Gosset 3 21 polytope

Gosset 3 21 polytope

In 7-dimensional geometry, the 321 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. Coxeter named it 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: :

This polytope, along with the 7-simplex, can tessellate 7-dimensional space, represented by 331 and 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 [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
** (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 k 21 polytope
*Gosset 1 32 polytope
*Gosset 2 31 polytope


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