Schläfli symbol

In geometry, the Schläfli symbol is a notation of the form {p,q,r,...} that defines regular polytopes and tessellations.

The Schläfli symbol is named after 19th-century Swiss mathematician Ludwig Schläfli who made important contributions in geometry and other areas.

The Schläfli symbol is a recursive description, starting with a p-sided regular polygon as {p}. For example, {3} is an equilateral triangle, {4} is a square and so on.

A regular polyhedron that has q regular p-sided polygon faces around each vertex is represented by {p,q}. For example, the cube has 3 squares around each vertex and is represented by {4,3}.

A regular 4-dimensional polytope, with r {p,q} regular polyhedral cells around each edge is represented by {p,q,r}. For example a tesseract, {4,3,3}, has 3 cubes, {4,3}, around an edge.

In general a regular polytope {p,q,r,...,y,z} has z {p,q,r,...,y} facets around every peak, where a peak is a vertex in a polyhedron, an edge in a 4-polytope, a face in a 5-polytope, a cell in a 6-polytope, and an (n-3)-face in an n-polytope.

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