10-polytope

10-simplex (Hendecaxennon)	10-orthoplex (Decacross)	10-cube (Dekeract)	10-demicube (Demidekeract)
Graphs of three regular 10-polytopes and one semiregular form.

In geometry, a ten-dimensional polytope, or 10-polytope, is a polytope in 10-dimensional space, each 8-polytope ridge being shared by exactly two 9-polytope facets.

A proposed name for 10-polytope is polyxennon (plural: polyxenna), created from poly- xenna (a variation on ennea meaning nine) and -on.

[edit] Regular and uniform 10-polytopes by fundamental Coxeter groups

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.

Regular and 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:
     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. Demihypercube D₁₀ family: [3^7,1,1] -
   • 767 uniform 10-polytopes as permutations of rings in the group diagram, including:
     1. {3^1,7,1} - 10-demicube or demidekeract, 1_7,1 - ; also as h{4,3⁸} .
     2. {3^7,1,1} - decacross, 7_1,1 -

[edit] 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	A~9	p[310]
2	B~9	[4,37,4]
3	C~9	h[4,37,4] [4,36,31,1]
4	D~9	q[4,37,4] [31,1,35,31,1]

Regular and uniform tessellations include:

• Regular 9-hypercubic honeycomb, with symbols {4,3⁷,4}, =
• Uniform alternated 8-hypercubic honeycomb with symbols h{4,3⁷,4}, =

[edit] See also

• List of regular polytopes#Higher dimensions
• polygon
• polyhedron
• polychoron
• 5-polytope
• 6-polytope
• 7-polytope
• 8-polytope
• 9-polytope

[edit] 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, editied 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

[edit] External links

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