# 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 has 660 symmetries. It has Schläfli symbol {3,5,3}.

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.
TypeAbstract regular 4-polytope
Cells11 hemi-icosahedra
Faces55 {3}
Edges55
Vertices11
Vertex figure(hemi-dodecahedron)
Schläfli symbol{3,5,3}
Symmetry groupL2(11) (order 660)
Dualself-dual
PropertiesRegular

It was discovered in 1977 by Branko Grünbaum, who constructed it by pasting hemi-icosahedra together, three at each 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 can be drawn as a regular figure in 11-space, although then its hemi-icosahedral cells are skew; that is, each cell is not contained within a flat 3-dimensional subspace.

## 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.
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