# FRACTAL GEOMETRY

Cite this article as:

Trubetskova E. G., Trubetskov D. I. FRACTAL GEOMETRY. Izvestiya VUZ, Applied Nonlinear Dynamics, 2016, vol. 24, iss. 6, pp. 4-38 DOI: 10.18500/0869-6632-2016-24-6-4-38

The article deals with the bases of fractal geometry and fates of its creators. The biographies and the discoveries of Felix Hausdorff and Abram Besicovitch – the main characters of the great play called fractal geometry – are presented with the possible degree of detail. There is no doubt that the author and director of this play was Benoit Mandelbrot. The article presents his biography and brief descriptions of the lives of his predecessors: Henri Poincare, Maurice Gaston Julia and Pierre Gaston Jose Fatou. Special attention is paid to the Poincare’s discovery of homoclinic tangle bundle.

The biography of Lewis Fry Richardson, whose posthumous publication of the results of the length’s measurement of the UK coastline laid the basis of modern fractal geometry, is presented. A brief presentation of Richardson’s results on the turbulence’s theory and possible contemporary approaches to the solution of the Navier–Stokes equations are given. It led to the conclusion that turbulence should be described in terms of fractal geometry.

There was presented Richardson’s empirical theory, that was applied towards the mea-

surement of the coastline. The definition of topological dimension and the derivation of the Hausdorff–Besicovitch dimension are given. Two definitions of fractals, offered by Mandelbrot, are provided. Within the framework of these definitions Cantor set, triadic Koch curve, Sier-

pinski carpet and cloth, Menger sponge, Peano curve, Weierstrass curve are analyzed.

The continual family of Alfred Renyi dimensions, a particular case of which is the Haus-dorff–Besicovitch dimension, is described. The biography of the scientist is also given.

There was presented the description of linear and nonlinear fractals which is based on linguistic metaphor comparing the linear fractals with Indo-European family of languages, and non-linear – with the Sino-Tibetan one. The basics of nonlinear «dialect», which singles out a square «dialect», created by Julia and Fatou, is qualitatively described. The diffusion-limited aggregation, which corresponds to the model of fractal growth, is also given from the qualitative viewpoint.

The article ends with the biography of Jean Perrin and exposition of his amazing prediction of the practical value of the sets, which began to be known as fractals.

DOI: 10.18500/0869-6632-2016-24-6-4-38

*Paper reference:* Trubetskov D.I., Trubetskova E.G. Fractal geometry. *Izvestiya VUZ. Applied Nonlinear Dynamics.* 2016. Vol. 24. Issue 6. P. 4–38.

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## BibTeX

author = {Elena Gennadjevna Trubetskova and Dmitry Ivanovich Trubetskov},

title = {FRACTAL GEOMETRY},

year = {2016},

journal = {Izvestiya VUZ. Applied Nonlinear Dynamics},

volume = {24},number = {6},

url = {http://andjournal.sgu.ru/en/articles/fractal-geometry},

address = {Саратов},

language = {russian},

doi = {10.18500/0869-6632-2016-24-6-4-38},pages = {4--38},issn = {0869-6632},

keywords = {fractal,Hausdorff–Besicovitch dimension,Mandelbrot set,turbulence,chaos,linear and nonlinear fractals,diffusion limited aggregation.},

abstract = {The article deals with the bases of fractal geometry and fates of its creators. The biographies and the discoveries of Felix Hausdorff and Abram Besicovitch – the main characters of the great play called fractal geometry – are presented with the possible degree of detail. There is no doubt that the author and director of this play was Benoit Mandelbrot. The article presents his biography and brief descriptions of the lives of his predecessors: Henri Poincare, Maurice Gaston Julia and Pierre Gaston Jose Fatou. Special attention is paid to the Poincare’s discovery of homoclinic tangle bundle. The biography of Lewis Fry Richardson, whose posthumous publication of the results of the length’s measurement of the UK coastline laid the basis of modern fractal geometry, is presented. A brief presentation of Richardson’s results on the turbulence’s theory and possible contemporary approaches to the solution of the Navier–Stokes equations are given. It led to the conclusion that turbulence should be described in terms of fractal geometry. There was presented Richardson’s empirical theory, that was applied towards the mea- surement of the coastline. The definition of topological dimension and the derivation of the Hausdorff–Besicovitch dimension are given. Two definitions of fractals, offered by Mandelbrot, are provided. Within the framework of these definitions Cantor set, triadic Koch curve, Sier- pinski carpet and cloth, Menger sponge, Peano curve, Weierstrass curve are analyzed. The continual family of Alfred Renyi dimensions, a particular case of which is the Haus-dorff–Besicovitch dimension, is described. The biography of the scientist is also given. There was presented the description of linear and nonlinear fractals which is based on linguistic metaphor comparing the linear fractals with Indo-European family of languages, and non-linear – with the Sino-Tibetan one. The basics of nonlinear «dialect», which singles out a square «dialect», created by Julia and Fatou, is qualitatively described. The diffusion-limited aggregation, which corresponds to the model of fractal growth, is also given from the qualitative viewpoint. The article ends with the biography of Jean Perrin and exposition of his amazing prediction of the practical value of the sets, which began to be known as fractals. DOI: 10.18500/0869-6632-2016-24-6-4-38 Paper reference: Trubetskov D.I., Trubetskova E.G. Fractal geometry. Izvestiya VUZ. Applied Nonlinear Dynamics. 2016. Vol. 24. Issue 6. P. 4–38. Download full version }}