Wiki English
Wiki EnglishWiki DeutschWiki Nederlands

Uniform 10-Polytope

In ten-dimensional geometry, a 10-polytope is a 10-dimensional polytope whose boundary consists of 9-polytope facets, exactly two such facets meeting at each 8-polytope ridge.

Graphs of three regular and related uniform polytopes.
Uniform 10-Polytope
10-simplex 		Uniform 10-Polytope
Truncated 10-simplex 		Uniform 10-Polytope
Rectified 10-simplex
Uniform 10-Polytope
Cantellated 10-simplex 		Uniform 10-Polytope
Runcinated 10-simplex
Uniform 10-Polytope
Stericated 10-simplex 		Uniform 10-Polytope
Pentellated 10-simplex 		Uniform 10-Polytope
Hexicated 10-simplex
Uniform 10-Polytope
Heptellated 10-simplex 		Uniform 10-Polytope
Octellated 10-simplex 		Uniform 10-Polytope
Ennecated 10-simplex
Uniform 10-Polytope
10-orthoplex 		Uniform 10-Polytope
Truncated 10-orthoplex 		Uniform 10-Polytope
Rectified 10-orthoplex
Uniform 10-Polytope
10-cube 		Uniform 10-Polytope
Truncated 10-cube 		Uniform 10-Polytope
Rectified 10-cube
Uniform 10-Polytope
10-demicube 		Uniform 10-Polytope
Truncated 10-demicube

A uniform 10-polytope is one which is vertex-transitive, and constructed from uniform facets.

Regular 10-polytopes

Regular 10-polytopes can be represented by the Schläfli symbol {p,q,r,s,t,u,v,w,x}, with x {p,q,r,s,t,u,v,w} 9-polytope facets around each peak.

There are exactly three such convex regular 10-polytopes:

  1. {3,3,3,3,3,3,3,3,3} - 10-simplex
  2. {4,3,3,3,3,3,3,3,3} - 10-cube
  3. {3,3,3,3,3,3,3,3,4} - 10-orthoplex

There are no nonconvex regular 10-polytopes.

Euler characteristic

The topology of any given 10-polytope is defined by its Betti numbers and torsion coefficients.

The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 10-polytopes, whatever their underlying topology. This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers.

Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes, and this led to the use of torsion coefficients.

Uniform 10-polytopes by fundamental Coxeter groups

Uniform 10-polytopes with reflective symmetry can be generated by these three Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagrams:

# Coxeter group Coxeter-Dynkin diagram
1 A10 [39] Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
2 B10 [4,38] Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
3 D10 [37,1,1] Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope

Selected regular and uniform 10-polytopes from each family include:

  1. Simplex family: A10 [39] - Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
    • 527 uniform 10-polytopes as permutations of rings in the group diagram, including one regular:
      1. {39} - 10-simplex - Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
  2. Hypercube/orthoplex family: B10 [4,38] - Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
    • 1023 uniform 10-polytopes as permutations of rings in the group diagram, including two regular ones:
      1. {4,38} - 10-cube or dekeract - Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
      2. {38,4} - 10-orthoplex or decacross - Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
      3. h{4,38} - 10-demicube Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope.
  3. Demihypercube D10 family: [37,1,1] - Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
    • 767 uniform 10-polytopes as permutations of rings in the group diagram, including:
      1. 17,1 - 10-demicube or demidekeract - Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
      2. 71,1 - 10-orthoplex - Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope

The A10 family

The A10 family has symmetry of order 39,916,800 (11 factorial).

There are 512+16-1=527 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. 31 are shown below: all one and two ringed forms, and the final omnitruncated form. Bowers-style acronym names are given in parentheses for cross-referencing.

# Graph Coxeter-Dynkin diagram
Schläfli symbol
Name 		Element counts
9-faces 8-faces 7-faces 6-faces 5-faces 4-faces Cells Faces Edges Vertices
1 Uniform 10-Polytope

Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t0{3,3,3,3,3,3,3,3,3}
10-simplex (ux)

11 55 165 330 462 462 330 165 55 11
2 Uniform 10-Polytope

Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t1{3,3,3,3,3,3,3,3,3}
Rectified 10-simplex (ru)

495 55
3 Uniform 10-Polytope

Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t2{3,3,3,3,3,3,3,3,3}
Birectified 10-simplex (bru)

1980 165
4 Uniform 10-Polytope

Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t3{3,3,3,3,3,3,3,3,3}
Trirectified 10-simplex (tru)

4620 330
5 Uniform 10-Polytope

Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t4{3,3,3,3,3,3,3,3,3}
Quadrirectified 10-simplex (teru)

6930 462
6 Uniform 10-Polytope

Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t0,1{3,3,3,3,3,3,3,3,3}
Truncated 10-simplex (tu)

550 110
7 Uniform 10-Polytope

Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t0,2{3,3,3,3,3,3,3,3,3}
Cantellated 10-simplex

