8-polytope

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In geometry, an eight-dimensional polytope, or 8-polytope, is a polytope in 8-dimensional space. Each 6-polytope ridge being shared by exactly two 7-polytope facets.

A proposed name for 8-polytope is polyzetton, (plural polyexa), created from poly, zetta 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,v}, with v {p,q,r,s,t,u} 7-polytope facets around each peak.

There are 3 finite regular 8-polytopes:

• {3,3,3,3,3,3,3} - 8-simplex
• {4,3,3,3,3,3,3} - octeract or 8-measure polytope
• {3,3,3,3,3,3,4} - octacross or 8-cross-polytope

[edit] Prismatic forms

There are 10 categorical uniform prismatic forms:

1. {} x {p,q,r,s,t,u}
2. {p} x {q,r,s,t,u}
3. {p,q} x {r,s,t,u}
4. {p,q,r} x {s,t,u}
5. {} x {p} x {q,r,s,t}
6. {} x {p,q} x {r,s,t}
7. {p} x {q} x {r,s,t}
8. {p} x {q,r} x {s,t}
9. {} x {p} x {q} x {r,s}
10. {} x {p} x {q} x {r} x {s}

[edit] Semiregular form

Thorold Gosset's 1900 published list of semiregular polytopes included one in 8-space, the E8 polytope, named now by the E₈ Coxeter group, and Coxeter-Dynkin diagram .

It has 240 vertices with 18440 facets: 2160 heptacrosses and 17280 simplexes.

[edit] See also

• polygon
• polyhedron
• polychoron
• 5-polytope
• 6-polytope
• 7-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