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Uniform 9-polytope

Type of geometric object

Uniform 9-polytope

Type of geometric object

[[File:9-demicube.svg150px]]
[9-demicube](9-demicube)[[File:Truncated 9-demicube.png150px]]
Truncated 9-demicube

In nine-dimensional geometry, a nine-dimensional polytope or 9-polytope is a polytope contained by 8-polytope facets. Each 7-polytope ridge being shared by exactly two 8-polytope facets.

A uniform 9-polytope is one which is vertex-transitive, and constructed from uniform 8-polytope facets.

Regular 9-polytopes

Regular 9-polytopes can be represented by the Schläfli symbol {p,q,r,s,t,u,v,w}, with w {p,q,r,s,t,u,v} 8-polytope facets around each peak.

There are exactly three such convex regular 9-polytopes:

  1. {3,3,3,3,3,3,3,3} - 9-simplex
  2. {4,3,3,3,3,3,3,3} - 9-cube
  3. {3,3,3,3,3,3,3,4} - 9-orthoplex

There are no nonconvex regular 9-polytopes.

Euler characteristic

The topology of any given 9-polytope is defined by its Betti numbers and torsion coefficients.

The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, 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.

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.

Uniform 9-polytopes by fundamental Coxeter groups

Uniform 9-polytopes with reflective symmetry can be generated by these three Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagrams:

Coxeter groupCoxeter-Dynkin diagram
A9[38]
B9[4,37]
D9[36,1,1]

Selected regular and uniform 9-polytopes from each family include:

  • Simplex family: A9 [38] -
    • 271 uniform 9-polytopes as permutations of rings in the group diagram, including one regular:
      1. {38} - 9-simplex or deca-9-tope or decayotton -
  • Hypercube/orthoplex family: B9 [4,38] -
    • 511 uniform 9-polytopes as permutations of rings in the group diagram, including two regular ones:
      1. {4,37} - 9-cube or enneract -
      2. {37,4} - 9-orthoplex or enneacross -
  • Demihypercube D9 family: [36,1,1] -
    • 383 uniform 9-polytope as permutations of rings in the group diagram, including:
      1. {31,6,1} - 9-demicube or demienneract, 161 - ; also as h{4,38} .
      2. {36,1,1} - 9-orthoplex, 611 -

The A9 family

The A9 family has symmetry of order 3628800 (10 factorial).

There are 256+16-1=271 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. These are all enumerated below. Bowers-style acronym names are given in parentheses for cross-referencing.

