Uniform 10-polytope

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.

The topology of any given 10-polytope is defined by its Betti numbers and torsion coefficients.Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy, Princeton, 2008.

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.

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

Selected regular and uniform 10-polytopes from each family include:

1. Simplex family: A₁₀ [3⁹] - {{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
2. * 527 uniform 10-polytopes as permutations of rings in the group diagram, including one regular:
3. *# {3⁹} - 10-simplex - {{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
4. Hypercube/orthoplex family: B₁₀ [4,3⁸] - {{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
5. * 1023 uniform 10-polytopes as permutations of rings in the group diagram, including two regular ones:
6. *# {4,3⁸} - 10-cube or dekeract - {{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
7. *# {3⁸,4} - 10-orthoplex or decacross - {{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node}}
8. *# h{4,3⁸} - 10-demicube {{CDD|node_h|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}.
9. Demihypercube D₁₀ family: [3^7,1,1] - {{CDD|nodes|split2|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
10. * 767 uniform 10-polytopes as permutations of rings in the group diagram, including:
11. *# 1_7,1 - 10-demicube or demidekeract - {{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
12. *# 7_1,1 - 10-orthoplex - {{CDD|nodes|split2|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}

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.

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.

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.

There are four fundamental affine Coxeter groups that generate regular and uniform tessellations in 9-space:

Regular and uniform tessellations include:

• Regular 9-hypercubic honeycomb, with symbols {4,3⁷,4}, {{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node}}
• Uniform alternated 9-hypercubic honeycomb with symbols h{4,3⁷,4}, {{CDD|node_h|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node}}

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.

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

• 6₂₁ honeycomb: {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea_1}}
• 2₆₁ honeycomb: {{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}
• 1₆₂ honeycomb: {{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}

{{reflist}}

• 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}} [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
• {{KlitzingPolytopes|polyxenna.htm|10D|uniform polytopes (polyxenna)}}

• [http://www.steelpillow.com/polyhedra/ditela.html Polytope names]
• [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions], Jonathan Bowers
• [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]
• {{PolyCell | urlname = glossary.html| title = Glossary for hyperspace}}

{{Polytopes}}

Category:10-polytopes