Cross-polytope

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Cross-polytopes of dimension 2 to 5
2-orthoplex.svg Octahedron.png
2 dimensions
square
3 dimensions
octahedron
Schlegel wireframe 16-cell.png 5-cube t4.svg
4 dimensions
16-cell
5 dimensions
5-orthoplex

In geometry, a cross-polytope, [1] 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.

Contents

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 1-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 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 a n-dimensional cross-polytope is a Turán graph T(2n, n).

4 dimensions

The 4-dimensional cross-polytope also goes by the name hexadecachoron or 16-cell . It is one of the six convex regular 4-polytopes. These 4-polytopes 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. [3]

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

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):

[4]

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
Schläfli Coxeter-Dynkin
diagrams
Vertices Edges Faces Cells 4-faces5-faces6-faces7-faces8-faces9-faces10-faces
0 β0 Point
0-orthoplex
.( )CDel node.png
1          
1 β1 Line segment
1-orthoplex
Cross graph 1.svg { }CDel node 1.png
CDel node f1.png
21         
2 β2
111
square
2-orthoplex
Bicross
Cross graph 2.png {4}
2{ } = { }+{ }
CDel node 1.pngCDel 4.pngCDel node.png
CDel node f1.pngCDel 2.pngCDel node f1.png
441        
3 β3
011
octahedron
3-orthoplex
Tricross
3-orthoplex.svg {3,4}
{31,1}
3{ }
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel split1.pngCDel nodes.png
CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.png
61281       
4 β4
111
16-cell
4-orthoplex
Tetracross
4-orthoplex.svg {3,3,4}
{3,31,1}
4{ }
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
CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.png
82432161      
5 β5
211
5-orthoplex
Pentacross
5-orthoplex.svg {33,4}
{3,3,31,1}
5{ }
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
CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.png
10408080321     
6 β6
311
6-orthoplex
Hexacross
6-orthoplex.svg {34,4}
{33,31,1}
6{ }
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
CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.png
1260160240192641    
7 β7
411
7-orthoplex
Heptacross
7-orthoplex.svg {35,4}
{34,31,1}
7{ }
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
CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.png
14842805606724481281   
8 β8
511
8-orthoplex
Octacross
8-orthoplex.svg {36,4}
{35,31,1}
8{ }
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
CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.png
1611244811201792179210242561  
9 β9
611
9-orthoplex
Enneacross
9-orthoplex.svg {37,4}
{36,31,1}
9{ }
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
CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.png
18144672201640325376460823045121 
10 β10
711
10-orthoplex
Decacross
10-orthoplex.svg {38,4}
{37,31,1}
10{ }
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
CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.png
2018096033608064134401536011520512010241
...
nβn
k11
n-orthoplex
n-cross
{3n  2,4}
{3n  3,31,1}
n{}
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
CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.png...CDel 2.pngCDel node f1.png
2n0-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. [5]

Generalized orthoplex

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 CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.png..CDel 3.pngCDel node.pngCDel 4.pngCDel pnode.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. [6] 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.

Generalized orthoplexes
p = 2p = 3p = 4p = 5p = 6p = 7p = 8
Complex bipartite graph square.svg
2{4}2 = {4} = CDel node 1.pngCDel 4.pngCDel node.png
K2,2
Complex polygon 2-4-3-bipartite graph.png
2{4}3 = CDel node 1.pngCDel 4.pngCDel 3node.png
K3,3
Complex polygon 2-4-4 bipartite graph.png
2{4}4 = CDel node 1.pngCDel 4.pngCDel 4node.png
K4,4
Complex polygon 2-4-5-bipartite graph.png
2{4}5 = CDel node 1.pngCDel 4.pngCDel 5node.png
K5,5
6-generalized-2-orthoplex.svg
2{4}6 = CDel node 1.pngCDel 4.pngCDel 6node.png
K6,6
7-generalized-2-orthoplex.svg
2{4}7 = CDel node 1.pngCDel 4.pngCDel 7node.png
K7,7
8-generalized-2-orthoplex.svg
2{4}8 = CDel node 1.pngCDel 4.pngCDel 8node.png
K8,8
Complex tripartite graph octahedron.svg
2{3}2{4}2 = {3,4} = CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
K2,2,2
3-generalized-3-orthoplex-tripartite.svg
2{3}2{4}3 = CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 3node.png
K3,3,3
4-generalized-3-orthoplex.svg
2{3}2{4}4 = CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 4node.png
K4,4,4
5-generalized-3-orthoplex.svg
2{3}2{4}5 = CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 5node.png
K5,5,5
6-generalized-3-orthoplex.svg
2{3}2{4}6 = CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 6node.png
K6,6,6
7-generalized-3-orthoplex.svg
2{3}2{4}7 = CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 7node.png
K7,7,7
8-generalized-3-orthoplex.svg
2{3}2{4}8 = CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 8node.png
K8,8,8
Complex multipartite graph 16-cell.svg
2{3}2{3}2
{3,3,4} = CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
K2,2,2,2
3-generalized-4-orthoplex.svg
2{3}2{3}2{4}3
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 3node.png
K3,3,3,3
4-generalized-4-orthoplex.svg
2{3}2{3}2{4}4
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 4node.png
K4,4,4,4
5-generalized-4-orthoplex.svg
2{3}2{3}2{4}5
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 5node.png
K5,5,5,5
6-generalized-4-orthoplex.svg
2{3}2{3}2{4}6
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 6node.png
K6,6,6,6
7-generalized-4-orthoplex.svg
2{3}2{3}2{4}7
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 7node.png
K7,7,7,7
8-generalized-4-orthoplex.svg
2{3}2{3}2{4}8
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 8node.png
K8,8,8,8
2-generalized-5-orthoplex.svg
2{3}2{3}2{3}2{4}2
{3,3,3,4} = CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
K2,2,2,2,2
3-generalized-5-orthoplex.svg
2{3}2{3}2{3}2{4}3
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 3node.png
K3,3,3,3,3
4-generalized-5-orthoplex.svg
2{3}2{3}2{3}2{4}4
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 4node.png
K4,4,4,4,4
5-generalized-5-orthoplex.svg
2{3}2{3}2{3}2{4}5
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 5node.png
K5,5,5,5,5
6-generalized-5-orthoplex.svg
2{3}2{3}2{3}2{4}6
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 6node.png
K6,6,6,6,6
7-generalized-5-orthoplex.svg
2{3}2{3}2{3}2{4}7
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 7node.png
K7,7,7,7,7
8-generalized-5-orthoplex.svg
2{3}2{3}2{3}2{4}8
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 8node.png
K8,8,8,8,8
2-generalized-6-orthoplex.svg
2{3}2{3}2{3}2{3}2{4}2
{3,3,3,3,4} = CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
K2,2,2,2,2,2
3-generalized-6-orthoplex.svg
2{3}2{3}2{3}2{3}2{4}3
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 3node.png
K3,3,3,3,3,3
4-generalized-6-orthoplex.svg
2{3}2{3}2{3}2{3}2{4}4
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 4node.png
K4,4,4,4,4,4
5-generalized-6-orthoplex.svg
2{3}2{3}2{3}2{3}2{4}5
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 5node.png
K5,5,5,5,5,5
6-generalized-6-orthoplex.svg
2{3}2{3}2{3}2{3}2{4}6
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 6node.png
K6,6,6,6,6,6
7-generalized-6-orthoplex.svg
2{3}2{3}2{3}2{3}2{4}7
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 7node.png
K7,7,7,7,7,7
8-generalized-6-orthoplex.svg
2{3}2{3}2{3}2{3}2{4}8
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 8node.png
K8,8,8,8,8,8

