Uniform 10-polytope

Graphs of three regular and related uniform polytopes.

10-simplex

Truncated 10-simplex

Rectified 10-simplex

Cantellated 10-simplex

Runcinated 10-simplex

Stericated 10-simplex

Pentellated 10-simplex

Hexicated 10-simplex

Heptellated 10-simplex

Octellated 10-simplex

Ennecated 10-simplex

10-orthoplex

Truncated 10-orthoplex

Rectified 10-orthoplex

10-cube

Truncated 10-cube

Rectified 10-cube

10-demicube

Truncated 10-demicube

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.

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.[1]

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.[1]

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.[1]

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
1A10 [39]
2B10[4,38]
3D10[37,1,1]

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

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

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


t0{3,3,3,3,3,3,3,3,3}
10-simplex (ux)

11551653304624623301655511
2


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

495 55
3


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

1980 165
4


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

4620 330
5


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

6930 462
6


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

550 110
7


t0,2{3,3,3,3,3,3,3,3,3}
Cantellated 10-simplex

4455 495
8


t1,2{3,3,3,3,3,3,3,3,3}
Bitruncated 10-simplex

2475 495
9


t0,3{3,3,3,3,3,3,3,3,3}
Runcinated 10-simplex

15840 1320
10


t1,3{3,3,3,3,3,3,3,3,3}
Bicantellated 10-simplex

17820 1980
11


t2,3{3,3,3,3,3,3,3,3,3}
Tritruncated 10-simplex

6600 1320
12


t0,4{3,3,3,3,3,3,3,3,3}
Stericated 10-simplex

32340 2310
13


t1,4{3,3,3,3,3,3,3,3,3}
Biruncinated 10-simplex

55440 4620
14


t2,4{3,3,3,3,3,3,3,3,3}
Tricantellated 10-simplex

41580 4620
15


t3,4{3,3,3,3,3,3,3,3,3}
Quadritruncated 10-simplex

11550 2310
16


t0,5{3,3,3,3,3,3,3,3,3}
Pentellated 10-simplex

41580 2772
17


t1,5{3,3,3,3,3,3,3,3,3}
Bistericated 10-simplex

97020 6930
18


t2,5{3,3,3,3,3,3,3,3,3}
Triruncinated 10-simplex

110880 9240
19


t3,5{3,3,3,3,3,3,3,3,3}
Quadricantellated 10-simplex

62370 6930
20


t4,5{3,3,3,3,3,3,3,3,3}
Quintitruncated 10-simplex

13860 2772
21


t0,6{3,3,3,3,3,3,3,3,3}
Hexicated 10-simplex

34650 2310
22


t1,6{3,3,3,3,3,3,3,3,3}
Bipentellated 10-simplex

103950 6930
23


t2,6{3,3,3,3,3,3,3,3,3}
Tristericated 10-simplex

161700 11550
24


t3,6{3,3,3,3,3,3,3,3,3}
Quadriruncinated 10-simplex

138600 11550
25


t0,7{3,3,3,3,3,3,3,3,3}
Heptellated 10-simplex

18480 1320
26


t1,7{3,3,3,3,3,3,3,3,3}
Bihexicated 10-simplex

69300 4620
27


t2,7{3,3,3,3,3,3,3,3,3}
Tripentellated 10-simplex

138600 9240
28


t0,8{3,3,3,3,3,3,3,3,3}
Octellated 10-simplex

5940 495
29


t1,8{3,3,3,3,3,3,3,3,3}
Biheptellated 10-simplex

27720 1980
30


t0,9{3,3,3,3,3,3,3,3,3}
Ennecated 10-simplex

990 110
31
t0,1,2,3,4,5,6,7,8,9{3,3,3,3,3,3,3,3,3}
Omnitruncated 10-simplex
19958400039916800

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
t0{4,3,3,3,3,3,3,3,3}
10-cube (deker)
201809603360806413440153601152051201024
2
t0,1{4,3,3,3,3,3,3,3,3}
Truncated 10-cube (tade)
51200 10240
3
t1{4,3,3,3,3,3,3,3,3}
Rectified 10-cube (rade)
46080 5120
4
t2{4,3,3,3,3,3,3,3,3}
Birectified 10-cube (brade)
184320 11520
5
t3{4,3,3,3,3,3,3,3,3}
Trirectified 10-cube (trade)
322560 15360
6
t4{4,3,3,3,3,3,3,3,3}
Quadrirectified 10-cube (terade)
322560 13440
7
t4{3,3,3,3,3,3,3,3,4}
Quadrirectified 10-orthoplex (terake)
201600 8064
8
t3{3,3,3,3,3,3,3,4}
Trirectified 10-orthoplex (trake)
80640 3360
9
t2{3,3,3,3,3,3,3,3,4}
Birectified 10-orthoplex (brake)
20160 960
10
t1{3,3,3,3,3,3,3,3,4}
Rectified 10-orthoplex (rake)
2880 180
11
t0,1{3,3,3,3,3,3,3,3,4}
Truncated 10-orthoplex (take)
3060 360
12
t0{3,3,3,3,3,3,3,3,4}
10-orthoplex (ka)
102451201152015360134408064336096018020

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
10-demicube (hede)
532530024000648001155841424641228806144011520512
2
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{\tilde {A}}_{9}[3[10]]
2{\tilde {B}}_{9}[4,37,4]
3{\tilde {C}}_{9}h[4,37,4]
[4,36,31,1]
4{\tilde {D}}_{9}q[4,37,4]
[31,1,35,31,1]

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 noncompact hyperbolic Coxeter groups of rank 9, each generating uniform honeycombs in 9-space as permutations of rings of the Coxeter diagrams.

{\bar {Q}}_{9} = [31,1,34,32,1]:
{\bar {S}}_{9} = [4,35,32,1]:
E_{10} or {\bar {T}}_{9} = [36,2,1]:

Three honeycombs from the E_{10} family, generated by end-ringed Coxeter diagrams are:

References

  1. 1 2 3 Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy, Princeton, 2008.

External links

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