11-cell
| 11-cell | |
|---|---|
| File:Hemi-icosahedron coloured.svg 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 4-polytope |
| Cells | 11 hemi-icosahedron File:Hemi-icosahedron.png |
| Faces | 55 {3} |
| Edges | 55 |
| Vertices | 11 |
| Vertex figure | hemi-dodecahedron |
| Schläfli symbol | <math>\{\{3,5\}_5,\{5,3\}_5\}</math> |
| Symmetry group | order 660 Abstract L2(11) |
| Dual | self-dual |
| Properties | Regular |
In mathematics, the 11-cell 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. It has Schläfli type {3,5,3}, with 3 hemi-icosahedra (Schläfli type {3,5}) 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 of the 2-dimensional vector space over the finite field with 11 elements L2(11).
It was discovered in 1976 by Branko Grünbaum,Template:Sfn 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.Template:Sfn It has since been studied and illustrated by Séquin.Template:SfnTemplate:Sfn
Related polytopesEdit
The dual polytope of the 11-cell is the 57-cell.Template:Sfn
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 alsoEdit
- 5-simplex
- 57-cell
- Icosahedral honeycomb - regular hyperbolic honeycomb with same Schläfli type, {3,5,3}. (The 11-cell can be considered to be derived from it by identification of appropriate elements.)
CitationsEdit
ReferencesEdit
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- Peter McMullen, Egon Schulte, Abstract Regular Polytopes, Cambridge University Press, 2002. Template:ISBN
- The Classification of Rank 4 Locally Projective Polytopes and Their Quotients, 2003, Michael I Hartley
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