Fractal images from z <- za + c in the complex c-plane

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In this paper, we propose the generalized transformation function z <- z[superscript a] + c for generating fractal images. The self-squared function z <- z[superscript 2] + c, which is discussed extensively in the literature, is a special case of this function. A multitude of interesting, intriguing and rich families of fractals are generated by changing a single parameter, a. Direct relationships are observed between a and the visual characteristics of the fractal image in the c-plane. The exponent a can be represented as a = +-(n+e), where n and e are the integer and fractional parts, respectively. It is found that when a is a positive integer number, the resulting image contains lobular structures. The number of major lobes equals (n-1). When a 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 (n+1). A continuous variation of a between two consecutive integers results into a continuous proportional change between the two limiting fractal images. Several conjectures about the visual characteristics of the images and the value of a are stated.