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.

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[edit] Regular forms

Regular polypeta can be represented by the Schläfli symbol {p,q,r,s,t}, with t {p,q,r,s} polyteron facets around each cell.

There are 3 finite regular polypetons:

  • {3,3,3,3,3} - 6-simplex (Elements: petons=7 {3,3,3,3}, terons=21 {3,3,3}, cells=35 {3,3}, faces=35 {3}, edges=21, vertices=7)
  • {4,3,3,3,3} - hexateract or 6-measure polytope (Elements: P=12 {4,3,3,3}, T=60 {4,3,3}, C=160 {4,3}, F=240 {4}, E=192, V=64)
  • {3,3,3,3,4} - hexacross or 6-cross-polytope (Elements: P=64 {3,3,3,3}, T=192 {3,3,3}, C=240 {3,3}, F=160 {3}, E=60, V=12)

Each of these regular forms can generate 62 uniform polypetons from the Wythoff construction.

[edit] Prismatic forms

There are 5 categorical uniform prismatic forms:

  1. {} x {p,q,r,s} - 5-polytope prisms
  2. {p} x {p,r,s} - polygonal-polychoral duoprisms
  3. {p,q} x {r,s} - polyhedral duprisms
  4. {} x {p} x {q,r} - polygonal-polyhedral duoprism prism
  5. {p} x {q} x {r} - triprism

[edit] Semiregular form

Thorold Gosset's 1900 published list of semiregular polytopes included one in 6-space, the E6 polytope, named now by the E6 Coxeter group, and Coxeter-Dynkin diagram .

It has 27 vertices with 99 facets: 27 pentacrosses and 72 simplexes.

[edit] See also

[edit] References

  • T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900

[edit] External links

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