In the Quest for Invariant Structures Through Graph Theory, Groups and Mechanics: Methodological Aspects in the History of Applied Mathematics

Abstract

The purpose of this paper is to analyze a geometrical case study as a sample of an intended methodology based on invariant theory’s strategies, which have been developed particularly throughout the nineteenth century as one of the cornerstones of mathematics [15, p. 41], and whose resolution was reached by means of a combination of different disciplines: graph theory, mechanics and group theory, among others. This case study presents the “perfect squared rectangle problem”, that is an exhaustive classification of the dissection of a rectangle into a finite number of unequal squares. Despite its simplicity, in both description and mathematical resolution, it provides plausible elements of generalization from “the ‘applied field’ of mathematics” [8, p. 658], as a special case of applied mathematical toolkit [1, p. 715], related to the practice of invariant strategies that remain fixed through changes.