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# 11-cell

In mathematics, the 11-cell (or hendecachoron) is a self-dual abstract regular 4-polytope (four-dimensional polytope). Its 11 cells are hemi-icosahedral. It has 11 vertices, 55 edges and 55 faces. Its symmetry group is the projective special linear group L2(11), so it has660 symmetries. It has Schläfli symbol {3,5,3}.

It was discovered by Branko Grünbaum in 1977, who constructed it by pasting hemi-icosahedra together, three per edge until the shape closed up. It was independently discovered by H. S. M. Coxeter in 1984, who studied its structure and symmetry in greater depth.

The abstract "11-cell" contains the same number of vertices and edges as the 10-dimensional 10-simplex, and contains 1/3 of its 165 faces. Thus it could be drawn as a regular figure in 11-space, although its hemi-icosahedral cells would be skew (Cell vertices are not contained within the same 3 dimensional subspace).

* 57-cell
* Order-3 icosahedral honeycomb - regular honeycomb with same Schläfli symbol {3,5,3}.

References

* Peter McMullen, Egon Schulte, "Abstract Regular Polytopes", Cambridge University Press, 2002. ISBN 0-521-81496-0
* Coxeter, H.S.M., "A Symmetrical Arrangement of Eleven hemi-Icosahedra", Annals of Discrete Mathematics 20 pp103–114.

* [http://discovermagazine.com/2007/apr/jarons-world-shapes-in-other-dimensions J. Lanier, Jaron’s World. Discover, April 2007, pp 28-29.]
* [http://www.cs.berkeley.edu/~sequin/PAPERS/2007_ISAMA_11Cell.pdf] 2007 ISAMA paper: "Hyperseeing the Regular Hendecachoron", Carlo H. Séquin & Jaron Lanier

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