Fractals from z ← zα + c in the complex c-plane

Abstract

The self-squared function z ← z 2 + c, has been discussed extensively in the literature for generating fractals. In this article, we consider the generalized transformation function z ← z α + c for generating fractal images. A multitude of interesting, intriguing, and rich families of fractals are generated by changing a single parameter α. Direct relationships are observed between α and the visual characteristics of the fractal images in the c-plane. The exponent α can be represented as α = ± ( η + ϵ), where η and ϵ are the integer and fractional parts, respectively. It is found that when α is a positive integer number, the resulting image contains lobular structures. The number of major lobes equals ( η − 1). When α is a negative integer number, the generated fractal image is a planetary structure consisting of overlapping central planets surrounded by satellite structures. The number of satellite structures equals ( η + 1). A continuous variation of α between two consecutive integers results into a continuous proportional change between the two limiting fractal images. Several conjuctures about the visual characteristics of the images and the value of α are stated.