#GraphCoxeter-Dynkin diagram
Schläfli symbol
NameElement counts123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186187188189190191192193194195196197198199200201202203204205206207208209210211212213214215216217218219220221222223224225226227228229230231232233234235236237238239240241242243244245246247248249250251252253254255256257258259260261262263264265266267268269270271
8-faces7-faces6-faces5-faces4-facesCellsFacesEdgesVertices
[[File:9-simplex t0.svg60px]]10451202102522101204510
[[File:9-simplex t1.svg60px]]36045
[[File:9-simplex t2.svg60px]]1260120
[[File:9-simplex t3.svg60px]]2520210
[[File:9-simplex t4.svg60px]]3150252
[[File:9-simplex t01.svg60px]]40590
[[File:9-simplex t02.svg60px]]2880360
[[File:9-simplex t12.svg60px]]1620360
[[File:9-simplex t03.svg60px]]8820840
[[File:9-simplex t13.svg60px]]100801260
[[File:9-simplex t23.svg60px]]3780840
[[File:9-simplex t04.svg60px]]151201260
[[File:9-simplex t14.svg60px]]264602520
[[File:9-simplex t24.svg60px]]201602520
56701260
[[File:9-simplex t05.svg60px]]157501260
378003150
441004200
252003150
[[File:9-simplex t06.svg60px]]10080840
315002520
504004200
[[File:9-simplex t07.svg60px]]3780360
151201260
[[File:9-simplex t08.svg60px]]72090
[[File:9-simplex t012.svg60px]]3240720
189002520
126002520
[[File:9-simplex t123.svg60px]]113402520
478805040
604807560
529207560
277205040
415807560
[[File:9-simplex t234.svg60px]]226805040
661506300
12600012600
10710012600
10710012600
15120018900
8190012600
378006300
8190012600
7560012600
[[File:9-simplex t345.svg60px]]283506300
529205040
13860012600
11340012600
17640016800
23940025200
12600016800
11340012600
22680025200
20160025200
327605040
9450012600
239402520
831607560
642607560
14490012600
18900018900
13860012600
26460025200
718207560
176402520
5400720
252002520
579605040
756006300
226805040
10584015120
7560015120
7560015120
6804015120
21420025200
28350037800
26460037800
24570037800
13860025200
22680037800
18900037800
13860025200
20790037800
11340025200
22680025200
45360050400
40320050400
37800050400
40320050400
60480075600
52920075600
35280050400
52920075600
30240050400
15120025200
35280050400
27720050400
35280050400
49140075600
25200050400
15120025200
32760050400
12852015120
35910037800
30240037800
28350037800
47880050400
68040075600
60480075600
37800050400
56700075600
32130037800
68040075600
56700075600
64260075600
907200113400
26460037800
9828015120
30240037800
22680037800
42840050400
30240037800
9828015120
352805040
13608015120
10584015120
25200025200
34020037800
17640025200
25200025200
50400050400
45360050400
13608015120
37800037800
352805040
13608030240
49140075600
37800075600
37800075600
37800075600
34020075600
756000100800
1058400151200
982800151200
982800151200
907200151200
554400100800
907200151200
831600151200
756000151200
554400100800
907200151200
756000151200
554400100800
831600151200
453600100800
56700075600
1209600151200
1058400151200
1058400151200
982800151200
1134000151200
1701000226800
1587600226800
1474200226800
982800151200
1587600226800
1360800226800
982800151200
1474200226800
45360075600
1058400151200
907200151200
831600151200
1058400151200
1587600226800
1360800226800
907200151200
45360075600
1058400151200
1058400151200
45360075600
19656030240
60480075600
49140075600
49140075600
856800100800
1209600151200
1134000151200
655200100800
1058400151200
655200100800
60480075600
1285200151200
1134000151200
1209600151200
1814400226800
49140075600
19656030240
60480075600
856800100800
680400151200
1814400302400
1512000302400
1512000302400
1512000302400
1512000302400
1360800302400
1965600302400
2948400453600
2721600453600
2721600453600
2721600453600
2494800453600
1663200302400
2721600453600
2494800453600
2494800453600
2268000453600
1663200302400
2721600453600
2494800453600
2268000453600
1663200302400
2721600453600
1663200302400
907200151200
2116800302400
1814400302400
1814400302400
1814400302400
2116800302400
3175200453600
2948400453600
2948400453600
1814400302400
2948400453600
2721600453600
1814400302400
907200151200
2116800302400
1814400302400
2116800302400
3175200453600
907200151200
2721600604800
4989600907200
4536000907200
4536000907200
4536000907200
4536000907200
4536000907200
4082400907200
3326400604800
5443200907200
4989600907200
4989600907200
4989600907200
4989600907200
3326400604800
5443200907200
4989600907200
4989600907200
3326400604800
5443200907200
81648001814400
90720001814400
90720001814400
90720001814400
90720001814400
163296003628800

The B9 family

There are 511 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings.

Eleven cases are shown below: Nine rectified forms and 2 truncations. Bowers-style acronym names are given in parentheses for cross-referencing.

#GraphCoxeter-Dynkin diagram
Schläfli symbol
NameElement counts8-faces7-faces6-faces5-faces4-facesCellsFacesEdgesVertices1234567891011
[[File:9-cube.svg60px]]
t0{4,3,3,3,3,3,3,3}
[9-cube](9-cube) (enne)1814467220164032537646082304512
[[File:Truncated 9-cube.png60px]]
t0,1{4,3,3,3,3,3,3,3}
Truncated 9-cube (ten)23044608
[[File:Rectified 9-cube.png60px]]
t1{4,3,3,3,3,3,3,3}
Rectified 9-cube (ren)184322304
[[File:Birectified 9-cube.png60px]]
t2{4,3,3,3,3,3,3,3}
Birectified 9-cube (barn)645124608
[[File:Quintirectified 9-orthoplex.png60px]]
t3{4,3,3,3,3,3,3,3}
Trirectified 9-cube (tarn)967685376
[[File:Quadrirectified 9-orthoplex.png60px]]
t4{4,3,3,3,3,3,3,3}
Quadrirectified 9-cube (nav)
(Quadrirectified 9-orthoplex)806404032
[[File:Trirectified 9-orthoplex.png60px]]
t3{3,3,3,3,3,3,3,4}
Trirectified 9-orthoplex (tarv)403202016
[[File:Birectified 9-orthoplex.png60px]]
t2{3,3,3,3,3,3,3,4}
Birectified 9-orthoplex (brav)12096672
[[File:Rectified heptacross.png60px]]
t1{3,3,3,3,3,3,3,4}
Rectified 9-orthoplex (riv)2016144
[[File:Truncated 9-orthoplex.png60px]]
t0,1{3,3,3,3,3,3,3,4}
Truncated 9-orthoplex (tiv)2160288
[[File:Cross graph 9.png60px]]
t0{3,3,3,3,3,3,3,4}
[9-orthoplex](9-orthoplex) (vee)5122304460853764032201667214418

