ANALYSIS OF C-PLANE FRACTAL IMAGES FROM z ← zα + c FOR (α < 0)

Abstract

This paper expounds the definition of the c-plane fractal images from the complex mapping z ← zα + c (α < 0). A series of interesting and intriguing fractal images are generated by changing a single parameter α. When symbol α is a negative integer, the fractal image is a planetary structure consisting of a central planet surrounded by satellite structures. The number of satellite structures equals (1 + | α |). For negative non-integer values of α, additional embryonic satellite structure, proportional in size to the fractional part of α, are observed. The position and size of the planet and the positions of the satellite structures are analytically estimated. The four evolutions of the c-plane fractal image are given for the first time. The reasons resulting in the four evolutions of the c-plane fractal images are analyzed.