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.
Type	Abstract regular 4-polytope Abstract polytope In mathematics, an abstract polytope, informally speaking, is a structure which considers only the combinatorial properties of a traditional polytope, ignoring many of its other properties, such as angles, edge lengths, etc...
Cells	11 hemi-icosahedra Hemi-icosahedron A hemi-icosahedron is an abstract regular polyhedron, containing half the faces of a regular icosahedron. It can be realized as a projective polyhedron , which can be visualized by constructing the projective plane as a hemisphere where opposite points along the boundary are connected and dividing...
Faces	55 {3}
Edges	55
Vertices	11
Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...	(hemi-dodecahedron Hemi-dodecahedron A hemi-dodecahedron is an abstract regular polyhedron, containing half the faces of a regular dodecahedron. It can be realized as a projective polyhedron , which can be visualized by constructing the projective plane as a hemisphere where opposite points along the boundary are connected and... )
Schläfli symbol	{3,5,3}
Symmetry group Symmetry group The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...	L2(11) (order 660)
Dual	self-dual
Properties	Regular

In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the 11-cell (or hendecachoron) is a self-dual

Duality (mathematics)

In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often by means of an involution operation: if the dual of A is B, then the dual of B is A. As involutions sometimes have...

 abstract regular 4-polytope

Abstract polytope

In mathematics, an abstract polytope, informally speaking, is a structure which considers only the combinatorial properties of a traditional polytope, ignoring many of its other properties, such as angles, edge lengths, etc...

 (four-dimensional polytope

Polychoron

In geometry, a polychoron or 4-polytope is a four-dimensional polytope. It is a connected and closed figure, composed of lower dimensional polytopal elements: vertices, edges, faces , and cells...

). Its 11 cells are hemi-icosahedral

Hemi-icosahedron

A hemi-icosahedron is an abstract regular polyhedron, containing half the faces of a regular icosahedron. It can be realized as a projective polyhedron , which can be visualized by constructing the projective plane as a hemisphere where opposite points along the boundary are connected and dividing...

. It has 11 vertices, 55 edges and 55 faces. Its symmetry group is the projective special linear group L₂(11), so it has
660 symmetries. It has Schläfli symbol {3,5,3}.

It was discovered in 1977 by Branko Grünbaum

Branko Grünbaum

Branko Grünbaum is a Croatian-born mathematician and a professor emeritus at the University of Washington in Seattle. He received his Ph.D. in 1957 from Hebrew University of Jerusalem in Israel....

, 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.

Related polytopes

The abstract 11-cell contains the same number of vertices and edges as the 10-dimensional 10-simplex

10-simplex

In geometry, a 10-simplex is a self-dual regular 10-polytope. It has 11 vertices, 55 edges, 165 triangle faces, 330 tetrahedral cells, 462 5-cell 4-faces, 462 5-simplex 5-faces, 330 6-simplex 6-faces, 165 7-simplex 7-faces, 55 8-simplex 8-faces, and 11 9-simplex 9-faces...

, and contains 1/3 of its 165 faces. Thus it can be drawn as a regular figure in 11-space, although then its hemi-icosahedral cells are skew; that is, each cell is not contained within a flat 3-dimensional subspace

Euclidean subspace

In linear algebra, a Euclidean subspace is a set of vectors that is closed under addition and scalar multiplication. Geometrically, a subspace is a flat in n-dimensional Euclidean space that passes through the origin...

.

See also

• 57-cell
• Icosahedral honeycomb - regular hyperbolic honeycomb with same Schläfli symbol {3,5,3}. (The 11-cell can be considered to be derived from it by identification of appropriate elements.)

External links

• J. Lanier, Jaron’s World. Discover, April 2007, pp 28-29.
• http://www.cs.berkeley.edu/~sequin/PAPERS/2007_ISAMA_11Cell.pdf 2007 ISAMA paper: Hyperseeing the Regular Hendecachoron, Carlo H. Séquin & Jaron Lanier