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11-cell

{{Short description|Abstract regular 4-polytope}}

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!bgcolor=#e7dcc3 colspan=2|11-cell

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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.
bgcolor=#e7dcc3|TypeAbstract regular 4-polytope
bgcolor=#e7dcc3|Cells11 hemi-icosahedron
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bgcolor=#e7dcc3|Faces55 {3}
bgcolor=#e7dcc3|Edges55
bgcolor=#e7dcc3|Vertices11
bgcolor=#e7dcc3|Vertex figurehemi-dodecahedron
bgcolor=#e7dcc3|Schläfli symbol\{\{3,5\}_5,\{5,3\}_5\}
bgcolor=#e7dcc3|Symmetry grouporder 660
Abstract L2(11)
bgcolor=#e7dcc3|Dualself-dual
bgcolor=#e7dcc3|PropertiesRegular

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,{{Sfn|Grünbaum|1976|loc=Regularity of Graphs, Complexes and Designs}} 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.{{Sfn|Coxeter|1984|loc=A Symmetrical Arrangement of Eleven Hemi-Icosahedra}} It has since been studied and illustrated by Séquin.{{Sfn|Séquin & Lanier|2007|loc=Hyperseeing the Regular Hendacachoron}}{{Sfn|Séquin|2012|loc=A 10-Dimensional Jewel}}

Related polytopes

{{Dark mode invert|Image:10-simplex t0.svg of 10-simplex with 11 vertices, 55 edges]]}}

The dual polytope of the 11-cell is the 57-cell.{{Sfn|Séquin & Hamlin|2007|loc=The Regular 4-dimensional 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, {3,5,3}. (The 11-cell can be considered to be derived from it by identification of appropriate elements.)

Citations

{{Reflist}}

References

  • {{Citation | last=Grünbaum | first=Branko | author-link=Branko Grünbaum | year=1976 | title=Regularity of Graphs, Complexes and Designs | journal=Colloques Internationaux C.N.R.S. | publisher=Orsay | volume=260 | pages=191–197 | url=https://faculty.washington.edu/moishe/branko/BG111.Regularity%20of%20graphs,etc.pdf }}
  • {{Citation | last=Coxeter | first=H.S.M. | author-link=Harold Scott MacDonald Coxeter | year=1984 | title=A Symmetrical Arrangement of Eleven Hemi-Icosahedra | journal=Annals of Discrete Mathematics (20): Convexity and Graph Theory | series=North-Holland Mathematics Studies | publisher=North-Holland | volume=87 | pages=103–114 | doi=10.1016/S0304-0208(08)72814-7 | isbn=978-0-444-86571-7 | url=https://www.sciencedirect.com/science/article/pii/S0304020808728147

| url-access=subscription }}

  • Peter McMullen, Egon Schulte, Abstract Regular Polytopes, Cambridge University Press, 2002. {{ISBN|0-521-81496-0}}
  • [https://arxiv.org/abs/math/0310429 The Classification of Rank 4 Locally Projective Polytopes and Their Quotients], 2003, Michael I Hartley
  • {{citation | last1 = Séquin | first1 = Carlo H. | author1-link = Carlo H. Séquin | last2 = Lanier | first2 = Jaron | author2-link = Jaron Lanier | title = Hyperseeing the Regular Hendacachoron | year = 2007 | journal = ISAMA | publisher=Texas A & M | pages=159–166 | issue=May 2007 | url=https://people.eecs.berkeley.edu/~sequin/PAPERS/2007_ISAMA_11Cell.pdf | ref={{SfnRef|Séquin & Lanier|2007}}}}
  • {{citation | last1 = Séquin | first1 = Carlo H. | author1-link = Carlo H. Séquin | last2 = Hamlin | first2 = James F. | chapter = The regular 4-dimensional 57-cell | title = ACM SIGGRAPH 2007 sketches | doi = 10.1145/1278780.1278784 | location = New York, NY, USA | publisher = ACM | series = SIGGRAPH '07 | page = 3 | isbn = 978-1-4503-4726-6 | year = 2007| s2cid = 37594016 | url = https://people.eecs.berkeley.edu/%7Esequin/PAPERS/2007_SIGGRAPH_57Cell.pdf | ref={{SfnRef|Séquin & Hamlin|2007}}}}
  • {{citation | last=Séquin | first=Carlo H. | title=A 10-Dimensional Jewel | journal=Gathering for Gardner G4GX | place=Atlanta GA | year=2012 | url=https://people.eecs.berkeley.edu/%7Esequin/PAPERS/2012_G4GX_10D_jewel.pdf }}

External links

  • {{cite journal| first=Ivars | last=Peterson | journal=The Mathematical Tourist | title=The Fabulously Odd 11-Cell | date=26 April 2007 | url=https://mathtourist.blogspot.com/2007/04/fabulously-odd-11-cell.html }}
  • {{KlitzingPolytopes|../explain/gc.htm|Explanations|Grünbaum-Coxeter Polytopes}}
  • [http://discovermagazine.com/2007/apr/jarons-world-shapes-in-other-dimensions J. Lanier, Jaron’s World. Discover, April 2007, pp 28-29.]

Category:Regular 4-polytopes