![]() Some graphical examples of fractal patterns obtained with the help of the proposed methods are presented in Sect. Section 4 is devoted to remarks on the implementation of the algorithms on the GPU using shaders. The first two methods are based on notions from the literature, and the third one is a completely new method. 3, we introduce three approaches on how to combine root finding methods to generate fractal patterns. 2, we briefly introduce methods of generating fractal patterns by a single root finding method. In this paper, we present ways of combining several root finding methods to generate new fractal art patterns. Using different RFMs, we are able to obtain various patterns thus, combining them together could further enrich the set of fractal patterns and lead to completely new ones, which we had not been able to obtain previously. Finally, in we can find the use of different switching processes.Īll the above-mentioned methods of fractal generation that use RFMs, except the ones that use the switching processes, use only a single root finding method in the generation process. In the first method, the authors used different iteration methods known from fixed point theory then, in, a perturbation mapping was added to the feedback process. Recently, new methods of obtaining fractal patterns from root finding methods were presented. In, it was shown that the colouring method of the patterns also plays a significant role in obtaining interesting patterns. Another popular method of generating fractal patterns from root finding methods is the use of different convergence tests and different orbit traps. ![]() ![]() The most popular root finding method used is the Newton method, but other methods are also widely used: Halley’s method, the secant method, Aitken’s method or even whole families of root finding methods, such as Basic Family and Euler-Schröder Family. The most obvious method of obtaining various patterns of this type is the use of different root finding methods. For their generation, different methods are used. These patterns are used to obtain paintings, carpet or tapestry designs, sculptures or even in animation. Īnother example of complex fractal patterns is patterns obtained with the help of root finding methods (RFM). escape time algorithm and layering technique. They are generated using different techniques, e.g. Mandelbrot and Julia sets together with their variations are examples of this type of fractals. One of the fractal types widely used in arts is complex fractals, i.e. in the Iterated Function Systems only information about a finite number of contractive mappings is needed. While fractal patterns are very complex, only a small amount of information is needed to generate them, e.g. The computational experiments show that the dependence of the measures on the parameters used in the methods is in most cases a non-trivial, complex and non-monotonic function.įractals, since their introduction, have been used in arts to generate very complex and beautiful patterns. Moreover, we study the proposed methods using five different measures: average number of iterations, convergence area index, generation time, fractal dimension and Wada measure. All the proposed approaches allow us to obtain new and diverse fractal patterns that can be used, for instance, as textile or ceramics patterns. affine and s-convex combination, (2) the use of iteration processes from fixed point theory, (3) multistep polynomiography. We use three approaches to combine the methods: (1) the use of different combinations, e.g. ![]() In this paper, we propose the use of a combination of root finding methods in the generation of fractal patterns. In the generation methods of these fractals, only one root finding method is used. Fractal patterns generated in the complex plane by root finding methods are well known in the literature.
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