7-polytope

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In geometry, a seven-dimensional polytope, or 7-polytope, is a polytope in 7-dimensional space. Each 5-polytope ridge being shared by exactly two 6-polytope facets.

A proposed name for 7-polytope is polyexon (plural: polyexa), created from poly-, exa- and -on.

Contents 1 Regular and uniform 7-polytopes by fundamental Coxeter groups 2 Uniform prismatic forms 3 See also 4 References 5 External links

[edit] Regular and uniform 7-polytopes by fundamental Coxeter groups

Regular 7-polytopes can be generated from Coxeter groups represented by the Schläfli symbol {p,q,r,s,t,u} with u {p,q,r,s,t} 6-polytopes facets around each 4-face.

Uniform 7-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 7-polytopes, two linear and two bifurcating:

1. Simplex A₇ family: [3,3,3,3,3,3] -
   • 71 uniform 7-polytopes as permutations of rings in the group diagram, including one regular:
     1. {3,3,3,3,3,3} - octaexon or 7-simplex,
        • It has 8 vertices, 28 edges, 56 faces, 70 cells, 56 4-faces, 28 5-faces, and 8 6-faces. All elements are simplexes.
2. Hypercube/orthoplex C₇ family: [4,3,3,3,3,3] -
   • 63 uniform 7-polytopes as permutations of rings in the group diagram, including two regular ones. There's also an alternated regular one.
     1. {4,3,3,3,3,3} - hepteract or 7-hypercube
        • It has 128 vertices, 448 edges, 672 faces, 560 cells, 280 4-faces, 84 5-faces, and 14 6-faces. All elements are hypercubes.
     2. {3,3,3,3,3,4} - heptacross or 7-orthoplex
        • It has 14 vertices, 84 edges, 280 faces, 560 cells, 672 4-faces, 448 5-faces, and 128 6-faces. All elements are simplexes.
     3. h{4,3,3,3,3,3} - demihepteract or 7-demicube
        • It has 64 vertices, 672 edges, 2240 faces, 2800 cells, 1624 hypercells, 532 5-faces, and 78 6-faces. The 6-face facets are: 14 demihexeract and 64 6-simplexes.
3. Demihypercube B₇ family: [3^4,1,1] -
   •  ? uniform 7-polytope as permutations of rings in the group diagram, including:
     1. {3^1,4,1} - demihepteract
     2. {3^4,1,1} - heptacross
4. Semiregular E₇ family:[3^3,2,1] -
   •  ? uniform 7-polytopes as permutations of rings in the group diagram, including one semiregular:
     1. {3^3,2,1} - , Thorold Gosset's semiregular E7 polytope, 3_2,1:
        • It has 56 vertices, 756 edges, 4032 faces, 10080 cells, 12096 4-faces, 6048 5-faces, and 702 6-faces. The 6-face facets are of two types: 126 hexacrosses and 576 6-simplexes.

[edit] Uniform prismatic forms

There are 37 uniform prismatic forms based on Cartesian products of lower dimensional uniform polytopes:

• 1. A₆xA₁: [3⁵] x [ ]
  2. C₆xA₁: [4,3⁴] x [ ]
  3. B₆xA₁: [3^3,1,1] x [ ]
  4. E₆xA₁: [3^2,2,1] x [ ]
  5. A₅xD₂^p: [3,3,3] x [p]
  6. C₅xD₂^p: [4,3,3] x [p]
  7. B₅xD₂^p: [3^2,1,1] x [p]
  8. A₄xA₃: [3,3,3] x [3,3]
  9. A₄xC₃: [3,3,3] x [4,3]
  10. A₄xG₃: [3,3,3] x [5,3]
  11. C₄xA₃: [4,3,3] x [3,3]
  12. C₄xC₃: [4,3,3] x [4,3]
  13. C₄xG₃: [4,3,3] x [5,3]
  14. G₄xA₃: [5,3,3] x [3,3]
  15. G₄xC₃: [5,3,3] x [4,3]
  16. G₄xG₃: [5,3,3] x [5,3]
  17. F₄xA₃: [3,4,3] x [3,3]
  18. F₄xC₃: [3,4,3] x [4,3]
  19. F₄xG₃: [3,4,3] x [5,3]
  20. B₄xA₃: [3^1,1,1] x [3,3]
  21. B₄xC₃: [3^1,1,1] x [4,3]
  22. B₄xG₃: [3^1,1,1] x [5,3]
  23. A₄xD₂^pxA₁: [3,3,3] x [p] x [ ]
  24. C₄xD₂^pxA₁: [4,3,3] x [p] x [ ]
  25. F₄xD₂^pxA₁: [3,4,3] x [p] x [ ]
  26. G₄xD₂^pxA₁: [5,3,3] x [p] x [ ]
  27. B₄xD₂^pxA₁: [3^1,1,1] x [p] x [ ]
  28. A₃xA₃xA₁: [3,3] x [3,3] x [ ]
  29. A₃xC₃xA₁: [3,3] x [4,3] x [ ]
  30. A₃xG₃xA₁: [3,3] x [5,3] x [ ]
  31. C₃xC₃xA₁: [4,3] x [4,3] x [ ]
  32. C₃xG₃xA₁: [4,3] x [5,3] x [ ]
  33. G₃xA₃xA₁: [5,3] x [5,3] x [ ]
  34. A₃xD₂^pxD₂^q: [3,3] x [p] x [q]
  35. C₃xD₂^pxD₂^q: [4,3] x [p] x [q]
  36. G₃xD₂^pxD₂^q: [5,3] x [p] x [q]
  37. D₂^pxD₂^qxD₂^rA₁: [p] x [q] x [r] x [ ]

[edit] See also

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