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 Sierpinski 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.
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