Minimum modulus visualization of algebraic fractals

Abstract

Fractals are a family of shapes formed by irregular and fragmented patterns. They can be classified into two main groups: geometric and algebraic. Whereas the former are characterized by a fixed geometric replacement rule, the latter are defined by a recurrence function in the complex plane. The classical method for visualizing algebraic fractals considers the sequence of complex numbers originated from each point in the complex plane. Thus, each original point is colored depending on whether its generated sequence escapes to infinity. The present work introduces a novel visualization method for algebraic fractals. This method colors each original point by taking into account the complex number with minimum modulus within its generated sequence. The advantages of the novel method are twofold: on the one hand, it preserves the fractal view that the classical method offers of the escape set boundary and, on the other hand, it additionally provides interesting visual details of the prisoner set (the complement of the escape set). The novel method is comparatively evaluated with other classical and non-classical visualization methods of fractals, giving rise to aesthetic views of prisoner sets.