E7 polytope

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E7 polytope
Vertex-edge graph
Type	Uniform 7-polytope
Family	Semiregular E-polytope
Schläfli symbol	t₀{3^3,2,1}
Coxeter-Dynkin diagram
6-faces	702 total: 126 hexacrosses and 576 6-simplices
5-faces	6048 5-simplices
4-faces	12096 pentachorons
Cells	10080 tetrahedrons
Faces	4032 triangles
Edges	756
Vertices	56
Vertex figure	E6 polytope: {3^2,2,1}
Symmetry group	E₇, [3^3,2,1]
Properties	convex

The E7 polytope is a semiregular polytope, enumerated by Thorold Gosset in his 1900 paper. He called it an 7-ic semi-regular figure. It is called the Hess polytope for Edmund Hess who first discovered it.

Its construction is based on the E7 group. It is also named by Coxeter as 3₂₁ by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 3-node sequence.

It is also one of a family of 127 (2⁷-1) convex uniform polytopes in 7-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of ringed Coxeter-Dynkin diagrams.

[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, 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, ISBN 978-0-471-01003-6 [1]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] See p342 (figure 3.7c) by Peter mcMullen: (18-gonal node-edge graph of 3₂₁)