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

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.

Keywords

• fractal
• hausdorff–besicovitch dimension
• julia
• fatou
• richardson
• hausdorff
• besicovitch
• mandelbrot
• biography