_split1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/1f/CDel_nodes.png)
![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) |
8 |
24 |
32 |
16 |
1 |
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5 |
β5 211 |
5-orthoplex Pentacross |
![](//upload.wikimedia.org/wikipedia/commons/thumb/7/75/5-orthoplex.svg/50px-5-orthoplex.svg.png) |
{33,4} {3,3,31,1} 5{ } |
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png)
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/a/a1/CDel_split1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/1f/CDel_nodes.png)
![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) |
10 |
40 |
80 |
80 |
32 |
1 |
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6 |
β6 311 |
6-orthoplex Hexacross |
![](//upload.wikimedia.org/wikipedia/commons/thumb/8/89/6-orthoplex.svg/50px-6-orthoplex.svg.png) |
{34,4} {33,31,1} 6{ } |
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png)
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/a/a1/CDel_split1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/1f/CDel_nodes.png)
![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) |
12 |
60 |
160 |
240 |
192 |
64 |
1 |
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7 |
β7 411 |
7-orthoplex Heptacross |
![](//upload.wikimedia.org/wikipedia/commons/thumb/4/45/7-orthoplex.svg/50px-7-orthoplex.svg.png) |
{35,4} {34,31,1} 7{ } |
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png)
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/a/a1/CDel_split1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/1f/CDel_nodes.png)
![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) |
14 |
84 |
280 |
560 |
672 |
448 |
128 |
1 |
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8 |
β8 511 |
8-orthoplex Octacross |
![](//upload.wikimedia.org/wikipedia/commons/thumb/0/07/8-orthoplex.svg/50px-8-orthoplex.svg.png) |
{36,4} {35,31,1} 8{ } |
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png)
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/a/a1/CDel_split1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/1f/CDel_nodes.png)
![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) |
16 |
112 |
448 |
1120 |
1792 |
1792 |
1024 |
256 |
1 |
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9 |
β9 611 |
9-orthoplex Enneacross |
![](//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/9-orthoplex.svg/50px-9-orthoplex.svg.png) |
{37,4} {36,31,1} 9{ } |
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png)
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/a/a1/CDel_split1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/1f/CDel_nodes.png)
![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) |
18 |
144 |
672 |
2016 |
4032 |
5376 |
4608 |
2304 |
512 |
1 |
|
10 |
β10 711 |
10-orthoplex Decacross |
![](//upload.wikimedia.org/wikipedia/commons/thumb/4/42/10-orthoplex.svg/50px-10-orthoplex.svg.png) |
{38,4} {37,31,1} 10{ } |
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png)
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/a/a1/CDel_split1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/1f/CDel_nodes.png)
![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) |
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{} |
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ...![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png)
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ...![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/a/a1/CDel_split1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/1f/CDel_nodes.png)
![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ...![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) |
2n 0-faces, ... k-faces ..., 2n (n−1)-faces |
Close
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.[6]
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 ![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png)
![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png)
..![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png)
![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png)
. 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
. A p-generalized n-orthoplex has pn vertices. Generalized orthoplexes have regular simplexes (real) as facets.[7] 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.
More information
,
...
