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 forms 2 Prismatic forms 3 Semiregular form 4 See also 5 References 6 External links

[edit] Regular forms

Regular polyexa can be represented by the Schläfli symbol {p,q,r,s,t,u}, with u {p,q,r,s,t} 6-polytope facets around each hypercell.

There are 3 finite regular 7-polytopes:

• {3,3,3,3,3,3} - 7-simplex
• {4,3,3,3,3,3} - hepteract or 7-measure polytope
• {3,3,3,3,3,4} - heptacross or 7-cross-polytope

[edit] Prismatic forms

There are 7 categorical uniform prismatic forms:

1. {} x {p,q,r,s,t} - 6-polytope prism s
2. {p} x {q,r,s,t} - polygonal-polyteric duoprisms
3. {p,q} x {r,s,t} - polyhedral-polychoral duoprisms
4. {} x {p} x {q,r,s} polygonal-polychoral duoprism prisms
5. {} x {p,q} x {r,s} polyhedral duoprism prisms
6. {p} x {q} x {r,s} - bipolygonal-polyhedral duoprism
7. {} x {p} x {q} x {r} - triprism prism

[edit] Semiregular form

Thorold Gosset's 1900 published list of semiregular polytopes included one in 7-space, the E7 polytope, named now by the E₇ Coxeter group, and Coxeter-Dynkin diagram .

It has 56 vertices with 702 facets: 126 hexacrosses and 576 simplexes.

[edit] See also

• 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

[edit] External links

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