ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)


For citation:

Vdovina G. M., Trubetskov D. I. The 100th anniversary of fractal geometry: From Julia and Fatou through Hausdorff and Besicovitch to Mandelbrot. Izvestiya VUZ. Applied Nonlinear Dynamics, 2020, vol. 28, iss. 2, pp. 208-222. DOI: 10.18500/0869-6632-2020-28-2-208-222

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
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Russian
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Article
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510; 514

The 100th anniversary of fractal geometry: From Julia and Fatou through Hausdorff and Besicovitch to Mandelbrot

Autors: 
Vdovina Galina Mihajlovna, Saratov State University
Trubetskov Dmitriy Ivanovich, Saratov State University
Abstract: 

The purpose of this article is to present the biographies of the main creators of fractal geometry from the moment the first ideas arose, when the term «fractal» did not exist, to the present day. The main subjects of the article are Julia, Fatou, Richardson, Hausdorff, Besicovitch and Mandelbrot. The fates of these people are rich in dramatic events, some are tragic. Methods. The content of this paper is based on the analysis of various works on the biographical facts of the founders of fractal geometry; these facts are scattered in literature relevant to the issue under consideration. Results. It is both clear and surprising that the first articles, that could be called the basis of fractal geometry, appeared when there were no computers. The main subjects here are Julia and Fatou. In fact, a hundred years ago they discovered what today is known as the Mandelbrot fractal. Mandelbrot managed to visually represent it. Another impetus for the study of fractals was made by Richardson, who showed that, for example, a winding coastline cannot be measured by methods suitable for smooth curves. The mathematical concept of fractal dimension was introduced by Hausdorff and developed in the works of Besikovitch. Some facts from the life of Julia, Fatou, Richardson and Besikovitch are presented in the article. Special attention is given to the tragic fate of Hausdorff, who survived the Nazi genocide during World War II. It led Hausdorff to the decision to commit suicide. The biography of Mandelbrot is described briefly. His interest in what he later called a fractal is related to his work in economics and telephone signal analysis. In the final part of the article, we quote some authors who were convinced that there are no fractals in the real world, just as there are no perfect lines and circles. However, mathematical models that describe reality help us better understand it. Conclusion. This article is not a consistent overview of the topic; it is a collection of small essays that develops into an integrative view on the centennial life of fractal geometry

 

Acknowledgements. This work was supported by Russian Foundation for Basic Research, grant no. 18-02-00666.

Reference: 

  1. Mandelbrot B. The Fractal Geometry of Nature. New York, W.H. Freeman and Company. 1983. 468 p.

2. Demenok S.L. Superfractal. Saint Petersburg, Strata, 2015. 196 p. (in Russian).

3. Art-fractal. Collections of Articles. Transl. from eng., fr. by E.B. Nikolaeva. Saint Petersburg, Strata, 2015, 156 p. (in Russian).

4. Gulia G. Memoire sur l’Iteration des Fonctions Rationnelles. Journal de Mathematiques Pures et Appliquees, 1918, vol. 1, pp. 47–245.

5. Fatou P. Sur les Equations Fonctionnelles. Bulletin Societe. Math. France. 1919, vol. 47, pp. 161–271.

6. Bassa M. Una Nueva Manera de Ver el Mundo: La Geometr´ıa Fractal. Navarra, RBA Coleccionables, 2011, 142 p.

7. Mandelbrot B. Fractals and the Rebirth of Iteration Theory, pp. 151–160. In: Peitgen H.-O., Richter P.H. The Beauty of Fractals, Berlin, Springer. 1986. 199 p.

8. Photo Archive. Available at: https://www.wikiwand.com/en/ (accessed 3 november 2019).

9. Gaston Julia. Available at: http://www.bibmath.net/bios/index.php?action=affiche&quoi=julia (accessed 3 november 2019).

10. Richardson L.F. The problem of contiguity: An appendix of statistics of deadly quarrels. General Systems Yearbook. 1961, vol. 6, pp. 139–187.

11. Richardson L.F. Weather Prediction by Numerical Process. Cambridge, Cambridge University Press. 1922. 66 p.

12. Duran A.J. The Poetry of Numbers. Revealing Beauty in Maths. Navarra, RBA Coleccionables, 2010, 151 p.

13. Kostitsyn V.I. Rectors of Perm State University. 1916–2016. Perm: Perm State University, 2016, 352 p. (in Russian).

14. Trubetskov D.I., Trubetskova E.G. Fractal geometry. Izvestiya VUZ. Applied Nonlinear Dynamics, 2016, vol. 24, no. 6, pp. 5–38. (in Russian). DOI: 10.18500/0869-6632-2016-24-6-4-38

15. Rozenberg G.S., Chuprunov E.V., Gelashvili D.B., Iudin D.I. Nature’s geometry has a fractal face. Vestnik of Lobachevsky University of Nizhni Novgorod, 2011, no. 1, pp. 411–417 (in Russian).

16. Beveridge C. Cracking Mathematics: You, this book and 4,000 years of theories. London, Cassel, 2016, 402 p.

17. Pineiro G. The Sphere that Wanted to be Infinite: The paradoxes of measurement. Barcelona, RBA Coleccionables, 2017, 141 p.

18. Makarenko N.G. Fractals, attractors, neural networks and all that. Scientific session MEPHI – 2002. The 4th Russian scientific technical conference «Neuroinformatics – 2002» Lectures on Neuroinformatics. Part 2. Moscow, MEPHI, 2002, pp. 121–169 (in Russian).

19. Vovk I.S., Grinchenko V.T., Matsypuka V.T., Snarskii A.A. Dozen Lectures about Fractals: From the Object of Admiration to the Instrument of Knowledge. Moscow, LENAND, 2018, 264 p. (in Russian).

20. Feder J. Fractals. New York and London, Plenum Press, 1988, 283 p.

21. Schroeder M. Fractals, Chaos, Power Laws. New York, W.H. Freeman and Company, 1991, 429 p.

22. Zel’dovich Ya.B., Sokolov D.D. Fractals, similarity, intermediate asymptotics. Sov. Phys. Usp, 1985, vol. 28, no. 7, pp. 608–616. DOI: 10.3367/UFNr.0146.198507d.0493.

23. Morozov A.D. Introduction to the Fractal Theory. Nizhny Novgorod, Nizhny Novgorod University, 1999, 140 p. (in Russian).

24. Crownover R.M. Introduction to Fractals and Chaos. Boston – London, Jones and Bartlett Publishers, 1995, 306 p.

25. Drobinin N. The Game «Chaos and fractals», Kvant, 1997, no. 4, pp. 2–8 (in Russian).

26. D’iudni A.K.The Mandelbrot set and its related Julia sets. In the World of Science, 1988, no. 1, pp. 88–92 (in Russian).

 

Received: 
25.11.2019
Accepted: 
04.01.2020
Published: 
30.04.2020