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Fractals as objects with nontrivial structures at all scales

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  • Lacan, Francis
  • Tresser, Charles

Abstract

Toward the middle of 2001, the authors started arguing that fractals are important when discussing the operational resilience of information systems and related computer sciences issues such as artificial intelligence. But in order to argue along these lines it turned out to be indispensable to define fractals so as to let one recognize as fractals some sets that are very far from being self similar in the (usual) metric sense. This paper is devoted to define (in a loose sense at least) fractals in ways that allow for instance all the Cantor sets to be fractals and that permit to recognize fractality (the property of being fractal) in the context of the information technology issues that we had tried to comprehend. Starting from the meta-definition of a fractal as an “object with non-trivial structure at all scales” that we had used for long, we ended up taking these words seriously. Accordingly we define fractals in manners that depend both on the structures that the fractals are endowed with and the chosen sets of structure compatible maps, i.e., we approach fractals in a category-dependent manner. We expect that this new approach to fractals will contribute to the understanding of more of the fractals that appear in exact and other sciences than what can be handled presently.

Suggested Citation

  • Lacan, Francis & Tresser, Charles, 2015. "Fractals as objects with nontrivial structures at all scales," Chaos, Solitons & Fractals, Elsevier, vol. 75(C), pages 218-242.
    • Handle: RePEc:eee:chsofr:v:75:y:2015:i:c:p:218-242
    DOI: 10.1016/j.chaos.2015.02.003
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    1. Benoit Mandelbrot, 2015. "The Variation of Certain Speculative Prices," World Scientific Book Chapters, in: Anastasios G Malliaris & William T Ziemba (ed.), THE WORLD SCIENTIFIC HANDBOOK OF FUTURES MARKETS, chapter 3, pages 39-78, World Scientific Publishing Co. Pte. Ltd..
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    1. Balankin, Alexander S., 2020. "Fractional space approach to studies of physical phenomena on fractals and in confined low-dimensional systems," Chaos, Solitons & Fractals, Elsevier, vol. 132(C).
    2. Balankin, Alexander S. & Mena, Baltasar, 2023. "Vector differential operators in a fractional dimensional space, on fractals, and in fractal continua," Chaos, Solitons & Fractals, Elsevier, vol. 168(C).
    3. Didier Samayoa & Liliana Alvarez-Romero & José Alfredo Jiménez-Bernal & Lucero Damián Adame & Andriy Kryvko & Claudia del C. Gutiérrez-Torres, 2024. "Torricelli’s Law in Fractal Space–Time Continuum," Mathematics, MDPI, vol. 12(13), pages 1-13, June.

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