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The vertices and edges form the Perkel graph, the unique distance-regular graph with
The Classification of Rank 4 Locally Projective Polytopes and Their Quotients
2003, Michael I Hartley
Siggraph 2007: 11-cell and 57-cell by Carlo Sequin
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* Regular 4-polytopes {{Polychora-stub
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the 57-cell (pentacontaheptachoron) is a self-dual abstract regular 4-polytope ( four-dimensional polytope). Its 57 cells are hemi-dodecahedra. It also has 57 vertices, 171 edges and 171 two-dimensional faces.
The symmetry order is 3420, from the product of the number of cells (57) and the symmetry of each cell (60). The symmetry abstract structure is the 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 ...
of the 2-dimensional vector space over the finite field of 19 elements, L2(19).
It has Schläfli type with 5 hemi-dodecahedral cells around each edge. It was discovered by .
Perkel graph
The vertices and edges form the Perkel graph, the unique distance-regular graph with intersection array
In the mathematical field of graph theory, a distance-regular graph is a regular graph such that for any two Vertex (graph theory), vertices and , the number of vertices at distance (graph theory), distance from and at distance from depends ...
, discovered by .
See also
* 11-cell – abstract regular polytope with hemi-icosahedral cells. *120-cell
In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, hec ...
– regular 4-polytope with dodecahedral cells
* Order-5 dodecahedral honeycomb - regular hyperbolic honeycomb with same Schläfli type, . (The 57-cell can be considered as being derived from it by identification of appropriate elements.)
References
*. * *. *The Classification of Rank 4 Locally Projective Polytopes and Their Quotients
2003, Michael I Hartley
External links
Siggraph 2007: 11-cell and 57-cell by Carlo Sequin
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* Regular 4-polytopes {{Polychora-stub