# Uniform 10-polytope

Short description: Type of geometrical object
 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
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}

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}

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}

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}

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
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}
51200 10240
3
t1{4,3,3,3,3,3,3,3,3}
46080 5120
4
t2{4,3,3,3,3,3,3,3,3}
184320 11520
5
t3{4,3,3,3,3,3,3,3,3}
322560 15360
6
t4{4,3,3,3,3,3,3,3,3}
322560 13440
7
t4{3,3,3,3,3,3,3,3,4}
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 × 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 $\displaystyle{ {\tilde{A}}_9 }$ [3[10]]
2 $\displaystyle{ {\tilde{B}}_9 }$ [4,37,4]
3 $\displaystyle{ {\tilde{C}}_9 }$ h[4,37,4]
[4,36,31,1]
4 $\displaystyle{ {\tilde{D}}_9 }$ 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 paracompact hyperbolic Coxeter groups of rank 9, each generating uniform honeycombs in 9-space as permutations of rings of the Coxeter diagrams.

 $\displaystyle{ {\bar{Q}}_9 }$ = [31,1,34,32,1]: $\displaystyle{ {\bar{S}}_9 }$ = [4,35,32,1]: $\displaystyle{ E_{10} }$ or $\displaystyle{ {\bar{T}}_9 }$ = [36,2,1]:

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

• 621 honeycomb:
• 261 honeycomb:
• 162 honeycomb:

## References

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