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9-polytope

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.

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 -

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}, =

See also

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

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

External links

• Polytope names
• Polytopes of Various Dimensions
• Multi-dimensional Glossary
• Glossary for hyperspace