In 8-dimensional geometry, there are 135 uniform polytopes with A<sub>8</sub> symmetry. There is one self-dual regular form, the 8-simplex with 9 vertices.
Each can be visualized as symmetric orthographic projections in Coxeter planes of the A<sub>8</sub> Coxeter group, and other subgroups.
Graphs
Symmetric orthographic projections of these 135 polytopes can be made in the A<sub>8</sub>, A<sub>7</sub>, A<sub>6</sub>, A<sub>5</sub>, A<sub>4</sub>, A<sub>3</sub>, A<sub>2</sub> Coxeter planes. A<sub>k</sub> has [k+1] symmetry.
These 135 polytopes are each shown in these 7 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.
References
- H.S.M. Coxeter:
- H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
- Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, wiley.com,
- (Paper 22) H.S.M. Coxeter, Regular and Semi-Regular Polytopes I, [Math. Zeit. 46 (1940) 380âÂÂ407, MR 2,10]
- (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559âÂÂ591]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3âÂÂ45]
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
External links