The fractal application formalism in mathematical description of the structure

V. I. Bolshakov, V. N. Volchuk, Yu. I. Dubrov


Formulation of the problem. The practice of fractal formalism for identifying corresponding structures has shown that in this case, often there is the inadequacy of their perception as a fractal. This is probably due to the fact that any object recognition occurs on the basis of formation of the model (if recognition is made directly by man, the pattern formed in his mind). An inadequate recognizable model leads to the loss dramatically. This is often observed in the endeavors of researchers, to create mathematical models (MM) in the field of materials science, in which the structure, such as metal, intractable deterministic description, they tried to synthesize MM, using statistical analysis. Objective. In this regard, we suggest that if it is the dimension will be determined at the stage of determination of the type MM, its appearance will be apparent. Results and discussion. Unexpectedly for many researchers in accessing MM such a system, which, for example, represents the structure of a material, it was not possible to obtain an one-to-one correspondence between her and every structure of MM. Identifying the reasons for this discrepancy has led scientists to determine the factors influencing this disparity, the main ones were: factors not included in the MM, significantly affecting the function; MM does not display non-linearity states identify the object space (SIOS). Most likely, this is the root cause nonlinearity SIOS appearance of fractal dimensions. Conclusions. Thus, the pattern presented in the form of a function, be sure to include metric SIOS. Once again, we note that the reason for the inadequacy of the model often lies not in the shortcomings of the model, and in the nature of the phenomenon under study, which can be identified by the use of fractal formalism. After an extensive literature review, the authors of this article had the impression that the fractal dimension of the relation with the physical nature of the object displayed in the undeservedly little publications that reflect the practical applicability of this phenomenon. Probably the latter due to the fact that for each material, with its corresponding fractal characteristics, there is one and only one, only his inherent fractal dimension.


mathematical model; structure; fractal dimension; linear space; function



Mandelbrot B.B. The Fractal Geometry of Nature / B.B. Mandelbrot. − New-York, San Francisco : Freeman Publ., 1982.− 480 p. – Режим доступа:

Rényi A. Probability Theory / A. Rényi. − Amsterdam. The Netherlands : North-Holland Publ., 1970. − 670 р. – Режим доступа:

Hentschel H.G. The infinite number of generalized dimensions of fractals and strange attractors. Physica D / H.G. Hentschel, I.E. Procaccia. − 1983. − Vol. 8 (3). − Pp. 435-444. − Режим доступа: http://www.sciencedirectр.com/ science/article/pii/016727898390235X.

Свечников А. В. Материалы с кластерной структурой - новые свойства, новые возможности / А. В. Свечников // Сучасне матеріалознавство XXI сторіччя: збірник Національної Академії наук України. - Київ : Наукова думка, 1998. - С. 352−369. − Режим доступа:

Дубров Ю. Пути идентификации периодических многокритериальных технологий / Ю. Дубров, В. Большаков, В. Волчук. – Саарбюкхен, Германия: Palmarium Academic Publishing, 2015. – 236 p. − Режим доступа: Пути-идентификации-периодических-многокритериальных-технологий/isbn978 -3-659-60262-7.

Большаков Вад. І. Часткова компенсація неповноти формальної аксіоматики при ідентифікації структури металу / Вад. І. Большаков, В. І. Большаков, В. М. Волчук [та ін.] // Вісник Національної Академії наук України. - 2014. - № 12. – С. 45-48. − Режим доступа:

Большаков В. И. Особенности применения мультифрактального формализма в материаловедении / В. И. Большаков, В. Н. Волчук, Ю. И. Дубров // Доповіді Національної Академії наук України. - 2008. - №11. - С. 99-107. − Режим доступа:

Федер Е. Фракталы: монография / Е. Федер. – Москва : Мир, 1991. - 254 с. − Режим доступа:Динамические %20системы%20и%20Хаос/Федер%20Е.,%20Фракталы,%201991.pdf

Фрактальные множества, функции, распределения : монография / [А. Ф. Турбин, Н. В. Працевитый]. - Киев : Наукова думка, 1992. - 208 с. − Режим доступа: VDPU&S21FMT=&S21ALL=(%3C.%3EK%3D%D0%91%D0%90%D0%97%D0%98$%3C.%3E)&FT_REQUEST=&FT_PREFIX=&Z21ID=&S21STN=1&S21REF=&S21CNR=20

Schaefer D.W. Structure of Random Porous Materials: Silica Aerogel. / D.W. Schaefer, K. D. Keefer // Phys. Rev. Lett. − 1986. − Vоl. 56. − Рp. 2199−2202. − Режим доступа: ett.56.2199.

