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The Sierpinski curve viewed by numerical computations with infinities and infinitesimals

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  • Caldarola, Fabio

Abstract

The Sierpinski curve is one of the most known space-filling curves and one with the highest number of applications. We present a recently proposed computational methodology based on the infinite quantity called grossone to investigate the behavior of two different constructions of the Sierpinski curve. We emphasize that, adopting this point of view, we have infinitely many Sierpinski curves depending, contrarily to traditional analysis, on each specific starting configuration. Of particular interest are some power series expansions in the new infinitesimal quantities emerging from the study of the considered curves.

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  • Caldarola, Fabio, 2018. "The Sierpinski curve viewed by numerical computations with infinities and infinitesimals," Applied Mathematics and Computation, Elsevier, vol. 318(C), pages 321-328.
    • Handle: RePEc:eee:apmaco:v:318:y:2018:i:c:p:321-328
    DOI: 10.1016/j.amc.2017.06.024
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    References listed on IDEAS

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    1. Sergeyev, Yaroslav D., 2009. "Evaluating the exact infinitesimal values of area of Sierpinski’s carpet and volume of Menger’s sponge," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 3042-3046.
    2. Amodio, P. & Iavernaro, F. & Mazzia, F. & Mukhametzhanov, M.S. & Sergeyev, Ya.D., 2017. "A generalized Taylor method of order three for the solution of initial value problems in standard and infinity floating-point arithmetic," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 141(C), pages 24-39.
    3. Sergeyev, Yaroslav D., 2007. "Blinking fractals and their quantitative analysis using infinite and infinitesimal numbers," Chaos, Solitons & Fractals, Elsevier, vol. 33(1), pages 50-75.
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    Cited by:

    1. Tohmé, Fernando & Caterina, Gianluca & Gangle, Rocco, 2020. "Computing Truth Values in the Topos of Infinite Peirce’s α-Existential Graphs," Applied Mathematics and Computation, Elsevier, vol. 385(C).
    2. Fiaschi, Lorenzo & Cococcioni, Marco, 2021. "Non-Archimedean game theory: A numerical approach," Applied Mathematics and Computation, Elsevier, vol. 409(C).
    3. Renato Leone & Giovanni Fasano & Massimo Roma & Yaroslav D. Sergeyev, 2020. "Iterative Grossone-Based Computation of Negative Curvature Directions in Large-Scale Optimization," Journal of Optimization Theory and Applications, Springer, vol. 186(2), pages 554-589, August.
    4. Caldarola, Fabio & Maiolo, Mario, 2021. "A mathematical investigation on the invariance problem of some hydraulic indices," Applied Mathematics and Computation, Elsevier, vol. 409(C).

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