Schläfli

Ludwig Karl Schläfli (1814–1895) was a Swiss mathematician renowned for his work in geometry, especially on spaces of dimensions beyond the familiar three. He made foundational contributions to the theory of regular polytopes and tessellations in higher dimensions and introduced a compact notation that bears his name.

Schläfli is best known for the Schläfli symbol, a notation {p, q, r, …} used to describe regular polytopes and their tessellations in any dimension. In this system, the entries encode how many elements of one dimension meet around elements of the previous dimension, such as how many p-sided facets meet at a ridge and how many ridges meet at a face in higher-dimensional polytopes. This notation generalizes the familiar two-dimensional and three-dimensional cases and provides a concise way to classify regular polytopes and uniform tilings across dimensions.

In his work, Schläfli studied the structure of n-dimensional spaces, deriving properties of regular polytopes and exploring their incidence relations and symmetries. His investigations laid the groundwork for later developments in multidimensional geometry and influenced subsequent researchers in the field, even as the full potential of higher-dimensional geometry would be realized many decades after his time.

Schläfli published on the theory of multidimensional spaces and related geometric topics, contributing to the early formal framework for understanding polytopes beyond three dimensions. Today, his name is primarily associated with the notation and concepts that underpin the study of regular polytopes and their tessellations in higher-dimensional geometry.