4455 495
8 Uniform 10-Polytope

Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t1,2{3,3,3,3,3,3,3,3,3}
Bitruncated 10-simplex

2475 495
9 Uniform 10-Polytope

Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t0,3{3,3,3,3,3,3,3,3,3}
Runcinated 10-simplex

15840 1320
10 Uniform 10-Polytope

Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t1,3{3,3,3,3,3,3,3,3,3}
Bicantellated 10-simplex

17820 1980
11 Uniform 10-Polytope

Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t2,3{3,3,3,3,3,3,3,3,3}
Tritruncated 10-simplex

6600 1320
12 Uniform 10-Polytope

Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t0,4{3,3,3,3,3,3,3,3,3}
Stericated 10-simplex

32340 2310
13 Uniform 10-Polytope

Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t1,4{3,3,3,3,3,3,3,3,3}
Biruncinated 10-simplex

55440 4620
14 Uniform 10-Polytope

Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t2,4{3,3,3,3,3,3,3,3,3}
Tricantellated 10-simplex

41580 4620
15

Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t3,4{3,3,3,3,3,3,3,3,3}
Quadritruncated 10-simplex

11550 2310
16 Uniform 10-Polytope

Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t0,5{3,3,3,3,3,3,3,3,3}
Pentellated 10-simplex

41580 2772
17

Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t1,5{3,3,3,3,3,3,3,3,3}
Bistericated 10-simplex

97020 6930
18

Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t2,5{3,3,3,3,3,3,3,3,3}
Triruncinated 10-simplex

110880 9240
19 Uniform 10-Polytope

Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t3,5{3,3,3,3,3,3,3,3,3}
Quadricantellated 10-simplex

62370 6930
20

Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t4,5{3,3,3,3,3,3,3,3,3}
Quintitruncated 10-simplex

13860 2772
21 Uniform 10-Polytope

Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t0,6{3,3,3,3,3,3,3,3,3}
Hexicated 10-simplex

34650 2310
22

Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t1,6{3,3,3,3,3,3,3,3,3}
Bipentellated 10-simplex

103950 6930
23

Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t2,6{3,3,3,3,3,3,3,3,3}
Tristericated 10-simplex

161700 11550
24

Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t3,6{3,3,3,3,3,3,3,3,3}
Quadriruncinated 10-simplex

138600 11550
25 Uniform 10-Polytope

Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t0,7{3,3,3,3,3,3,3,3,3}
Heptellated 10-simplex

18480 1320
26

Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t1,7{3,3,3,3,3,3,3,3,3}
Bihexicated 10-simplex

69300 4620
27

Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t2,7{3,3,3,3,3,3,3,3,3}
Tripentellated 10-simplex

138600 9240
28 Uniform 10-Polytope

Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t0,8{3,3,3,3,3,3,3,3,3}
Octellated 10-simplex

5940 495
29

Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t1,8{3,3,3,3,3,3,3,3,3}
Biheptellated 10-simplex

27720 1980
30 Uniform 10-Polytope

Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t0,9{3,3,3,3,3,3,3,3,3}
Ennecated 10-simplex

990 110
31 Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t0,1,2,3,4,5,6,7,8,9{3,3,3,3,3,3,3,3,3}
Omnitruncated 10-simplex 		199584000 39916800

The B10 family

There are 1023 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings.

Twelve cases are shown below: ten single-ring (rectified) forms, and two truncations. Bowers-style acronym names are given in parentheses for cross-referencing.

# Graph Coxeter-Dynkin diagram
Schläfli symbol
Name 		Element counts
9-faces 8-faces 7-faces 6-faces 5-faces 4-faces Cells Faces Edges Vertices
1 Uniform 10-Polytope Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t0{4,3,3,3,3,3,3,3,3}
10-cube (deker) 		20 180 960 3360 8064 13440 15360 11520 5120 1024
2 Uniform 10-Polytope Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t0,1{4,3,3,3,3,3,3,3,3}
Truncated 10-cube (tade) 		51200 10240
3 Uniform 10-Polytope Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t1{4,3,3,3,3,3,3,3,3}
Rectified 10-cube (rade) 		46080 5120
4 Uniform 10-Polytope Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t2{4,3,3,3,3,3,3,3,3}
Birectified 10-cube (brade) 		184320 11520
5 Uniform 10-Polytope Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t3{4,3,3,3,3,3,3,3,3}
Trirectified 10-cube (trade) 		322560 15360
6 Uniform 10-Polytope Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t4{4,3,3,3,3,3,3,3,3}
Quadrirectified 10-cube (terade) 		322560 13440
7 Uniform 10-Polytope Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t4{3,3,3,3,3,3,3,3,4}
Quadrirectified 10-orthoplex (terake) 		201600 8064
8 Uniform 10-Polytope Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t3{3,3,3,3,3,3,3,4}
Trirectified 10-orthoplex (trake) 		80640 3360
9 Uniform 10-Polytope Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t2{3,3,3,3,3,3,3,3,4}
Birectified 10-orthoplex (brake) 		20160 960
10 Uniform 10-Polytope Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t1{3,3,3,3,3,3,3,3,4}
Rectified 10-orthoplex (rake) 		2880 180
11 Uniform 10-Polytope Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t0,1{3,3,3,3,3,3,3,3,4}
Truncated 10-orthoplex (take) 		3060 360
12 Uniform 10-Polytope Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
t0{3,3,3,3,3,3,3,3,4}
10-orthoplex (ka) 		1024 5120 11520 15360 13440 8064 3360 960 180 20

The D10 family

The D10 family has symmetry of order 1,857,945,600 (10 factorial × 29).