Witten Т.А. Diffusion-limited aggregation / Т.А. Witten, L. M. Sander // Phys. Rev. − Ser.B. − 1983. − Vol. 27. − Pp. 5686. − Режим доступа:

Calcagni G. Particle-physics constraints on multifractal spacetimes / G. Calcagni, G. Giuseppe Nardelli, D. Rodríguez-Fernández // Phys. Rev. − Ser. D. − 2016. − Vol. 93. − Рр. 25005. − Режим доступа: 10.1103/PhysRevD.93.025005.

Большаков В. И. Материаловедческие аспекты применения вейвлетно-мультифрактального подхода для оценки структуры и свойств малоуглеродистой стали / В. И. Большаков, В. Н. Волчук // Металлофизика и новейшие технологии. - 2011. – Т. 33. - № 3. - С. 347–360. − Режим доступа:

Большаков В. И. Об оценке применимости языка фрактальной геометрии для описания качественных трансформаций материалов / В. И. Большаков, Ю. И. Дубров // Доповіді Національної Академії наук України. - № 4. - 2002. - С. 116-121. − Режим доступа:


Mandelbrot B.B. The Fractal Geometry of Nature. New-York, San Francisco: Freeman Publ., 1982, 480 p.

Rényi A. Probability Theory. Amsterdam. The Netherlands: North-Holland Publ., 1970, 670 р.

Hentschel H.G. and Procaccia I.E.The infinite number of generalized dimensions of fractals and strange attractors. Physica D. 1983, vol. 8 (3), pp. 435-444.

Svechnikov A.V. Materialy s klasternoy strukturoy - novyye svoystva, novyye vozmozhnosti [Materials with cluster structure - new features, new opportunities]. Modern materials XXI century: a collection of National Academy of Sciences of Ukraine, department of Physics and Engineering. Kyiv : Naukova Dumka, 1998, рp. 352−369. (in Russian).

Dubrov Yu., Bolshakov V. and Volchuk V. Puti identifikatsii periodicheskikh mnogokriterial'nykh tekhnologiy [Road periodic identification of multi-criteria Technology]. Saarbrücken, Deutschland: Palmarium Academic Publishing, 2015, 236 p. (in Russian).

Bolshakov Vad.I., Bolshakov V.I., Volchuk V.N. [and oth]. Chastkova kompensatsiya nepovnoty formalʹnoyi aksiomatyky pry identyfikatsiyi struktury metalu [The partial compensation of incompleteness of formal axiomatics in the identification of the metal structure]. Vіsnik Nacіonal'noї Akademії nauk Ukraїni [Bulletin of the National Academy of Sciences of Ukraine]. 2014, no. 12, pp. 45-48. (in Ukrainian).

Bolshakov V.I., Volchuk V.N. and Dubrov Yu.I. Osobennosti primeneniya mul'tifraktal'nogo formalizma v materialovedenii [Features of the multifractal formalism in materials]. Dopovіdі Nacіonal'noї Akademії nauk Ukraїni [Reports of the National Academy of Sciences of Ukraine]. 2008, no.11, pp. 99−107. (in Russian).

Feder E. Fraktaly [Fractals]. Moscow : Mir, 1991, 254 p. (in Russian).

Turbin A.F. and Pratsevity N.V. Fraktal'nyye mnozhestva, funktsii, raspredeleniya [Fractal sets, functions, distribution]. Kyiv : Naukova Dumka, 1992, 208 р. (in Ukrainian).

Schaefer D.W. and Keefer K.D. Structure of Random Porous Materials: Silica Aerogel. Phys. Rev. Lett., 1986, vоl. 56, pp. 2199−2202.

Witten Т.А. and Sander L.M. Diffusion-limited aggregation. Phys. Rev. Ser. B. 1983, vol. 27, pp. 5686.

Calcagni G., Giuseppe Nardelli G. and Rodríguez-Fernández D. Particle-physics constraints on multifractal spacetimes. Phys. Rev. D. 2016, vol. 93, pр. 25005.

Bolshakov V.I. and Volchuk V.N. Materialovedcheskiye aspekty primeneniya veyvletno-mul'tifraktal'nogo podkhoda dlya otsenki struktury i svoystv malouglerodistoy stali [Materialovedchesky aspects of wavelet-multifractal approach for assessing the structure and properties of low-carbon steel]. Metallofizika i novejshie tehnologii [Metal Physics and Advanced Technologies]. 2011, vol. 33, no. 3, pp. 347−360. (in Russian).

Bolshakov V.I. and Dubrov Yu.I. Ob otsenke primenimosti yazyka fraktal'noy geometrii dlya opisaniya kachestvennykh transformatsiy materialov [An estimate of the applicability of the language of fractal geometry to describe Ria-quality transformation of materials]. Dopovіdі Nacіonal'noї Akademії nauk Ukraїni [Reports National Academy of Sciences of Ukraine]. No. 4, 2002, pp. 116-121. (in Russian).

GOST Style Citations