11-cell

Abstract regular 4-polytope 11-cell240pxThe 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.TypeAbstract regular 4-polytopeCells11 hemi-icosahedron150pxFaces55 {3}Edges55Vertices11Vertex figurehemi-dodecahedronSchläfli symbolSymmetry grouporder 660Abstract L2(11)Dualself-dualPropertiesRegular 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. 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, L2(11). It was discovered in 1976 by Branko Grünbaum,Grünbaum 1976 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.Coxeter 1984 A Symmetrical Arrangement of Eleven Hemi-Icosahedra It has since been studied and illustrated by Carlo H. Séquin.Séquin & Lanier 2007 Hyperseeing the Regular HendecachoronSéquin 2012 A 10-Dimensional Jewel Looking only at the vertices and cells, its abstract structure is geometric configuration (116) and can be defined with a cyclic configuration, with a generator "line" as {0,1,2,4,5,7}11. (Sequential lines increment vertex indices by 1 modulo 11.)November 2025 Needed to support notability Related polytopes 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 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 References Grünbaum Branko Branko Grünbaum 1976 Regularity of Graphs, Complexes and Designs Colloques Internationaux C.N.R.S. CNRS Éditions Orsay 260 191–197 https://faculty.washington.edu/moishe/branko/BG111.Regularity%20of%20graphs,etc.pdf Coxeter H.S.M. Harold Scott MacDonald Coxeter 1984 A Symmetrical Arrangement of Eleven Hemi-Icosahedra Annals of Discrete Mathematics (20): Convexity and Graph Theory North-Holland Mathematics Studies North-Holland 87 103–114 10.1016/S0304-0208(08)72814-7 978-0-444-86571-7 https://www.sciencedirect.com/science/article/pii/S0304020808728147 subscription Peter McMullen, Egon Schulte, Abstract Regular Polytopes, Cambridge University Press, 2002. 0-521-81496-0 The Classification of Rank 4 Locally Projective Polytopes and Their Quotients, 2003, Michael I Hartley Séquin Carlo H. Carlo H. Séquin Lanier Jaron Jaron Lanier Hyperseeing the Regular Hendecachoron 2007 ISAMA Texas A & M 159–166 May 2007 https://people.eecs.berkeley.edu/~sequin/PAPERS/2007_ISAMA_11Cell.pdf Séquin & Lanier 2007 Séquin Carlo H. Carlo H. Séquin Hamlin James F. The regular 4-dimensional 57-cell ACM SIGGRAPH 2007 sketches 10.1145/1278780.1278784 New York, NY, USA ACM SIGGRAPH '07 3 978-1-4503-4726-6 2007 37594016 https://people.eecs.berkeley.edu/%7Esequin/PAPERS/2007_SIGGRAPH_57Cell.pdf Séquin & Hamlin 2007 Séquin Carlo H. A 10-Dimensional Jewel Gathering for Gardner G4GX Atlanta GA 2012 https://people.eecs.berkeley.edu/%7Esequin/PAPERS/2012_G4GX_10D_jewel.pdf External links 11-cell Ivars Peterson The Mathematical Tourist The Fabulously Odd 11-Cell 26 April 2007 https://mathtourist.blogspot.com/2007/04/fabulously-odd-11-cell.html ../explain/gc.htm Explanations Grünbaum-Coxeter Polytopes Category:Regular 4-polytopes