Schläfli symbol

The Schläfli symbol is a notation of the form {p,q,r, ...} that defines regular polytopes and tessellations. It is built recursively: {p} describes a convex regular polygon with p sides, so {3} is an equilateral triangle and {4} is a square, and a star polygon uses a fraction such as {5/2} for the pentagram. A regular polyhedron with q regular p-sided faces around each vertex is written {p,q}, so the cube is {4,3}, and a four dimensional polytope with r polyhedral cells around each edge is written {p,q,r}, so the tesseract is {4,3,3}. The vertex figure of {p,q,r,...,y,z} is {q,r,...,y,z}, and the dual polytope is written with the entries in reverse order. Depending on the angle defect of the construction, a symbol can stand for a finite convex polytope, an infinite tessellation of Euclidean space or a tessellation of hyperbolic space. It is named after the nineteenth century Swiss mathematician Ludwig Schläfli, whose notation was almost unknown in his lifetime and was rediscovered independently by several others, including Thorold Gosset.