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Uniform 10-polytope

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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.

Contents

  • Regular 10-polytopes 1
  • Euler characteristic 2
  • Uniform 10-polytopes by fundamental Coxeter groups 3
  • The A10 family 4
  • The B10 family 5
  • The D10 family 6
  • Regular and uniform honeycombs 7
    • Regular and uniform hyperbolic honeycombs 7.1
  • References 8
  • External links 9

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:
    • {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:
    • {4,38} - 10-cube or dekeract -
    • {38,4} - 10-orthoplex or decacross -
    • 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:
    • 17,1 - 10-demicube or demidekeract -
    • 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}
    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}
    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 × 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

  4. ^ a b c 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
    • Richard Klitzing, 10D, uniform polytopes (polyxenna)

    External links

    • Polytope names
    • Polytopes of Various Dimensions, Jonathan Bowers
    • Multi-dimensional Glossary
    • Glossary for hyperspace, George Olshevsky.
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