Generalized orthoplexes
| p = 2 | | p = 3 | p = 4 | p = 5 | p = 6 | p = 7 | p = 8 |
![{\displaystyle \mathbb {R} ^{2}}](//wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd) |
2{4}2 = {4} = ![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) K2,2 |
![{\displaystyle \mathbb {\mathbb {C} } ^{2}}](//wikimedia.org/api/rest_v1/media/math/render/svg/3a922cb48f84ced84102524db4a1d478030d2e3a) |
2{4}3 = ![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/4/4a/CDel_3node.png) K3,3 |
2{4}4 = ![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/0/07/CDel_4node.png) K4,4 |
2{4}5 = ![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/f/fd/CDel_5node.png) K5,5 |
2{4}6 = ![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/5/56/CDel_6node.png) K6,6 |
2{4}7 = ![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/2/24/CDel_7node.png) K7,7 |
2{4}8 = ![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9c/CDel_8node.png) K8,8 |
![{\displaystyle \mathbb {R} ^{3}}](//wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5) |
2{3}2{4}2 = {3,4} = ![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) K2,2,2 |
![{\displaystyle \mathbb {\mathbb {C} } ^{3}}](//wikimedia.org/api/rest_v1/media/math/render/svg/f09d7fd24e58ce1cfaedbbdd676e720a8348250d) |
2{3}2{4}3 = ![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/4/4a/CDel_3node.png) K3,3,3 |
2{3}2{4}4 = ![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/0/07/CDel_4node.png) K4,4,4 |
2{3}2{4}5 = ![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/f/fd/CDel_5node.png) K5,5,5 |
2{3}2{4}6 = ![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/5/56/CDel_6node.png) K6,6,6 |
2{3}2{4}7 = ![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/2/24/CDel_7node.png) K7,7,7 |
2{3}2{4}8 = ![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9c/CDel_8node.png) K8,8,8 |
![{\displaystyle \mathbb {R} ^{4}}](//wikimedia.org/api/rest_v1/media/math/render/svg/e4abb9b9dab94f7b25a4210364f0f9032704bfb9) |
2{3}2{3}2 {3,3,4} = ![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) K2,2,2,2 |
![{\displaystyle \mathbb {\mathbb {C} } ^{4}}](//wikimedia.org/api/rest_v1/media/math/render/svg/29c7ae8352bc5bdf95430cc6f0b5c473a050da9a) |
2{3}2{3}2{4}3
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/4/4a/CDel_3node.png) K3,3,3,3 |
2{3}2{3}2{4}4
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/0/07/CDel_4node.png) K4,4,4,4 |
2{3}2{3}2{4}5
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/f/fd/CDel_5node.png) K5,5,5,5 |
2{3}2{3}2{4}6
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/5/56/CDel_6node.png) K6,6,6,6 |
2{3}2{3}2{4}7
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/2/24/CDel_7node.png) K7,7,7,7 |
2{3}2{3}2{4}8
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9c/CDel_8node.png) K8,8,8,8 |
![{\displaystyle \mathbb {R} ^{5}}](//wikimedia.org/api/rest_v1/media/math/render/svg/d52801f10586755156e5f9983b0241399d62f0fd) |
2{3}2{3}2{3}2{4}2 {3,3,3,4} = ![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) K2,2,2,2,2 |
![{\displaystyle \mathbb {\mathbb {C} } ^{5}}](//wikimedia.org/api/rest_v1/media/math/render/svg/751b546e40271f2c1b0597bbf4a06efcf676f65b) |
2{3}2{3}2{3}2{4}3
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/4/4a/CDel_3node.png) K3,3,3,3,3 |
2{3}2{3}2{3}2{4}4
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/0/07/CDel_4node.png) K4,4,4,4,4 |
2{3}2{3}2{3}2{4}5
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/f/fd/CDel_5node.png) K5,5,5,5,5 |
2{3}2{3}2{3}2{4}6
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/5/56/CDel_6node.png) K6,6,6,6,6 |
2{3}2{3}2{3}2{4}7
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/2/24/CDel_7node.png) K7,7,7,7,7 |
2{3}2{3}2{3}2{4}8
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9c/CDel_8node.png) K8,8,8,8,8 |
![{\displaystyle \mathbb {R} ^{6}}](//wikimedia.org/api/rest_v1/media/math/render/svg/547484e275658bac48b1ad8f5407446612d4a65c) |
2{3}2{3}2{3}2{3}2{4}2 {3,3,3,3,4} = ![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) K2,2,2,2,2,2 |
![{\displaystyle \mathbb {\mathbb {C} } ^{6}}](//wikimedia.org/api/rest_v1/media/math/render/svg/8bdbd57d6cdf90c5f1906ea46d3f2e314c021a12) |
2{3}2{3}2{3}2{3}2{4}3
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/4/4a/CDel_3node.png) K3,3,3,3,3,3 |
2{3}2{3}2{3}2{3}2{4}4
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/0/07/CDel_4node.png) K4,4,4,4,4,4 |
2{3}2{3}2{3}2{3}2{4}5
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/f/fd/CDel_5node.png) K5,5,5,5,5,5 |
2{3}2{3}2{3}2{3}2{4}6
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/5/56/CDel_6node.png) K6,6,6,6,6,6 |
2{3}2{3}2{3}2{3}2{4}7
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/2/24/CDel_7node.png) K7,7,7,7,7,7 |
2{3}2{3}2{3}2{3}2{4}8
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/9/9c/CDel_8node.png) K8,8,8,8,8,8 |
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