Schläfli symbol

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,...,p_n-1} if the facets have Schläfli symbol {p₁,p₂,...,p_n-2} and the vertex figures have Schläfli symbol {p₂,p₃,...,p_n-1}.

The Schläfli symbol of a line segment is {}. If a polytope has Schläfli symbol {p₁,p₂,...,p_n-1} then its dual polytope has Schläfli symbol {p_n-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.