| 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 4-polytope |
| Cells | 11 hemi-icosahedron |
| Faces | 55 {3} |
| Edges | 55 |
| Vertices | 11 |
| Vertex figure | hemi-dodecahedron |
| Schläfli symbol | { { 3 , 5 } 5 , { 5 , 3 } 5 } {\displaystyle \{\{3,5\}_{5},\{5,3\}_{5}\}} |
| Symmetry group | order 660Abstract 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, 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. It has since been studied and illustrated by Séquin.
Related polytopes
The dual polytope of the 11-cell is the 57-cell.5
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, {3,5,3}. (The 11-cell can be considered to be derived from it by identification of appropriate elements.)
Citations
- Grünbaum, Branko (1976), "Regularity of Graphs, Complexes and Designs" (PDF), Colloques Internationaux C.N.R.S., 260, Orsay: 191–197
- Coxeter, H.S.M. (1984), "A Symmetrical Arrangement of Eleven Hemi-Icosahedra", Annals of Discrete Mathematics (20): Convexity and Graph Theory, North-Holland Mathematics Studies, 87, North-Holland: 103–114, doi:10.1016/S0304-0208(08)72814-7, ISBN 978-0-444-86571-7
- Peter McMullen, Egon Schulte, Abstract Regular Polytopes, Cambridge University Press, 2002. ISBN 0-521-81496-0
- The Classification of Rank 4 Locally Projective Polytopes and Their Quotients, 2003, Michael I Hartley
- Séquin, Carlo H.; Lanier, Jaron (2007), "Hyperseeing the Regular Hendacachoron" (PDF), ISAMA (May 2007), Texas A & M: 159–166
- Séquin, Carlo H.; Hamlin, James F. (2007), "The regular 4-dimensional 57-cell", ACM SIGGRAPH 2007 sketches (PDF), SIGGRAPH '07, New York, NY, USA: ACM, p. 3, doi:10.1145/1278780.1278784, ISBN 978-1-4503-4726-6, S2CID 37594016
- Séquin, Carlo H. (2012), "A 10-Dimensional Jewel" (PDF), Gathering for Gardner G4GX, Atlanta GA
External links
- Peterson, Ivars (26 April 2007). "The Fabulously Odd 11-Cell". The Mathematical Tourist.
- Klitzing, Richard. "Explanations Grünbaum-Coxeter Polytopes".
- J. Lanier, Jaron’s World. Discover, April 2007, pp 28-29.
References
Grünbaum 1976, Regularity of Graphs, Complexes and Designs. - Grünbaum, Branko (1976), "Regularity of Graphs, Complexes and Designs" (PDF), Colloques Internationaux C.N.R.S., 260, Orsay: 191–197 https://faculty.washington.edu/moishe/branko/BG111.Regularity%20of%20graphs,etc.pdf ↩
Coxeter 1984, A Symmetrical Arrangement of Eleven Hemi-Icosahedra. - Coxeter, H.S.M. (1984), "A Symmetrical Arrangement of Eleven Hemi-Icosahedra", Annals of Discrete Mathematics (20): Convexity and Graph Theory, North-Holland Mathematics Studies, 87, North-Holland: 103–114, doi:10.1016/S0304-0208(08)72814-7, ISBN 978-0-444-86571-7 https://www.sciencedirect.com/science/article/pii/S0304020808728147 ↩
Séquin & Lanier 2007, Hyperseeing the Regular Hendacachoron. - Séquin, Carlo H.; Lanier, Jaron (2007), "Hyperseeing the Regular Hendacachoron" (PDF), ISAMA (May 2007), Texas A & M: 159–166 https://people.eecs.berkeley.edu/~sequin/PAPERS/2007_ISAMA_11Cell.pdf ↩
Séquin 2012, A 10-Dimensional Jewel. - Séquin, Carlo H. (2012), "A 10-Dimensional Jewel" (PDF), Gathering for Gardner G4GX, Atlanta GA https://people.eecs.berkeley.edu/%7Esequin/PAPERS/2012_G4GX_10D_jewel.pdf ↩
Séquin & Hamlin 2007, The Regular 4-dimensional 57-cell. - Séquin, Carlo H.; Hamlin, James F. (2007), "The regular 4-dimensional 57-cell", ACM SIGGRAPH 2007 sketches (PDF), SIGGRAPH '07, New York, NY, USA: ACM, p. 3, doi:10.1145/1278780.1278784, ISBN 978-1-4503-4726-6, S2CID 37594016 https://people.eecs.berkeley.edu/%7Esequin/PAPERS/2007_SIGGRAPH_57Cell.pdf ↩