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
Pentallated 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 contained by 9-polytope facets. Each 8-polytope ridge being shared by exactly two 9-polytope facets.

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

A proposed name for 10-polytope is polyxennon (plural: polyxenna), created from poly- xenna (a variation on ennea meaning nine) and -on.

Contents

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 Euler characteristic for 10-polytopes that are topological 9-spheres (including all convex 10-polytopes) is zero. χ=V-E+F-C+f4-f5+f6-f7+f8-f9=0.

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]
2 B10 [4,38]
3 D10 [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)

 11 55 165 330 462 462 330 165 55 11
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}
ennicated 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		 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
t0{4,3,3,3,3,3,3,3,3}
10-cube (deker)		 20 180 960 3360 8064 13440 15360 11520 5120 1024
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)		 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 x 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)		 532 5300 24000 64800 115584 142464 122880 61440 11520 512
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

External links