E8 polytope

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E8 polytope
Vertex-edge graph
Type	Uniform 8-polytope
Family	Semiregular E-polytope
Schläfli symbol	t₀{3^4,2,1}
Coxeter-Dynkin diagram
7-faces	19440 total: 2160 heptacrosses 17280 7-simplices
6-faces	207360 6-simplices
5-faces	483840 5-simplices
4-faces	483840 pentachorons
Cells	241920 tetrahedrons
Faces	60480 triangles
Edges	6720
Vertices	240
Vertex figure	E7 polytope: {3^3,2,1}
Symmetry group	E₈, [3^4,2,1]
Properties	convex

The E8 polytope is a semiregular polytope, the highest finite (and nonprismatic) semiregular figure. Discovered by Thorold Gosset, it is also called the Gossett polytope. Gosset called it an 8-ic semi-regular figure in his 1900 paper, with "semiregular" meaning that all of its facets are regular polytopes: 2,160 7-orthotopes and 17,280 7-simplices.

Its construction is based on the E8 group. It is also named by Coxeter as 4₂₁ by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 4-node sequence.

It is one of a family of 255 (2⁸-1) convex uniform polytopes in eight dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of ringed Coxeter-Dynkin diagrams.

1 Construction
2 References
3 See also
4 External links

[edit] Construction

The 240 vertices of the E₈ polytope can be constructed in two set, 112 (2²C₂⁸) with integer coordinates obtained from $(\pm 1,\pm 1,0,0,0,0,0,0)\,$ by taking an arbitrary combination of signs and an arbitrary permutation of coordinates, and 128 roots (2⁷) with half-integer coordinates obtained from $\left(\pm\tfrac12,\pm\tfrac12,\pm\tfrac12,\pm\tfrac12,\pm\tfrac12,\pm\tfrac12,\pm\tfrac12,\pm\tfrac12\right) \,$ by taking an even number of minus signs (or, equivalently, requiring that the sum of all the eight coordinates be even).

[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
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 p347 (figure 3.8c) by Peter mcMullen: (30-gonal node-edge graph of 4₂₁)