6-polytope

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In geometry, a six-dimensional polytope, or 6-polytope, is a polytope in 6-dimensional space. Each polychoral ridge being shared by exactly two 5-polytope facets.

A proposed name for 6-polytope is polypeton (plural: polypeta), created from poly-, peta- and -on.

Contents 1 Regular and Uniform 6-polytopes by fundamental Coxeter groups 2 Uniform prismatic forms 3 See also 4 References 5 External links

[edit] Regular and Uniform 6-polytopes by fundamental Coxeter groups

Regular 6-polytopes can be generated from Coxeter groups represented by the Schläfli symbol {p,q,r,s,t} with t {p,q,r,s} polyteron facets around each cell.

Uniform 6-polytopes can be generated by fundamental finite Coxeter groups and represented by permutations of rings of the Coxeter-Dynkin diagrams.

There are four fundamental finite Coxeter groups that generate regular and uniform 6-polytopes, two linear and two bifurcating:

1. Simplex A₆ family: [3,3,3,3,3] -
   • 35 uniform 6-polytope as permutations of rings in the group diagram, including one regular:
     1. {3,3,3,3,3} - heptapeton, 6-simplex,
        • It has 7 vertices, 21 edges, 35 faces, 35 cells, 21 4-faces, and 7 5-faces. All elements are simplexes.
2. Hypercube/orthoplex C₆ family: [4,3,3,3,3] -
   • 63 uniform truncations as permutations of rings in the group diagram, including two regular ones:
     1. {4,3,3,3,3} - hexeract or 6-hypercube
        • It has 64 vertices, 192 edges, 240 faces, 160 cells, 70 4-faces, and 12 5-faces. All elements are hypercubes.
     2. {3,3,3,3,4} - hexacross or 6-orthoplex
        • It has 12 vertices, 60 edges, 160 faces, 240 cells, 192 4-faces, 64 5-faces. All elements are simplexes.
3. Demihypercube B₆ family: [3^3,1,1] -
   • 46 uniform 6-polytopes as permutations of rings in the group diagram, including one from the demihypercube family:
     1. {3^3,1,1} - demihexeract. ; also as h{4,3,3,3,3}
        • It has 32 vertices, 240 edges, 640 faces, 640 cells, 252 4-faces, and 44 5-faces. The facets are: 12 demipenteract and 32 5-simplexes.
4. Semiregular E₆ family: [3^2,2,1] -
   • 39 uniform 6-polytope as permutations of rings in the group diagram, including one semiregular:
     1. {3^2,2,1} - , Thorold Gosset's semiregular E6 polytope, 2_2,1:
        • It has 27 vertices, 216 edges, 720 faces, 1080 cells, 648 4-faces, and 99 5-faces. The 99 regular facets are of two types: 27 pentacrosses and 72 5-simplexes.

[edit] Uniform prismatic forms

There are 18 categorical uniform prismatic forms based on Cartesian products of lower dimensional uniform polytopes. Each combination of at least one ring on every connected group produces a uniform prismatic 6-polytope.

• 1. A₅xA₁: [3,3,3,3] x [ ] -
  2. C₅xA₁:[4,3,3,3] x [ ] -
  3. B₅xA₁: [3^2,1,1] x [ ] -
  4. A₄xD₂^p: [3,3,3]x[p] -
  5. C₄xD₂^p: [4,3,3]x[p] -
  6. F₄xD₂^p: [3,4,3]x[p] -
  7. G₄xD₂^p: [5,3,3]x[p] -
  8. B₄xD₂^p: [3^1,1,1] x [p] -
  9. A₃xA₃: [3,3]x[3,3] -
  10. A₃xC₃: [3,3]x[4,3] -
  11. A₃xG₃: [3,3]x[5,3] -
  12. C₃xC₃: [4,3]x[4,3] -
  13. C₃xG₃: [4,3]x[5,3] -
  14. G₃xA₃: [5,3]x[5,3] -
  15. A₃xD₂^qxA₁: [3,3] x [p] x [ ] -
  16. C₃xD₂^qxA₁: [4,3] x [p] x [ ] -
  17. G₃xD₂^qxA₁: [5,3] x [p] x [ ] -
  18. D₂^pxD₂^qxD₂^r: [p] x [q] x [r] -

[edit] See also

• List of regular polytopes#Higher dimensions
• polygon
• polyhedron
• polychoron
• 5-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
• Multi-dimensional Glossary
• Glossary for hyperspace