176 Progress of Theoretical Physics Supplement No. 150, 2003 Self-Similar Planar Fractals Based on Branching Trees and Bushes

Branching trees and bushes are obtained from a segment by an infinite sequence of two elementary transformations — rotation for a positive angle ϑ < π and stretching with a factor r < 1. Trees and bushes themselves are not self-similar, but the resulting limiting sets of points are. Typical questions about tree fractals are: at what relation between r and ϑ the branches of the tree will meet (overlap), and what will be the limiting surrounding curve, when there is no overlapping. By summation of complex geometric progressions, we find an explicit connection between r and ϑ for this boundary case. We obtain polynomial equations and solve them exactly, when this is possible, but in most cases numerically. The results are of interest for different natural sciences and medicine. §1. The dimension of a self-similar fractal Our purpose is to discuss a special class of planar fractal sets, which have many applications. We cannot go here into details of general Hausdorff-Bezikovich dimen-sion, which is usually used for the definition of fractals, so we will restrict ourselves to the case of self-similar fractals and their dimension. A compact set in a metric space is called self-similar (in a strong sense) with a similarity coefficient p, when it can be divided into N congruent sets, each of which is exactly p-times smaller copy of the original set. As an example, we can take a triangle in R2, or a cube in R3, divided into an arbitrary number of equal pieces (Fig. 1). As in all such examples the ratio lnNln p is equal to the dimension of the considered object, the generalizing idea is, to define the dimension of any self-similar compact set of points with this ratio. We are giving the initial steps of obtaining two planar self-similar fractals, the first of which is known in mathematics as the Sierpinski triangle (Fig. 2). As on each step we obtain three new copies of the N=27, p=3N=16, p=