Six mathematical gems from the history of Distance Geometry

TL;DR

This paper explores the rich history of Distance Geometry by presenting key mathematical results and proofs, highlighting their significance and interconnections in the development of the field.

Contribution

It provides a historical overview and elementary proofs of six fundamental theorems in Distance Geometry, emphasizing their mathematical beauty and importance.

Findings

Proves Heron's formula and Cauchy's rigidity theorem

Generalizes Heron's formula to higher dimensions

Characterizes semi-metric spaces and relates distance to positive semidefinite matrices

Abstract

This is a partial account of the fascinating history of Distance Geometry. We make no claim to completeness, but we do promise a dazzling display of beautiful, elementary mathematics. We prove Heron's formula, Cauchy's theorem on the rigidity of polyhedra, Cayley's generalization of Heron's formula to higher dimensions, Menger's characterization of abstract semi-metric spaces, a result of Goedel on metric spaces on the sphere, and Schoenberg's equivalence of distance and positive semidefinite matrices, which is at the basis of Multidimensional Scaling.