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International Conference on Numerical Computations: Theory and Algorithms

NUMTA 2019: Numerical Computations: Theory and Algorithms pp 208–222Cite as

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

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11974))

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.

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Notes

  1. 1.

    The word “quadratic” refers here to the degree of homogeneity of the variables of the form (so each term of the form has degree two); whereas the adjective “binary” indicates the number of variables involved in the form.

  2. 2.

    This means that the determinant \( \Delta \) is non zero: \( \Delta \, = \,\text{mn}^{'} \, - \,\text{m}^{'} \text{n}\, \ne \,\text{0} \).

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Acknowledgements

This research was supported by the Research Group of Creativity and Innovation in Mathematics, National University of Villa Maria, Argentina.

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Authors and Affiliations

  1. National University of Cordoba, Cordoba, Argentina

    Sandra Visokolskis

  2. National University of Villa Maria, Cordoba, Argentina

    Carla Trillini

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  1. Sandra Visokolskis
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  2. Carla Trillini
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Correspondence to Sandra Visokolskis .

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  1. University of Calabria, Rende, Italy

    Yaroslav D. Sergeyev

  2. University of Calabria, Rende, Italy

    Dmitri E. Kvasov

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Visokolskis, S., Trillini, C. (2020). In the Quest for Invariant Structures Through Graph Theory, Groups and Mechanics: Methodological Aspects in the History of Applied Mathematics. In: Sergeyev, Y., Kvasov, D. (eds) Numerical Computations: Theory and Algorithms. NUMTA 2019. Lecture Notes in Computer Science(), vol 11974. Springer, Cham. https://doi.org/10.1007/978-3-030-40616-5_16

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  • DOI: https://doi.org/10.1007/978-3-030-40616-5_16

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