In
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 one-to-one fashion, often (but not always) by means of an involution operation: if the dual of is , then the ...
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
In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to mo ...
, 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 associa ...
L
2(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 descent[H. S. M. Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington t ...](_blank)
in 1984, who studied its structure and symmetry in greater depth.
Related polytopes
Orthographic projection
Orthographic projection (also orthogonal projection and analemma) is a means of representing three-dimensional objects in two dimensions. Orthographic projection is a form of parallel projection in which all the projection lines are orthogona ...
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
In five-dimensional geometry, a 5- simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos−1(), or approximately 78.46°.
The 5 ...
*
57-cell
*
Icosahedral honeycomb - regular hyperbolic honeycomb with same
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to mo ...
. (The 11-cell can be considered to be derived from it by identification of appropriate elements.)
References
* Peter McMullen,
Egon Schulte
Egon Schulte (born January 7, 1955 in Heggen ( Kreis Olpe), Germany) is a mathematician and a professor of Mathematics at Northeastern University in Boston. He received his Ph.D. in 1980 from the Technical University of Dortmund
TU Dortmund ...
, ''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