Editing 57-cell (section)

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!bgcolor=#e7dcc3 colspan=2|57-cell
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|bgcolor=#e7dcc3|Type||[[Abstract polytope|Abstract regular 4-polytope]]
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|bgcolor=#e7dcc3|Cells||57 [[hemi-dodecahedron|hemi-dodecahedra]]<BR>[[Image:Hemi-dodecahedron.png|150px]]
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|bgcolor=#e7dcc3|Faces||171 {5}
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|bgcolor=#e7dcc3|Edges||171
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|bgcolor=#e7dcc3|Vertices||57
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|bgcolor=#e7dcc3|[[Vertex figure]]||[[hemi-icosahedron]]
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|bgcolor=#e7dcc3|[[Schläfli symbol|Schläfli type]]||{5,3,5}
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|bgcolor=#e7dcc3|[[Symmetry group]]||order 3420<BR>Abstract [[projective special linear group|L<sub>2</sub>(19)]]
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|bgcolor=#e7dcc3|Dual||[[Self-dual polytope|self-dual]]
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|bgcolor=#e7dcc3|Properties||Regular
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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}}.