57-cell
Encyclopedia
Some drawings of Perkel graph. |
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| 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 | 57 hemi-dodecahedra 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...  |
| Faces | 171 {5} |
| Edges | 171 |
| Vertices | 57 |
| 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-icosahedron 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... ) |
| Schläfli symbol | {5,3,5} |
| 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(19) (order 3420) |
| 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 57-cell 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 57 cell
Cell (geometry)
In geometry, a cell is a three-dimensional element that is part of a higher-dimensional object.- In polytopes :A cell is a three-dimensional polyhedron element that is part of the boundary of a higher-dimensional polytope, such as a polychoron or honeycomb For example, a cubic honeycomb is made...
s are hemi-dodecahedra
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...
. It also has 57 vertices, 171 edges and 171 faces. Its symmetry group is the projective special linear group L2(19), so it has 3420 symmetries.
It has Schläfli symbol {5,3,5} with 5 hemi-dodecahedral cells around each edge. It was discovered by H. S. M. Coxeter in 1982.
Perkel graph
The vertices and edges form the Perkel graph, the unique distance-regular graph with intersection arrayIntersection array
In the mathematical field of graph theory, given a distance-regular graph G of diameter d, by definition there are integers bi and ci such that given any two vertices x and y in G at distance i, y has bi neighbours at distance i + 1 from x and ci neighbours at distance...
{6,5,2;1,1,3}, discovered in 1979 by Manley Perkel. http://www.math.wright.edu/People/manley.perkel/Vita
See also
- 11-cell11-cellIn mathematics, the 11-cell is a self-dual abstract regular 4-polytope . Its 11 cells are hemi-icosahedral. It has 11 vertices, 55 edges and 55 faces. Its symmetry group is the projective special linear group L2, so it has660 symmetries...
– abstract regular polytope with hemi-icosahedral cells. - 120-cell – regular 4-polytope with dodecahedral cells
- Order-5 dodecahedral honeycomb - regular hyperbolic honeycomb with same Schläfli symbol {5,3,5}. (The 57-cell can be considered as being derived from it by identification of appropriate elements.)