9-polytope
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In geometry, a nine-dimensional polytope, or 9-polytope, is a polytope in 9-dimensional space. Each 7-polytope ridge being shared by exactly two 8-polytope facets.
A proposed name for 9-polytope is polyyotton (plural: polyyotta), created from poly-, yotta- and -on.
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[edit] Regular forms
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 3 finite regular 9-polytopes:
- {3,3,3,3,3,3,3,3} - 9-simplex or deca-9-tope or decayotton -
- {4,3,3,3,3,3,3,3} - enneract or 9-hypercube -
- {3,3,3,3,3,3,3,4} - enneacross or 9-orthoplex -
[edit] Regular and uniform honeycombs
There's one regular honeycomb of Euclidean 8-space, the 8-hypercubic honeycomb, with symbols {4,36,4}, = 
There's one semiregular honeycomb (with regular polytope facets) called E8 lattice, discovered by Thorold Gosset, constructed in 8-space with E8 polytope vertex figure and octacross and 8-simplex facets.
There's also an alternated 8-hypercubic honeycomb, a uniform tessellation, with symbols h{4,36,4}, 
=
[edit] See also
- List of regular polytopes#Higher dimensions
- polygon
- polyhedron
- polychoron
- 5-polytope
- 6-polytope
- 7-polytope
- 8-polytope
- 10-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
