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The vertices and edges form the
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
*
* 4-polytopes {{Polychora-stub
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 57-cell (pentacontakaiheptachoron) 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 57 cell
Cell most often refers to:
* Cell (biology), the functional basic unit of life
Cell may also refer to:
Locations
* Monastic cell, a small room, hut, or cave in which a religious recluse lives, alternatively the small precursor of a monastery ...
s 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 ...
, L2(19).
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 5 hemi-dodecahedral cells around each edge. It was discovered by .
Perkel graph
The vertices and edges form the Perkel graph
In mathematics, the Perkel graph, named after Manley Perkel, is a 6-regular graph with 57 vertices and 171 edges. It is the unique distance-regular graph with intersection array (6, 5, 2; 1, 1, 3).Coolsaet, K. and Degraer, J. "A Computer Assisted ...
, the unique distance-regular graph with intersection array , discovered by .
See also
*11-cell
In mathematics, the 11-cell (or hendecachoron) 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 symbol , with 3 hemi-icosahedra ...
– 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, he ...
– regular 4-polytope with dodecahedral cells
* Order-5 dodecahedral 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 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
*
* 4-polytopes {{Polychora-stub