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Evaluating the exact infinitesimal values of area of Sierpinski’s carpet and volume of Menger’s sponge

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  • Sergeyev, Yaroslav D.

Abstract

Very often traditional approaches studying dynamics of self-similarity processes are not able to give their quantitative characteristics at infinity and, as a consequence, use limits to overcome this difficulty. For example, it is well known that the limit area of Sierpinski’s carpet and volume of Menger’s sponge are equal to zero. It is shown in this paper that recently introduced infinite and infinitesimal numbers allow us to use exact expressions instead of limits and to calculate exact infinitesimal values of areas and volumes at various points at infinity even if the chosen moment of the observation is infinitely faraway on the time axis from the starting point. It is interesting that traditional results that can be obtained without the usage of infinite and infinitesimal numbers can be produced just as finite approximations of the new ones. The importance of the possibility to have this kind of quantitative characteristics for E-Infinity theory is emphasized.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:chsofr:v:42:y:2009:i:5:p:3042-3046
    DOI: 10.1016/j.chaos.2009.04.013
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    References listed on IDEAS

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    1. Iovane, G., 2006. "Cantorian spacetime and Hilbert space: Part I—Foundations," Chaos, Solitons & Fractals, Elsevier, vol. 28(4), pages 857-878.
    2. He, Ji-Huan, 2006. "Application of E-infinity theory to turbulence," Chaos, Solitons & Fractals, Elsevier, vol. 30(2), pages 506-511.
    3. El Naschie, M.S., 2005. "A guide to the mathematics of E-infinity Cantorian spacetime theory," Chaos, Solitons & Fractals, Elsevier, vol. 25(5), pages 955-964.
    4. Iovane, G. & Chinnici, M. & Tortoriello, F.S., 2008. "Multifractals and El Naschie E-infinity Cantorian space–time," Chaos, Solitons & Fractals, Elsevier, vol. 35(4), pages 645-658.
    5. 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. 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.
    2. Lolli, Gabriele, 2015. "Metamathematical investigations on the theory of Grossone," Applied Mathematics and Computation, Elsevier, vol. 255(C), pages 3-14.
    3. Renato De Leone & Giovanni Fasano & Yaroslav D. Sergeyev, 2018. "Planar methods and grossone for the Conjugate Gradient breakdown in nonlinear programming," Computational Optimization and Applications, Springer, vol. 71(1), pages 73-93, September.
    4. De Leone, Renato, 2018. "Nonlinear programming and Grossone: Quadratic Programing and the role of Constraint Qualifications," Applied Mathematics and Computation, Elsevier, vol. 318(C), pages 290-297.
    5. Kauffman, Louis H., 2015. "Infinite computations and the generic finite," Applied Mathematics and Computation, Elsevier, vol. 255(C), pages 25-35.
    6. 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.
    7. Herrmann, Richard, 2015. "A fractal approach to the dark silicon problem: A comparison of 3D computer architectures – Standard slices versus fractal Menger sponge geometry," Chaos, Solitons & Fractals, Elsevier, vol. 70(C), pages 38-41.
    8. 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.
    9. Cococcioni, Marco & Pappalardo, Massimo & Sergeyev, Yaroslav D., 2018. "Lexicographic multi-objective linear programming using grossone methodology: Theory and algorithm," Applied Mathematics and Computation, Elsevier, vol. 318(C), pages 298-311.

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