Uniform 10-polytope explained

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, with x 9-polytope facets around each peak.

There are exactly three such convex regular 10-polytopes:

1. - 10-simplex
2. - 10-cube
3. - 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	[3<sup>9</sup>]
2	B10	[4,3<sup>8</sup>]
3	D10	[3<sup>7,1,1</sup>]

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. - 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. - 10-cube or dekeract -
     2. - 10-orthoplex or decacross -
     3. h - 10-demicube .
3. Demihypercube D₁₀ family: [3^7,1,1] -
   • 767 uniform 10-polytopes as permutations of rings in the group diagram, including:
     1. 1_7,1 - 10-demicube or demidekeract -
     2. 7_1,1 - 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 10-simplex (ux)	11	55	165	330	462	462	330	165	55	11
2		t1 Rectified 10-simplex (ru)									495	55
3		t2 Birectified 10-simplex (bru)									1980	165
4		t3 Trirectified 10-simplex (tru)									4620	330
5		t4 Quadrirectified 10-simplex (teru)									6930	462
6		t0,1 Truncated 10-simplex (tu)									550	110
7		t0,2 Cantellated 10-simplex									4455	495
8		t1,2 Bitruncated 10-simplex									2475	495
9		t0,3 Runcinated 10-simplex									15840	1320
10		t1,3 Bicantellated 10-simplex									17820	1980
11		t2,3 Tritruncated 10-simplex									6600	1320
12		t0,4 Stericated 10-simplex									32340	2310
13		t1,4 Biruncinated 10-simplex									55440	4620
14		t2,4 Tricantellated 10-simplex									41580	4620
15		t3,4 Quadritruncated 10-simplex									11550	2310
16		t0,5 Pentellated 10-simplex									41580	2772
17		t1,5 Bistericated 10-simplex									97020	6930
18		t2,5 Triruncinated 10-simplex									110880	9240
19		t3,5 Quadricantellated 10-simplex									62370	6930
20		t4,5 Quintitruncated 10-simplex									13860	2772
21		t0,6 Hexicated 10-simplex									34650	2310
22		t1,6 Bipentellated 10-simplex									103950	6930
23		t2,6 Tristericated 10-simplex									161700	11550
24		t3,6 Quadriruncinated 10-simplex									138600	11550
25		t0,7 Heptellated 10-simplex									18480	1320
26		t1,7 Bihexicated 10-simplex									69300	4620
27		t2,7 Tripentellated 10-simplex									138600	9240
28		t0,8 Octellated 10-simplex									5940	495
29		t1,8 Biheptellated 10-simplex									27720	1980
30		t0,9 Ennecated 10-simplex									990	110
31		t0,1,2,3,4,5,6,7,8,9 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 10-cube (deker)	20	180	960	3360	8064	13440	15360	11520	5120	1024
2		t0,1 Truncated 10-cube (tade)									51200	10240
3		t1 Rectified 10-cube (rade)									46080	5120
4		t2 Birectified 10-cube (brade)									184320	11520
5		t3 Trirectified 10-cube (trade)									322560	15360
6		t4 Quadrirectified 10-cube (terade)									322560	13440
7		t4 Quadrirectified 10-orthoplex (terake)									201600	8064
8		t3 Trirectified 10-orthoplex (trake)									80640	3360
9		t2 Birectified 10-orthoplex (brake)									20160	960
10		t1 Rectified 10-orthoplex (rake)									2880	180
11		t0,1 Truncated 10-orthoplex (take)									3060	360
12		t0 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<sup>[10]]
2	{\tilde{B}} _{9}	[4,3<sup>7</sup>,4]
3	{\tilde{C}} _{9}	h[4,3<sup>7</sup>,4] [4,3<sup>6</sup>,3<sup>1,1</sup>]
4	{\tilde{D}} _{9}	q[4,3<sup>7</sup>,4] [3<sup>1,1</sup>,3<sup>5</sup>,3<sup>1,1</sup>]

Regular and uniform tessellations include:

• Regular 9-hypercubic honeycomb, with symbols,
• Uniform alternated 9-hypercubic honeycomb with symbols h,

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.

{\bar{Q}} _{9} = [3<sup>1,1</sup>,3<sup>4</sup>,3<sup>2,1</sup>]:

{\bar{S}} _{9} = [4,3<sup>5</sup>,3<sup>2,1</sup>]:

E _{10} or {\bar{T}} _{9} = [3<sup>6,2,1</sup>]:

Three honeycombs from the

E _{10} family, generated by end-ringed Coxeter diagrams are:

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

References

• 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, 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

External links

• Polytope names
• Polytopes of Various Dimensions, Jonathan Bowers
• Multi-dimensional Glossary

Notes and References

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