Schläflisymbol

A Schläfli symbol is a notation used in geometry to describe regular polytopes and tessellations. It is written in the form {p, q, ...}, where each letter represents a specific characteristic of the geometric object. For a regular polygon, the Schläfli symbol is simply {p}, indicating a polygon with p sides. For example, {3} represents an equilateral triangle, and {4} represents a square.

In three dimensions, for regular polyhedra, the symbol is {p, q}. Here, p refers to the number of sides of each face, and q refers to the number of faces that meet at each vertex. The Platonic solids have Schläfli symbols such as {3,3} for the tetrahedron, {4,3} for the cube, {3,4} for the octahedron, {5,3} for the dodecahedron, and {3,5} for the icosahedron.

The notation can be extended to higher dimensions. A four-dimensional regular polytope has a Schläfli symbol of the form {p, q, r}. The first three numbers describe the structure of the cells (3D faces) and how they meet. For example, the 16-cell is denoted as {3,3,4}. The Schläfli symbol uniquely defines a regular polytope or tessellation. It is a fundamental concept in the study of geometric symmetry and classification.