ABSTRACT
Mathematical fractals exhibit exact self-similarity across all spatial or temporal scales, such that successive magnications reveal an identical structure. A self-similar object is composed of N copies of itself (with possible translations and rotations), each of which is scaled down by a scale ratio d in all directions of the DE dimensional available space. More formally, consider a set S of points at positions x x x xDE
=( , , , )1 2 in Euclidean space of dimension DE. Under a similarity transform with a scale ratio d (0 < d < 1), the set S becomes d S with points at positions δ δ δ δx x x xDE
= ( 1 2, , , ). A bounded set S is self-similar when S is the union of N nonoverlapping subsets, each of which is identical (under translations and rotations) to d S. A basic example of a self-similar fractal is the Cantor set (Cantor 1883). Consider a line segment ([0, 1]), divide it into thirds, and remove the central part. Repeat the procedure on the two remaining thirds, and after an innite number of iterations, one converges to a set of points or Cantor set, also referred to as Cantor dust (Figure 3.3).
