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, the 11-cell is a self-dualabstract regular 4-polytope ( four-dimensional polytope). Its 11 cells are hemi-icosahedral. It has 11 vertices, 55 edges and 55 faces. It has Schläfli type , with 3 hemi-icosahedra (Schläfli type ) around each edge.
It has symmetry order 660, computed as the product of the number of cells (11) and the symmetry of each cell (60). The symmetry structure is the abstract group
projective special linear group
In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associa ...
of the 2-dimensional vector space over the finite field with 11 elements L2(11).
It was discovered in 1976 by
Branko Grünbaum
Branko Grünbaum (; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentH. S. M. Coxeter in 1984, who studied its structure and symmetry in greater depth. It has since been studied and illustrated by Séquin.
Related polytopes
The dual polytope of the 11-cell is the 57-cell.
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 10-space, although then its hemi-icosahedral cells are skew; that is, each cell is not contained within a flat 3-dimensional subspace.
See also
* 5-simplex
* 57-cell
* Icosahedral honeycomb - regular hyperbolic honeycomb with same Schläfli type, . (The 11-cell can be considered to be derived from it by identification of appropriate elements.)