The D9 family

The D9 family has symmetry of order 92,897,280 (9 factorial × 28).

This family has 3×128−1=383 Wythoffian uniform polytopes, generated by marking one or more nodes of the D9 Coxeter-Dynkin diagram. Of these, 255 (2×128−1) are repeated from the B9 family and 128 are unique to this family, with the eight 1 or 2 ringed forms listed below. Bowers-style acronym names are given in parentheses for cross-referencing.

#Coxeter plane graphsCoxeter-Dynkin diagramSchläfli symbolBase point(Alternately signed)Element countsCircumradB9D9D8D7D6D5D4D3A7A5A387654321012345678
[[File:9-demicube t0 B9.svg60px]][[File:9-demicube t0 D9.svg60px]][[File:9-demicube t0 D8.svg60px]][[File:9-demicube t0 D7.svg60px]][[File:9-demicube t0 D6.svg60px]][[File:9-demicube t0 D5.svg60px]][[File:9-demicube t0 D4.svg60px]][[File:9-demicube t0 D3.svg60px]][[File:9-demicube t0 A7.svg60px]][[File:9-demicube t0 A5.svg60px]][[File:9-demicube t0 A3.svg60px]][9-demicube](9-demicube) (henne)(1,1,1,1,1,1,1,1,1)2742448
[[File:9-demicube t01 B9.svg60px]][[File:9-demicube t01 D9.svg60px]][[File:9-demicube t01 D8.svg60px]][[File:9-demicube t01 D7.svg60px]][[File:9-demicube t01 D6.svg60px]][[File:9-demicube t01 D5.svg60px]][[File:9-demicube t01 D4.svg60px]][[File:9-demicube t01 D3.svg60px]][[File:9-demicube t01 A7.svg60px]][[File:9-demicube t01 A5.svg60px]][[File:9-demicube t01 A3.svg60px]]Truncated 9-demicube (thenne)(1,1,3,3,3,3,3,3,3)
[[File:9-demicube t02 B9.svg60px]][[File:9-demicube t02 D9.svg60px]][[File:9-demicube t02 D8.svg60px]][[File:9-demicube t02 D7.svg60px]][[File:9-demicube t02 D6.svg60px]][[File:9-demicube t02 D5.svg60px]][[File:9-demicube t02 D4.svg60px]][[File:9-demicube t02 D3.svg60px]][[File:9-demicube t02 A7.svg60px]][[File:9-demicube t02 A5.svg60px]][[File:9-demicube t02 A3.svg60px]]Cantellated 9-demicube(1,1,1,3,3,3,3,3,3)
[[File:9-demicube t03 B9.svg60px]][[File:9-demicube t03 D9.svg60px]][[File:9-demicube t03 D8.svg60px]][[File:9-demicube t03 D7.svg60px]][[File:9-demicube t03 D6.svg60px]][[File:9-demicube t03 D5.svg60px]][[File:9-demicube t03 D4.svg60px]][[File:9-demicube t03 D3.svg60px]][[File:9-demicube t03 A7.svg60px]][[File:9-demicube t03 A5.svg60px]][[File:9-demicube t03 A3.svg60px]]Runcinated 9-demicube(1,1,1,1,3,3,3,3,3)
[[File:9-demicube t04 B9.svg60px]][[File:9-demicube t04 D9.svg60px]][[File:9-demicube t04 D8.svg60px]][[File:9-demicube t04 D7.svg60px]][[File:9-demicube t04 D6.svg60px]][[File:9-demicube t04 D5.svg60px]][[File:9-demicube t04 D4.svg60px]][[File:9-demicube t04 D3.svg60px]][[File:9-demicube t04 A7.svg60px]][[File:9-demicube t04 A5.svg60px]][[File:9-demicube t04 A3.svg60px]]Stericated 9-demicube(1,1,1,1,1,3,3,3,3)
[[File:9-demicube t05 B9.svg60px]][[File:9-demicube t05 D9.svg60px]][[File:9-demicube t05 D8.svg60px]][[File:9-demicube t05 D7.svg60px]][[File:9-demicube t05 D6.svg60px]][[File:9-demicube t05 D5.svg60px]][[File:9-demicube t05 D4.svg60px]][[File:9-demicube t05 D3.svg60px]][[File:9-demicube t05 A7.svg60px]][[File:9-demicube t05 A5.svg60px]][[File:9-demicube t05 A3.svg60px]]Pentellated 9-demicube(1,1,1,1,1,1,3,3,3)
[[File:9-demicube t06 B9.svg60px]][[File:9-demicube t06 D9.svg60px]][[File:9-demicube t06 D8.svg60px]][[File:9-demicube t06 D7.svg60px]][[File:9-demicube t06 D6.svg60px]][[File:9-demicube t06 D5.svg60px]][[File:9-demicube t06 D4.svg60px]][[File:9-demicube t06 D3.svg60px]][[File:9-demicube t06 A7.svg60px]][[File:9-demicube t06 A5.svg60px]][[File:9-demicube t06 A3.svg60px]]Hexicated 9-demicube(1,1,1,1,1,1,1,3,3)
[[File:9-demicube t07 B9.svg60px]][[File:9-demicube t07 D9.svg60px]][[File:9-demicube t07 D8.svg60px]][[File:9-demicube t07 D7.svg60px]][[File:9-demicube t07 D6.svg60px]][[File:9-demicube t07 D5.svg60px]][[File:9-demicube t07 D4.svg60px]][[File:9-demicube t07 D3.svg60px]][[File:9-demicube t07 A7.svg60px]][[File:9-demicube t07 A5.svg60px]][[File:9-demicube t07 A3.svg60px]]Heptellated 9-demicube(1,1,1,1,1,1,1,1,3)

