View source for 57-cell

{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|57-cell
<!--|-
|bgcolor=#ffffff align=center colspan=2|-->
|-
|bgcolor=#e7dcc3|Type||[[Abstract polytope|Abstract regular 4-polytope]]
|-
|bgcolor=#e7dcc3|Cells||57 [[hemi-dodecahedron|hemi-dodecahedra]]<BR>[[Image:Hemi-dodecahedron.png|150px]]
|-
|bgcolor=#e7dcc3|Faces||171 {5}
|-
|bgcolor=#e7dcc3|Edges||171
|-
|bgcolor=#e7dcc3|Vertices||57
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[hemi-icosahedron]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol|Schläfli type]]||{5,3,5}
|-
|bgcolor=#e7dcc3|[[Symmetry group]]||order 3420<BR>Abstract [[projective special linear group|L<sub>2</sub>(19)]]
|-
|bgcolor=#e7dcc3|Dual||[[Self-dual polytope|self-dual]]
|-
|bgcolor=#e7dcc3|Properties||Regular
|}
In [[mathematics]], the '''57-cell''' ('''pentacontaheptachoron''') is a [[duality (mathematics)|self-dual]] [[abstract polytope|abstract regular 4-polytope]] ([[4-polytope|four-dimensional polytope]]). Its 57 [[Cell (geometry)|cell]]s are [[hemi-dodecahedron|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]] of the 2-dimensional vector space over the finite field of 19 elements, L<sub>2</sub>(19). 

It has [[Schläfli symbol|Schläfli type]] {5,3,5} with 5 hemi-dodecahedral cells around each edge. It was discovered by {{harvs|first=H. S. M.|last=Coxeter|authorlink=Harold Scott MacDonald Coxeter|year=1982|txt}}.

== Perkel graph ==
[[File:Perkel graph embeddings.svg|thumb|left|[[Perkel graph]]s with 19-fold symmetry]]
The vertices and edges form the [[Perkel graph]], the unique distance-regular graph with [[intersection array]] {6,5,2;1,1,3}, discovered by {{harvs|first=Manley|last=Perkel|year=1979|txt}}.
{{clear|left}}

== See also ==
* [[11-cell]] – 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|Schläfli type]], {5,3,5}. (The 57-cell can be considered as being derived from it by identification of appropriate elements.)

== References ==
*{{citation
 | last = Coxeter | first = H. S. M. | authorlink = Harold Scott MacDonald Coxeter
 | doi = 10.1007/BF00149428
 | issue = 1
 | journal = [[Geometriae Dedicata]]
 | mr = 679218
 | pages = 87–99
 | title = Ten toroids and fifty-seven hemidodecahedra
 | volume = 13
 | year = 1982| s2cid = 120672023 }}.
*{{citation
 | last1 = McMullen | first1 = Peter | author1-link = Peter McMullen
 | last2 = Schulte | first2 = Egon | author2-link = Egon Schulte
 | doi = 10.1017/CBO9780511546686
 | isbn = 0-521-81496-0
 | mr = 1965665
 | pages = 185–186, 502
 | publisher = Cambridge University Press | location = Cambridge
 | series = Encyclopedia of Mathematics and its Applications
 | title = Abstract Regular Polytopes
 | url = https://books.google.com/books?id=JfmlMYe6MJgC&pg=PA185
 | volume = 92
 | year = 2002}}
*{{citation
 | last = Perkel | first = Manley
 | doi = 10.4153/CJM-1979-108-0
 | issue = 6
 | journal = [[Canadian Journal of Mathematics]]
 | mr = 553163
 | pages = 1307–1321
 | title = Bounding the valency of polygonal graphs with odd girth
 | volume = 31
 | year = 1979| doi-access = free
 }}.
*{{citation
 | last1 = Séquin | first1 = Carlo H. | author1-link = Carlo H. Séquin
 | last2 = Hamlin | first2 = James F.
 | contribution = The Regular 4-dimensional 57-cell
 | doi = 10.1145/1278780.1278784
 | location = New York, NY, USA
 | publisher = ACM
 | series = SIGGRAPH '07
 | title = ACM SIGGRAPH 2007 Sketches
 | contribution-url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 | year = 2007| s2cid = 37594016 | url = http://www.cs.berkeley.edu/~sequin/PAPERS/2007_SIGGRAPH_57Cell.pdf
 }}
* [https://arxiv.org/abs/math/0310429 The Classification of Rank 4 Locally Projective Polytopes and Their Quotients], 2003, Michael I Hartley

== External links ==
* [http://www.cs.berkeley.edu/~sequin/TALKS/2007_SIGGRAPH_57Cell.ppt Siggraph 2007: 11-cell and 57-cell by Carlo Sequin]
* {{Mathworld | urlname = PerkelGraph | title = Perkel graph}}
* [http://www.win.tue.nl/~aeb/graphs/Perkel.html Perkel graph]
* {{KlitzingPolytopes|../explain/gc.htm|Explanations|Grünbaum-Coxeter Polytopes}}

[[Category:Regular 4-polytopes]]
{{Polychora-stub}}