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: polyzetta), created from poly-, zetta- and -on.

1 Regular forms
2 Semiregular form
3 Uniform prismatic forms
4 See also
5 References
6 External links

[edit] Regular forms

Regular polyzetta 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 or ennea-9-tope or enneazetton.
{4,3,3,3,3,3,3} - octeract or 8-hypercube
{3,3,3,3,3,3,4} - octacross or 8-cross-polytope

[edit] Semiregular form

Thorold Gosset's 1900 published list of semiregular polytopes included one in 8-space, 4_2,1, related to the E₈, [3^4,2,1] Coxeter group. It has a Coxeter-Dynkin diagram .

It has 240 vertices, 6720 edges, 60480 faces (triangles), 241920 cells, 483840 4-faces, 483840 5-faces, 206360 6-faces, and 19440 7-faces. There are two types of 7-faces: 2160 heptacrosses and 17280 7-simplexes.

[edit] Uniform prismatic forms

There are many categorical uniform prismatic forms based on Cartesian products of lower dimensional polytopes, for example these regular products:

{p,q,r,s,t,u} x {} - {p,q,r,s,t,u} prism
{p,q,r,s,t} x {u}
{p,q,r,s} x {t,u} -
{p,q,r} x {s,t,u} -
{p,q,r,s} x {t} x {} -
{p,q,r} x {s,t} x {} -
{p,q,r} x {s} x {t} -
{p,q} x {r,s} x {t} -
{p,q} x {r} x {s} x {} -
{p} x {q} x {r} x {s} -

[edit] See also

[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