Gosset 4 21 polytope

The Gosset 4₂₁ polytope is an 8-dimensional semiregular uniform polytope composed of 17,280 7-simplex and 2,160 7-orthoplex facets.

It was discovered by Thorold Gosset, who described it in his 1900 paper as an "8-ic semi-regular figure." It is the last finite semiregular figure in his enumeration, semiregular to him meaning that it contained only regular facets.

Donald Coxeter called it 4₂₁ due to its bifurcating Coxeter-Dynkin diagram, having 3 branches of length 4, 2, and 1, and having a single ring on the final node of the 4 branch. As this graph is a representation of the simple Lie group E₈, the polytope is sometimes referred to as the E8 polytope.

For visualization this 8-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 240 vertices within a 30-gonal regular polygon (called a Petrie polygon). Its 6720 edges are drawn between the 240 vertices. Specific higher elements (faces, cells, etc) can also be extracted and drawn on this projection.

Construction

It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space.

The facet information can be extracted from its Coxeter-Dynkin diagram.
*

Removing the node on the short branch leaves the 7-simplex:
*

Removing the node on the end of the 2-length branch leaves the 7-orthoplex in its alternated form: (4₁₁)
*

Every simplex facet touches an 7-orthoplex facet, while alternate facets of the orthoplex touch either a simplex or another orthoplex.

The 240 vertices of the 4₂₁ 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 $left(pm\; frac12,pm\; frac12,pm\; frac12,pm\; frac12,pm\; frac12,pm\; frac12,pm\; frac12,pm\; frac12\; ight)\; ,$ by taking an even number of minus signs (or, equivalently, requiring that the sum of all the eight coordinates be even).

Related polytopes

The 4₂₁ polytope is last in a family called the k₂₁ polytopes. The first polytope in this family is the semiregular triangular prism which is constructed from three squares (2-orthoplexes) and two triangles (2-simplexes).

As a 4-dimensional regular complex polytope 3{3}3{3}3{3}3, Coxeter called the Witting polytope, after Alexander Witting. [Coxeter, H. S. M., "Regular Complex Polytopes", Cambridge University Press, (1974)., 12.5 The Witting polytope]

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 rings in this Coxeter-Dynkin diagram: :

Among these 255, two other special polytopes in this family are: 1₄₁ and 2₄₁, with a ring at the end of the other branches.

This polytope, along with the 8-simplex, can create a uniform tessellaton of hyperbolic 8-dimensional space, represented by symbol 4_3,1and Coxeter-Dynkin diagram::

See also

*Other notable 8-polytopes:
** 8-simplex
** 8-cube
** 8-orthoplex
** 8-demicube

Notes

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