Schl�fli symbol

Schl�fli symbol

In mathematics, the Schl�fli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope.

It is defined as follows. The Schl�fli symbol of a polygon with n edges is {n}. The Schl�fli symbol of a polyhedron is {p,q} if its faces are p-gons, and each vertex is surrounded by q faces. Note that the Schl�fli symbol is not well defined for polyhedra which are not (sufficiently) regular (such as the prism).

The Schl�fli 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}

Schl�fli symbols may also be defined for regular tessellations of euclidean or hyperbolic space in a similar way.

For higher dimensional polytopes, the Schl�fli symbol is defined recursively as {p₁,p₂,...,p_n-1} if the facets have Schl�fli symbol {p₁,p₂,...,p_n-2} and the vertex figures have Schl�fli symbol {p₂,p₃,...,p_n-1}.

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

The Schl�fli symbol is named after the 19th century mathematician Ludwig Schl�fli who made important contributions in geometry and other areas.