Chapter 5: Two Dimensions and Beyond

Section 4: Substitution Systems and Fractals

Complex maps

Many kinds of nonlinear transformations on complex numbers yield nested patterns. Sets of so-called Möbius transformations of the form z -> (a z + b)/(c z + d) always yield such patterns (and correspond to so-called modular groups when a d - b c ==1). Transformations of the form z -> {Sqrt[z - c], -Sqrt[z - c]} yield so-called Julia sets which form nested patterns for many values of c (see note below). In fact, a fair fraction of all possible transformations based on algebraic functions will yield nested patterns. For typically the continuity of such functions implies that only a limited number of shapes not related by limited variations in local magnification can occur at any scale.

From Stephen Wolfram: A New Kind of Science [citation]