Biomorphs via Modified Iterations

The aim of this paper is to present some modifications of the biomorphs generation algorithm introduced by Pickover in 1986. A biomorph stands for biological morphologies. It is obtained by a modified Julia set generation algorithm. The biomorph algorithm can be used in the creation of diverse and complicated forms resembling invertebrate organisms. In this paper the modifications of the biomorph algorithm in two directions are proposed. The first one uses different types of iterations (Picard, Mann, Ishikawa). The second one uses a sequence of parameters instead of one fixed parameter used in the original biomorph algorithm. Biomorphs generated by the modified algorithm are essentially different in comparison to those obtained by the standard biomorph algorithm, i.e., the algorithm with Picard iteration and one fixed constant

Mandelbrot and Julia Sets via Jungck-CR Iteration with s-convexity

In today’s world, fractals play an important role in many fields, e.g., image compression or encryption, biology, physics, and so on. One of the earliest studied fractal types was the Mandelbrot and Julia sets. These fractals have been generalized in many different ways. One of such generalizations is the use of various iteration processes from the fixed point theory. In this paper, we study the use of Jungck-CR iteration process, extended further by the use of s-convex combination. The Jungck-CR iteration process with s-convexity is an implicit three-step feedback iteration process. We prove new escape criteria for the generation of Mandelbrot and Julia sets through the proposed iteration process. Moreover, we present some graphical examples obtained by the use of escape time algorithm and the derived criteria

Comparing rendering methods for Julia sets

Sequential rendering methods for the graphical representation of Julia sets are compared. Two groups of methods are presented. In the first, the attractor of the Julia set is rendered and, in the second, the complement of the attractor is rendered. Examples of images obtained using these methods are also given