Abstract
The fractal images generated from the generalized transformation function z ←zα + c in the complex z-plane are analysed. The exponent a can assume any real or integer, either positive or negative, value. When the exponent α is a positive integer number, the fractal image has a lobular structure with number of lobes equal to α. When α is a negative integer number, the generated fractal image has a planetary structure with a central planet and ¦α¦ satellite structures around it. When α is varied continuously between two consecutive integer numbers, continuous and predictable changes are observed between the two limiting fractal images. Some conjectures regarding the visual characteristics of the fractal images and the value of α are included.
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