This family has 3×256−1=767 Wythoffian uniform polytopes, generated by marking one or more nodes of the D10 Coxeter-Dynkin diagram. Of these, 511 (2×256−1) are repeated from the B10 family and 256 are unique to this family, with 2 listed below. Bowers-style acronym names are given in parentheses for cross-referencing.

# Graph Coxeter-Dynkin diagram
Schläfli symbol
Name 		Element counts
9-faces 8-faces 7-faces 6-faces 5-faces 4-faces Cells Faces Edges Vertices
1 Uniform 10-Polytope Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
10-demicube (hede) 		532 5300 24000 64800 115584 142464 122880 61440 11520 512
2 Uniform 10-Polytope Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
Truncated 10-demicube (thede) 		195840 23040

Regular and uniform honeycombs

There are four fundamental affine Coxeter groups that generate regular and uniform tessellations in 9-space:

# Coxeter group Coxeter-Dynkin diagram
1 [3[10]] Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
2 [4,37,4] Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
3 h[4,37,4]
[4,36,31,1]		 Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope
4 q[4,37,4]
[31,1,35,31,1]		 Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope

Regular and uniform tessellations include:

Regular and uniform hyperbolic honeycombs

There are no compact hyperbolic Coxeter groups of rank 10, groups that can generate honeycombs with all finite facets, and a finite vertex figure. However, there are 3 paracompact hyperbolic Coxeter groups of rank 9, each generating uniform honeycombs in 9-space as permutations of rings of the Coxeter diagrams.

= [31,1,34,32,1]:
Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope 		= [4,35,32,1]:
Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope 		or = [36,2,1]:
Uniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-PolytopeUniform 10-Polytope

Three honeycombs from the family, generated by end-ringed Coxeter diagrams are:

References

  • T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
  • A. Boole Stott: Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
  • H.S.M. Coxeter:
    • H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller: Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, Londne, 1954
    • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
    • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
  • Klitzing, Richard. "10D uniform polytopes (polyxenna)".
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

Tags:

Uniform 10-Polytope Regular 10-polytopesUniform 10-Polytope Euler characteristicUniform 10-Polytope Uniform 10-polytopes by fundamental Coxeter groupsUniform 10-Polytope The A10 familyUniform 10-Polytope The B10 familyUniform 10-Polytope The D10 familyUniform 10-Polytope Regular and uniform honeycombsUniform 10-Polytope8-polytope9-polytopeFacet (mathematics)GeometryPolytopeRidge (geometry)

🔥 Trending searches on Wiki English:

Bernard HillJalen BrunsonThe Heart Part 6Nick Young (basketball)John Swinney2024 Indian general election in KarnatakaMurder of Reena VirkYouTube ShortsKirsten DunstWiki FoundationEvil (TV series)Laura LinneyTim ConnellyAyo EdebiriDonte DiVincenzoSanju SamsonShōgun (2024 miniseries)Narendra ModiDana WhiteAnimal (2023 Indian film)Katrina LawYoung SheldonAzerbaijanMalin AkermanUnder the Bridge (TV series)Glen PowellThe Big Bang TheoryDead Boy Detectives (TV series)Tyla (South African singer)Ipswich Town F.C.FascismKaley CuocoMarianne BachmeierList of European Cup and UEFA Champions League finalsAlgebraic notation (chess)Sydney SweeneyHoist with his own petardDemi MooreBob Hearts AbisholaMurder of XXXTentacionJapanSexual intercourseCallum TurnerList of massacres in China2024–25 UEFA Champions LeagueKaty PerryMichael Avenatti2024 Indian general election in GujaratGeneration XRajasthan RoyalsYouTube (YouTube channel)List of presidents of the United StatesRiley KeoughASAP RockyThe Family StarThe Gentlemen (2024 TV series)Push Ups (song)UEFA Euro 2024Anya Taylor-JoyJenna OrtegaHelldivers 2Gustav Magnar WitzøeBaby ReindeerChinaRick CarlisleFallout (video game)Ottoman EmpireRaindrop cakeBen AffleckTanga LoaElectronic voting in IndiaMatthew TkachukThe Garfield MoviePhoebe DynevorJake Paul vs. Mike TysonRusso-Ukrainian WarBenny BlancoShaitaan (2024 film)Wrexham A.F.C.🡆 More