Schläflitype

The Schläfli type is a classification system used in mathematics, particularly in geometry and combinatorics. It was introduced by the Swiss mathematician Ludwig Schläfli and is often used to identify regular polytopes and their symmetries. The Schläfli type is defined by a set of integers that describe the polytope's structure.

The general formula for the Schläfli type of a polytope is {p,q,r,...}, where p, q, r, etc. are integers that indicate the number of vertices, edges, and faces, respectively, that meet at each vertex. This formula extends the traditional way of representing polyhedra, which typically uses two numbers {p,q} to describe their structure. The Schläfli type provides a more comprehensive description of polytopes in higher-dimensional spaces.

The integers in the Schläfli type are not just limited to the number of vertices, edges, and faces. In the case of regular polytopes that have more than three dimensions, additional integers are used to describe the distances between the vertices and the curvature of the polytope's edges. Each polytope type can be uniquely represented by its set of integers, allowing mathematicians to identify and compare different polytopes.

Schläfli types are widely used in mathematics and physics to study and classify various types of polytopes. They are particularly useful in understanding the properties of higher-dimensional spaces and the symmetries of the molecules and crystals that make up the structure of materials.