Editing Regular polytope (section)

===Schläfli symbols===
{{main|Schläfli symbol}}

A concise symbolic representation for regular polytopes was developed by [[Ludwig Schläfli]] in the 19th century, and a slightly modified form has become standard. The notation is best explained by adding one dimension at a time.

*A [[convex polygon|convex]] [[regular polygon]] having ''n'' sides is denoted by {''n''}. So an equilateral triangle is {3}, a square {4}, and so on indefinitely. A regular ''n''-sided [[star polygon]] which winds ''m'' times around its centre is denoted by the fractional value {''n''/''m''}, where ''n'' and ''m'' are [[co-prime]], so a regular [[pentagram]] is {5/2}.
*A [[regular polyhedron]] having faces {''n''} with ''p'' faces joining around a vertex is denoted by {''n'', ''p''}. The nine [[regular polyhedra]] are [[Tetrahedron|{3, 3}]]; [[Octahedron|{3, 4}]]; [[Cube|{4, 3}]]; [[Regular icosahedron|{3, 5}]]; [[Regular dodecahedron|{5, 3}]]; [[Great icosahedron|{3, 5/2}]]; [[Great stellated dodecahedron|{5/2, 3}]]; [[Great dodecahedron|{5, 5/2}]]; and [[Small stellated dodecahedron|{5/2, 5}]]. {''p''} is the ''[[vertex figure]]'' of the polyhedron.
*A regular 4-polytope having cells {''n'', ''p''} with ''q'' cells joining around an edge is denoted by {''n'', ''p'', ''q''}. The vertex figure of the 4-polytope is a {''p'', ''q''}.
*A regular 5-polytope is an {''n'', ''p'', ''q'', ''r''}. And so on.