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.

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

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-faces8-faces7-faces6-faces5-faces4-facesCellsFacesEdgesVertices
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)

49555
3


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

1980165
4


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

4620330
5


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

6930462
6


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

550110
7


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

4455495
8


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

2475495
9


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

158401320
10


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

178201980
11


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

66001320
12


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

323402310
13


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

554404620
14


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

415804620
15


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

115502310
16


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

415802772
17


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

970206930
18


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

1108809240
19


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

623706930
20


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

138602772
21


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

346502310
22


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

1039506930
23


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

16170011550
24


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

13860011550
25


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

184801320
26


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

693004620
27


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

1386009240
28


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

5940495
29


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

277201980
30


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

990110
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)
19584023040

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]]
2[4,37,4]
3h[4,37,4]
[4,36,31,1]
4q[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.

= [31,1,34,32,1]:
= [4,35,32,1]:
or = [36,2,1]:

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

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
    • (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 OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 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
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