Fractal Geometry: Axioms, Fractal Derivative and Its Geometrical Meaning

Authors

  • V. K. Balkhanov Institute of Physical Materials Science of the Siberian Branch of the Russian Academy of Sciences, Ulan-Ude City, Russia

DOI:

https://doi.org/10.30564/jees.v1i1.475

Abstract

Physics success is largely determined by using mathematics. Physics often themselves create the necessary mathematical apparatus. This article shows how you can construct a fractal calculus - mathematics of fractal geometry. In modern scientific literature often write from a firm that "there is no strict definition of fractals", to the more moderate that "objects in a certain sense, fractal and similar." We show that fractal geometry is a strict mathematical theory, defined by their axioms. This methodology allows the geometry of axiomatised naturally define fractal integrals and differentials. Consistent application on your input below the axiom gives the opportunity to develop effective methods of measurement of fractal dimension, geometrical interpretation of fractal derivative gain and open dual symmetry.

Keywords:

Fractal geometry, Fractal dimension, Fractal calculus, Duality

References

[1] Mandelbrot B.B, Les objets fractals: forme, hazard et dimension, Paris: Flammarion, 1975.

[2] Gabriele A. Losa, Dušan Ristanović, Dejan Ristanović, Ivan Zaletel, Stefano Beltraminelli, From Fractal Geometry to Fractal Analysis, Applied Mathematics, 2016, 7, 346-354 Published Online March 2016 in SciRes. http://www.scirp.org/journal/am . http://dx.doi.org/10.4236/am.2016.74032.

[3] Dr. Vyomesh Pant, Poonam Pant, FRACTAL GEOMETRY: AN INTRODUCTION, Journal of Indian Research, v 1, no 2, 66-70, 2013.

[4] Kenneth Fakconer, Fractal geometry, matematical and applications, second edition, (wiley).

[5] SOUMITRO BANERJEE, NATURE’S GEOMETRY, Breakthrough, Vol.13, No.4, January 2009.

[6] Balkhanov V.K., Foundations of fractal geometry and fractal calculus, (Buryat State University Publishing House, Ulan-Ude, 2013). http://ipms.bscnet.ru/publications/src/2013/FractGeomet.pdf.

[7] Samko C.G., Kilbas A.A. and Marichev O.I., Integrals and derivatives of fractional order and some of their applications, (Nauka i Tekhnika, Minsk, 1987). Samko, S., Kilbas, A., and Marichev, O. (1993). Fractional integrals and derivatives: theory and applications.

[8] Grosberg A.Yu, Khokhlov A.P. Statistical Physics of macromolecules, (Science, Moscow, 1989).

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How to Cite

Balkhanov, V. K. (2019). Fractal Geometry: Axioms, Fractal Derivative and Its Geometrical Meaning. Journal of Environmental & Earth Sciences, 1(1), 1–5. https://doi.org/10.30564/jees.v1i1.475

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Article Type

Article