Cross-polytopes can be combined with their dual cubes to form compound polytopes:

See also

Citations

  1. Coxeter 1973, pp. 121–122, §7.21. illustration Fig 7-2B.
  2. Conway calls it an n-orthoplex for orthant complex.
  3. Coxeter 1973, pp. 120–124, §7.2.
  4. Coxeter 1973, p. 121, §7.2.2..
  5. Guy, Richard K. (1983), "An olla-podrida of open problems, often oddly posed", American Mathematical Monthly, 90 (3): 196–200, doi:10.2307/2975549, JSTOR   2975549 .
  6. Coxeter, Regular Complex Polytopes, p. 108

Related Research Articles

Cube A geometric 4-dimensional object with 6 square faces

In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

In elementary geometry, a polytope is a geometric object with flat sides (faces). It is a generalization in any number of dimensions of the three-dimensional polyhedron. Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. In this context, "flat sides" means that the sides of a (k + 1)-polytope consist of k-polytopes that may have (k – 1)-polytopes in common. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope.

Hypercube Convex polytope, the n-dimensional analogue of a square and a cube

In geometry, a hypercube is an n-dimensional analogue of a square and a cube. It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length. A unit hypercube's longest diagonal in n dimensions is equal to .

Regular polytope Polytope with highest degree of symmetry

In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension n.

16-cell Four-dimensional analog of the octahedron

In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,4}. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. It is also called C16, hexadecachoron, or hexdecahedroid.

5-polytope 5-dimensional geometric object

In geometry, a five-dimensional polytope is a polytope in five-dimensional space, bounded by (4-polytope) facets, pairs of which share a polyhedral cell.

Uniform 6-polytope

In six-dimensional geometry, a uniform polypeton is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes.

In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces.

5-demicube

In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube (penteract) with alternated vertices removed.

5-orthoplex

In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces.

In geometry, a complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one.

6-orthoplex

In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell 4-faces, and 64 5-faces.

7-orthoplex

In geometry, a 7-orthoplex, or 7-cross polytope, is a regular 7-polytope with 14 vertices, 84 edges, 280 triangle faces, 560 tetrahedron cells, 672 5-cells 4-faces, 448 5-faces, and 128 6-faces.

10-orthoplex

In geometry, a 10-orthoplex or 10-cross polytope, is a regular 10-polytope with 20 vertices, 180 edges, 960 triangle faces, 3360 octahedron cells, 8064 5-cells 4-faces, 13440 5-faces, 15360 6-faces, 11520 7-faces, 5120 8-faces, and 1024 9-faces.

Alternated hypercubic honeycomb

In geometry, the alternated hypercube honeycomb is a dimensional infinite series of honeycombs, based on the hypercube honeycomb with an alternation operation. It is given a Schläfli symbol h{4,3...3,4} representing the regular form with half the vertices removed and containing the symmetry of Coxeter group for n ≥ 4. A lower symmetry form can be created by removing another mirror on an order-4 peak.

Petrie polygon Skew polygon derived from a polytope

In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon in which every n – 1 consecutive sides belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a regular polyhedron is a skew polygon such that every two consecutive sides belongs to one of the faces. Petrie polygons are named for mathematician John Flinders Petrie.

2<sub> 21</sub> polytope

In 6-dimensional geometry, the 221 polytope is a uniform 6-polytope, constructed within the symmetry of the E6 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 6-ic semi-regular figure. It is also called the Schläfli polytope.

3<sub> 21</sub> polytope

In 7-dimensional geometry, the 321 polytope is a uniform 7-polytope, constructed within the symmetry of the E7 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 7-ic semi-regular figure.

4<sub> 21</sub> polytope

In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 8-ic semi-regular figure.

References

Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds