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

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Numerical Computations: Theory and Algorithms (NUMTA 2019)

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

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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: \( \Delta \, = \,\text{mn}^{'} \, - \,\text{m}^{'} \text{n}\, \ne \,\text{0} \).

References

  1. Archibald, T.: Transmitting disciplinary practice in applied mathematics? Textbooks 1900-1910. In: Epple, M., Hoff Kjeldsen, T., Siegmund-Schultze, R. (eds.) Mathematisches Forschungsinstitut Oberwolfach, Annual Report, vol. 12, pp. 714–719, Germany (2013)

    Google Scholar 

  2. Boole, G.: Exposition of a general theory of linear transformations. Part I. Cambridge Math. J. 3, 1–20 (1841)

    Google Scholar 

  3. Boole, G.: Exposition of a general theory of linear transformations. Part II. Cambridge Math. J. 3, 106–119 (1842)

    Google Scholar 

  4. Boole, G.: On the general theory of linear transformations. Cambridge Dublin Math. J. 6, 87–106 (1851)

    Google Scholar 

  5. Brooks, R., Smith, C., Stone, S., Tutte, W.T.: The dissection of rectangles into squares. Duke Math. J. 7, 312–340 (1940)

    Article  MathSciNet  Google Scholar 

  6. Cayley, A.: On linear transformations. Cambridge Dublin Math. J. 1, 104–122 (1846)

    Google Scholar 

  7. Dehn, M.: Über die Zerlegung von Rechtecken in Rechtecke. Math. Ann. 57, 314–332 (1903)

    Article  MathSciNet  Google Scholar 

  8. Epple, M., Hoff Kjeldsen, T., Siegmund-Schultze, R.: From ‘mixed’ to ‘applied’ mathematics: tracing an important dimension of mathematics and its history. In: Mathematisches Forschungsinstitut Oberwolfach, Annual Report, vol. 12, pp. 657–660. Germany (2013)

    Google Scholar 

  9. Gammerman, A., Vovk, V., Vapnik, V.: Learning by transduction. In: Cooper, G.F., Moral, S. (eds.) Proceedings of the Fourteenth Conference on Uncertainty in Artificial Intelligence, pp. 148–155. Morgan Kaufmann, San Francisco (1998)

    Google Scholar 

  10. Lambert, K.: A natural history of mathematics. George peacock and the making of English algebra. Isis 104(2), 278–302 (2013)

    Article  MathSciNet  Google Scholar 

  11. Moroń, Z.: O rozkladach prostokatów na kwadraty. Przleglad. Matem.-Fizyczny 3, 152–153 (1925)

    Google Scholar 

  12. Nagel, E.: The formation of modern conceptions of formal logic in the development of geometry. In: Nagel, E. (ed.) Teleology Revisited and Other Essays in the Philosophy and the History of Science, pp. 195–259. Columbia University Press, New York (1979)

    Google Scholar 

  13. Parshall, K.H.: Toward a history of nineteenth-century invariant theory. In: Rowe, D.E., McCleary, J. (eds.) The History of Modern Mathematics. Ideas and their Reception, vol. I, pp. 157–206. Academic Press and Harcourt Brace Jovanovich Publishers, Boston (1989)

    Chapter  Google Scholar 

  14. Peacock, G.: A Treatise on Algebra. J. & J. J. Deighton, Cambridge, C. J. G. & F. Rivington and Whittaker, Teacher & Arnot, London (1830)

    Google Scholar 

  15. Rota, G.-C.: What is invariant theory, really? In: Crapo, H., Senato, D. (eds.) Algebraic Combinatorics and Computer Science. A tribute to Gian-Carlo Rota, pp. 41–56. Springer, Italy (1998). https://doi.org/10.1007/978-88-470-2107-5_4

    Chapter  Google Scholar 

  16. Suppes, P.: New foundations of objective probability. Axioms for propensities. Stud. Log. Found. Math. 74, 515–529 (1973)

    Article  MathSciNet  Google Scholar 

  17. Tutte, W.T.: Graph Theory As I Have Known It. Clarendon Press/Oxford University Press, Oxford/New York (1998)

    MATH  Google Scholar 

  18. Visokolskis, A.S.: El fenómeno de la transducción en la matemática. Metáforas, analogías y cognición. In: Pochulu, M., Abrate, R., Visokolskis, A.S. (eds.) La metáfora en la educación. Descripción e implicaciones, pp. 37–53. Eduvim, Villa María (2009)

    Google Scholar 

  19. Visokolskis, A.S.: La noción de análisis como descubrimiento en la historia de la matemática. Propuesta de un modelo de descubrimiento creativo. Ph.D. Doctoral Dissertation. National University of Cordoba, Cordoba, Argentina (2016)

    Google Scholar 

  20. Visokolskis, A.S., Carrión, G.: Creative insights: dual cognitive processes in perspicuous diagrams. In: Sato, Y., Shams, Z. (eds.) Proceedings of the International Workshop on Set Visualization and Reasoning SetVR 2018, Set Visualization and Reasoning, pp. 28–43, Edinburgh (2018)

    Google Scholar 

  21. Visokolskis, A.S.: Filosofía de la creatividad en contextos matemáticos. Espacios transductivos como alternativa al dilema de Boden. Paper Presented at XX Jornadas Rolando Chuaqui Kettlun, Santiago, Chile, 27–30 August 2019

    Google Scholar 

  22. Wolfson, P.R.: George Boole and the origins of invariant theory. Hist. Math. 35, 37–46 (2008)

    Article  MathSciNet  Google Scholar 

  23. Woodger, J.H.: The Axiomatic Method in Biology. Cambridge University Press, Cambridge (1937)

    MATH  Google Scholar 

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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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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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