11-cell
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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the 11-cell (or hendecachoron) is a
self-dual In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a Injective function, one-to-one fashion, often (but not always) by means of an Involution (mathematics), involutio ...
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 symbol , with 3 hemi-icosahedra (Schläfli type ) 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 In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associat ...
L2(11). It was discovered in 1977 by
Branko Grünbaum Branko Grünbaum ( he, ברנקו גרונבאום; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentH. S. M. Coxeter in 1984, who studied its structure and symmetry in greater depth.


Related polytopes


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 In geometry, the icosahedral honeycomb is one of four compact, regular, space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli symbol there are three icosahedra around each edge, and 12 icosahedra around each vert ...
- regular hyperbolic honeycomb with same Schläfli symbol . (The 11-cell can be considered to be derived from it by identification of appropriate elements.)


References

* Peter McMullen, Egon Schulte, ''Abstract Regular Polytopes'', Cambridge University Press, 2002. * Coxeter, H.S.M., ''A Symmetrical Arrangement of Eleven hemi-Icosahedra'', Annals of Discrete Mathematics 20 pp103–114.
The Classification of Rank 4 Locally Projective Polytopes and Their Quotients
2003, Michael I Hartley


External links


J. Lanier, Jaron’s World. Discover, April 2007, pp 28-29.


2007 ISAMA paper: ''Hyperseeing the Regular Hendecachoron'', Carlo H. Séquin & Jaron Lanier, Als
Isama 2007, Texas A&m hyper-Seeing the Regular Hendeca-choron. (= 11-Cell)
* {{KlitzingPolytopes, ../explain/gc.htm, Explanations, Grünbaum-Coxeter Polytopes 4-polytopes