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In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope.
It is defined as follows. The Schläfli symbol of a polygon with n edges is {n}. The Schläfli symbol of a polyhedron is {p,q} if its faces are p-gons, and each vertex is surrounded by q faces. Note that the Schläfli symbol is not well defined for polyhedra which are not (sufficiently) regular (such as the prism).
The Schläfli symbols of the Platonic solids are:
- for the tetrahedron : {3,3}
- for the cube : {4,3}
- for the octahedron : {3,4}
- for the dodecahedron : {5,3}
- for the icosahedron : {3,5}
Schläfli symbols may also be defined for regular tessellations of euclidean or hyperbolic space in a similar way.
For higher dimensional polytopes, the Schläfli symbol is defined recursively as {p1,p2,...,pn-1} if the facets have Schläfli symbol {p1,p2,...,pn-2} and the vertex figures have Schläfli symbol {p2,p3,...,pn-1}.
The Schläfli symbol of a line segment is {}. If a polytope has Schläfli symbol {p1,p2,...,pn-1} then its dual polytope has Schläfli symbol {pn-1,...,p2,p1}.
Occasionally, you will see fractions in a Schläfli symbol. For example, there are several instances of 5/2 in the list of regular polytopes. The symbol {p/q} means a planar figure with p vertexes where every q-th vertex is connected. Thus, 5/2 is a five-pointed star shape.
The Schläfli symbol is named after the 19th century mathematician Ludwig Schläfli who made important contributions in geometry and other areas.