Regular and uniform honeycombs

Coxeter-Dynkin diagram correspondences between families and higher symmetry within diagrams. Nodes of the same color in each row represent identical mirrors. Black nodes are not active in the correspondence.

There are five fundamental affine Coxeter groups that generate regular and uniform tessellations in 8-space:

#Coxeter groupCoxeter diagramForms
1{\tilde{A}}_8[3[9]]
2{\tilde{C}}_8[4,36,4]
3{\tilde{B}}_8h[4,36,4]
[4,35,31,1]
4{\tilde{D}}_8q[4,36,4]
[31,1,34,31,1]
5{\tilde{E}}_8[35,2,1]

Regular and uniform tessellations include:

  • {\tilde{A}}_8 45 uniquely ringed forms
  • {\tilde{C}}_8 271 uniquely ringed forms
  • {\tilde{B}}_8: 383 uniquely ringed forms, 255 shared with {\tilde{C}}_8, 128 new
  • {\tilde{D}}_8, [31,1,34,31,1]: 155 unique ring permutations, and 15 are new, the first, , Coxeter called a quarter 8-cubic honeycomb, representing as q{4,36,4}, or qδ9.
  • {\tilde{E}}_8 511 forms

Regular and uniform hyperbolic honeycombs

There are no compact hyperbolic Coxeter groups of rank 9, groups that can generate honeycombs with all finite facets, and a finite vertex figure. However, there are 4 paracompact hyperbolic Coxeter groups of rank 9, each generating uniform honeycombs in 8-space as permutations of rings of the Coxeter diagrams.

{\bar{P}}_8 = [3,3[8]]:{\bar{Q}}_8 = [31,1,33,32,1]:{\bar{S}}_8 = [4,34,32,1]:{\bar{T}}_8 = [34,3,1]:

References

  • T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
  • 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, http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html
    • (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

References

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