E₇ polytope

E7 polytope
Orthogonal projection inside Petrie polygon
Type	Uniform 7-polytope
Family	Semiregular E-polytope
Schläfli symbol	{33,2,1}
Coxeter-Dynkin diagram
6-faces	702 total: 126 {33,1,1} 576 {35}
5-faces	6048: 4032 {33,0,1} 2016 {34}
4-faces	12096 {33}
Cells	10080 {3,3}
Faces	4032 {3}
Edges	756
Vertices	56
Vertex figure	E6 polytope: {32,2,1}
Petrie polygon	octadecagon
Coxeter group	E7, [33,2,1]
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 rings in this Coxeter-Dynkin diagram: .

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)

See also

• 7-polytope
• Semiregular E-polytope