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## Scientific Bulletin. Physical and Mathematical Research

#### Abstract

This article is devoted to the study of the future position of fractal measurements. Different methods of computer modeling of a wide range of classes of fractal geometric objects are described in detail, as well as the main methods of mathematical analysis of fractal size of virtual and real fractals are described. The article explains the differences between the concept of fractals, their properties, B. Mandelbrot's tariff, Hausdorf-Bezikovich scale, Minkowski-Buligan scale, topological measurement, the concept of fractal measurement and measurement in Euclidean geometry. This article provides basic information about fractals. A detailed description of the various methods of computer modeling of a wide range of classes of geometric and temporal fractal objects is given. It also describes the main well-known methods of mathematical analysis of the fractal size of virtual and real fractals. The basics of fractal theory, the concept of fractal size are described. Examples of numerous project fractals (Cox, Minkowski, Serpin, Lev, Kontor, etc.) are considered. Issues of chaos theory, the scheme of modeling fractals with the determination of the size of unusual attractors, as well as the calculation procedures for the determination of fractal dimensions are considered. Fractal measurements of several geometric objects have also been identified. These include the Cox curve, the Given curve, the Office set, the four-sided star-shaped fractal, the fractal dimension divided by the middle of the side of the triangle, and the fractal dimension divided by the middle of the height of the triangle. Hausdorf-Bezikovich and Minkowski-Buligan measurements were used to determine the fractal scale. Differences in approaches to fractal analysis of real existing and abstract mathematical fractals indicate that a special numerical algorithm must be used to calculate the capacitive size of the fractals. The results are obtained at each step of the iteration, which are presented in the form of formulas and graphs.

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#### References

1. Mandelbrot B.B. Les Objects Fractals: Forme, Hasard et Dimension.- Paris: Flammarion, 1975, 1984, 1989, 1995;
2. Mandelbrot B.B. Fractals: Forme, Chance and Dimension. - San - Francisco: Freeman, 1977. - 365 p.;
3. Mandelbrot B.B. The Fractals Geometry of Nature. – N.Y.: Freeman, 1982. - 468 p. (Рус. пер.:
4. Mandelbrot B.B. Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. – N.Y.: Springer-Verlag, 1997. - 551 p.;
5. Mandelbrot B.B. Fractales, Hasard et Finance (1959-1997). – Paris: Flammarion, 1997. - 246 p.
6. Balxanov V.K. Osnovы fraktalnoy geometri fraktalnogo ischisleniya. Otv. red. Ulan-Ude: Izd-vo Buryatskogo gosuniversiteta, 2013. – 224 s.
7. Bojokin S.V., Parshin D.A. Fraktal va multifraktal. – M.: Ijevsk: RXD, 2001.
8. Kronover R.M. Fraktal va xaos dinamicheskix sistemax // – M.: Postmarket, 2000.
9. Morozov A.D. Vvedeniye teoriyu fraktalov. - Nijniy Novgorod: NijGU, 1999.
10. Nazirov Sh.A., Anarova Sh.A., Nuraliyev F.M. Fraktallar nazariyasi asoslari. Toshkent-2017, «Navruz» nashriyoti, 128 b.
11. Kononyuk A.Ye. Diskretno-nepremernaya matematika. (Poverxnosti). — V 12-i kn. Kn.6. ch.2.— K.:Osvіta Ukraїni. 2016. – 618 s.
12. Bondarenko B.A. Generalized Pascal Triangles and Pyramids, their Fractals, Graphs, and Applications – USA, Santa Clara: Fibonacci Associations, The Third Edition. – 2010. – 296 p.
13. Gerald Elgar. Measure, Topology, and Fractal geometry. Second Edition. Springer Science+Business Media, LLC. 2008. – 272 p.
14. Kenneth Falconer. Fractal Geometry. Mathematical Foundations and Applications. Third Edition. University of St Andrews UK. Wiley. 2014. – 400 p.
15. Nuraliev F.M., Anarova Sh.A., Narzulloev O.M. Mathematical and software of fractal structures from combinatorial numbers. International Conference on Information Science and Communications Technologies ICISCT 2019 Applications, Trends and Opportunities 4th, 5th and 6th of November 2019, Tashkent University of Information Technologies TUIT

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