Star-Shaped Set Inversion Fractals

Abstract

In the paper, we generalized the idea of circle inversion to star-shaped sets and used the generalized inversion to replace the circle inversion transformation in the algorithm for the generation of the circle inversion fractals. In this way, we obtained the star-shaped set inversion fractals. The examples that we have presented show that we were able to obtain very diverse fractal patterns by using the proposed extension and that these patterns are different from those obtained with the circle inversion method. Moreover, because circles are star-shaped sets, the proposed generalization allows us to deform the circle inversion fractals in a very easy and intuitive way.

Pseudoinversion Fractals

“…They were based on circle inversion. Later in [2] a generalization from circles to the star-shaped sets was introduced. Some further generalizations, namely the use of iteration process from fixed-point theory and the use of q-systems, were presented in [3].…”

“…Many different methods of obtaining fractal patterns were proposed in the literature, e.g., dynamical systems [4], hyperbolic geometry [6], complex numbers [5] or iterated function systems [9]. One of the recent methods is the use of inversion transformation of the star-shaped sets [2,3]. This type of fractals are called inversion fractals.…”

Inversion Fractals and Iteration Processes in the Generation of Aesthetic Patterns

“…Moreover, he presented some examples of fractals obtained with star‐shaped set inversion. In this section, we briefly introduce the star‐shaped fractals presented in [Gda14].…”

“…In 2005, Leys in [Ley05] extended the idea of circle inversion to the sphere and presented sphere inversion fractals. Later, in 2014, Gdawiec in [Gda14] showed that we can generalize circle inversion to any star-shaped set. Moreover, he presented some examples of fractals obtained with star-shaped set inversion.…”

“…In this paper, we present extensions of the star‐shaped set inversion fractals [Gda14] which are based on different iteration schemes taken from fixed point theory and q ‐system numbers. The proposed extensions allow us to obtain new and diverse fractal patterns, which we had not been able to obtain previously.…”