11-cell

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11-cell
The 11 hemi-icosahedra with vertices labeled by indices 0..9,t. Faces are colored by the cell it connects to, defined by the small colored boxes.
Type	Abstract regular polychoron
Cells	11 hemi-icosahedra
Faces	55 {3}
Edges	55
Vertices	11
Vertex figure	(hemi-dodecahedron)
Schläfli symbol	{3,5,3}
Symmetry group	L₂(11) (order 660)
Dual	self-dual
Properties

In mathematics, the 11-cell (or hendecachoron) is a self-dual abstract regular polychoron (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 has 660 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.

Orthographic projection of 10-simplex with 11 vertices, 55 edges.

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).

[edit] See also

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

[edit] 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.