View source for 11-cell

{{Short description|Abstract regular 4-polytope}}
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|11-cell
|-
|bgcolor=#ffffff align=center colspan=2|[[Image:Hemi-icosahedron coloured.svg|240px]]<BR>''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|Type||[[Abstract polytope|Abstract regular 4-polytope]]
|-
|bgcolor=#e7dcc3|Cells||11 [[hemi-icosahedron]]<BR>[[Image:Hemi-icosahedron.png|150px]]
|-
|bgcolor=#e7dcc3|Faces||55 {3}
|-
|bgcolor=#e7dcc3|Edges||55
|-
|bgcolor=#e7dcc3|Vertices||11
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[hemi-dodecahedron]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||<math>\{\{3,5\}_5,\{5,3\}_5\}</math>
|-
|bgcolor=#e7dcc3|[[Symmetry group]]||order 660<BR>Abstract [[projective special linear group|L<sub>2</sub>(11)]]
|-
|bgcolor=#e7dcc3|Dual||[[Self-dual polytope|self-dual]]
|-
|bgcolor=#e7dcc3|Properties||Regular
|}
In [[mathematics]], the '''11-cell''' is a [[duality (mathematics)|self-dual]] [[abstract polytope|abstract regular 4-polytope]] ([[4-polytope|four-dimensional polytope]]). Its 11 cells are [[hemi-icosahedron|hemi-icosahedral]]. It has 11 vertices, 55 edges and 55 faces. It has [[Schläfli symbol|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 L<sub>2</sub>(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 [[Carlo H. 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|thumb|upright=1|[[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 10-space, although then its hemi-icosahedral cells are skew; that is, each cell is not contained within a flat 3-dimensional [[Euclidean subspace|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.)
* [[Projective linear group#Action on p points]] - the 11-cell can be used to construct the case where p = 11

== 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==
*{{wikiversity-inline|11-cell}}
*{{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}}

[[Category:Regular 4-polytopes]]