11-cell

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title: "11-cell" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["regular-4-polytopes"] description: "Abstract regular 4-polytope" topic_path: "general/regular-4-polytopes" source: "https://en.wikipedia.org/wiki/11-cell" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Abstract regular 4-polytope ::

::data[format=table]

11-cell
[[Image:Hemi-icosahedron coloured.svg
Type
Cells
Faces
Edges
Vertices
Vertex figure
Schläfli symbol
Symmetry group
Dual
Properties
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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, L(11).

It was discovered in 1976 by Branko Grünbaum, 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. It has since been studied and illustrated by Carlo H. Séquin.

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.)

Related polytopes

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.

Citations

References

• {{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.
• The Classification of Rank 4 Locally Projective Polytopes and Their Quotients, 2003, Michael I Hartley

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