Abstract
We consider configurations of lines of curvilinear three-web, that can be inscribed in a triangle formed by the lines of this web. In the case when the inscribed configuration is triangulating, it generates a fractal in each such triangle. This allows us to associate with smooth function of two variables a certain fractal that generalizes the well-known Sierpiński triangle. We introduce the concept of a regular fractal and prove that a regular fractal is obtained only for a regular three-web (generalization of the basic theorem on hexagonal three-webs). We also find the fractal dimensions of some regular fractals and formulate problems related to fractal dimension.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Shelekhov, A.M., Lazareva, V.B., Utkin, A.A. Curvilinear three-webs (Tvers. gos. univ., Tver, 2013) [in Russian].
Mandelbrot, B.B. The fractal geometry of nature (W. H. Freeman and Co., San Francisco, 1982).
Falconer, K. Fractal geometry: mathematical foundations and applications (John Wiley and Sons, New York, 1990).
Acknowledgments
The author expresses his gratitude to A.E. Milovidov for the production of pictures in this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Russian Text © The Author(s), 2019, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2019, No. 9, pp. 63–72.
About this article
Cite this article
Shelekhov, A.M. On Fractal Constructions on Curvilinear Three-Web. Russ Math. 63, 55–62 (2019). https://doi.org/10.3103/S1066369X19090068
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066369X19090068