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.

Its automorphism group has 660 elements. The automorphism group is isomorphic to the [[projective special linear group]] of the 2-dimensional vector space over the finite field with 11 elements, {{math|L{{sub|2}}(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}}

Looking only at the vertices and cells, its abstract structure is [[configuration (geometry)|geometric configuration]] (11<sub>6</sub>) and can be defined with a [[Configuration_(geometry)#Cyclic_configurations|cyclic configuration]], with a generator "line" as {0,1,2,4,5,7}<sub>11</sub>. (Sequential lines increment vertex indices by 1 modulo 11.){{citation needed|date=November 2025|reason=Needed to support notability}}

== 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]]