This study introduces an optimal fourth-order iterative method derived by combining two established methods, resulting in enhanced convergence when solving nonlinear equations. Through rigorous convergence analysis using both Taylor expansion and the Banach space framework, the fourth-order optimality condition is verified. We demonstrate the superior efficiency and stability of this new method compared to traditional alternatives. Numerical experiments confirm its effectiveness, showing a reduction in the average number of iterations and computational time. Visual analysis with polynomiographs confirms the method's robustness, focusing on convergence area index, iteration count, computational time, fractal dimension, and Wada measure of basins. These findings underscore the potential of this optimal method for tackling complex nonlinear problems in various scientific and engineering fields.

Abstract In this paper, we present a method for creating Escher‐like spherical patterns with regular polyhedron symmetries. Using the generators of the symmetry groups associated with regular polyhedra, we first provide fast algorithms to construct spherical tilings. Then, to obtain Escher‐like patterns, we specify texturing techniques to decorate the resulting tilings. Moreover, we present a strategy to create a novel dynamic effect of Escher‐like kaleidoscopes in which the motifs have a complete body. The method has the advantages of simple implementation, fast calculation, good graphics, and artistic effects, which can be used to create rich elegant spherical patterns.

Whirlpools , by the Dutch graphic artist M. C. Escher, is a woodcut print in which fish interlock as a double spiral tessellation. Inspired by this print, in this article we extend the idea and present a general method to create Escher-like interlocking spiral drawings of N whirlpools. To this end, we first introduce an algorithm for constructing regular spiral tiling T . Then, we design a suitable spiral tiling T and use N copies of T to compose an interlocking spiral tiling K of N whirlpools. To create Escher-like drawings similar to the print, we next specify realization details of using wallpaper templates to decorate K . To enhance the aesthetic appeal, we propose several measures to minimize motif overlaps of the spiral drawings. Technologically, we develop algorithms for generating Escher-like drawings that can be implemented using shaders. The method established is thus able to generate a great variety of exotic Escher-like interlocking spiral drawings.

Abstract In this paper, using both hand-drawn and computer-drawn graphics, we establish a method to generate advanced Escher-like spiral tessellations. We first give a way to achieve simple spiral tilings of cyclic symmetry. Then, we introduce several conformal mappings to generate three derived spiral tilings. To obtain Escher-like tessellations on the generated tilings, given pre-designed wallpaper motifs, we specify the tessellations’ implementation details. Finally, we exhibit a rich gallery of the generated Escher-like tessellations. According to the proposed method, one can produce a great variety of exotic Escher-like tessellations that have both good aesthetic value and commercial potential.

Applications of the fractal dimension include the analysis and interpretation of medical images. The article presents a method for determining image features that are based on fractal dimension. In the proposed method, an optimization process (modified semi-multifractal optimization algorithm) creates a division into sub-areas similarly to a multi-resolution method. Using this division, a characteristic spectrum based on the fractal dimensions is calculated. This spectrum is applied to the recognition method of X-ray images of teeth. The obtained experimental results showed that the proposed method can effectively recognize such images.

A fractal tiling ( f -tiling) is a kind of rarely explored tiling by similar polygonal tiles which possesses self-similarity and the boundary of which is a fractal. Based on a tiling by similar isosceles right triangles, Dutch graphic artist M. C. Escher created an ingenious print Square Limit in which fish are uniformly reduced in size as they approach the boundaries of the tiling. In this article, we present four families of f -tilings and propose an easy-to-implement method to achieve similar Escher-like drawings. By systematically investigating the local star-shaped structure of f -tilings, we first enumerate four families of f -tilings admitted by kite-shaped or dart-shaped prototiles. Then, we establish a fast binning algorithm for visualising f -tilings. To facilitate the creation of Escher-like drawings on the reported f -tilings, we next introduce one-to-one mappings between the square, and kite and dart, respectively. This treatment allows a pre-designed square template to be deformed into all prototiles considered in the article. Finally, we specify some technical implementations and present a gallery of the resulting Escher-like drawings. The method established in this article is thus able to generate a great variety of exotic Escher-like drawings.

Abstract Many algorithms that iteratively find solution of an equation require tuning. Due to the complex dependence of many algorithm’s elements, it is difficult to know their impact on the work of the algorithm. The article presents a simple root-finding algorithm with self-adaptation that requires tuning, similarly to evolutionary algorithms. Moreover, the use of various iteration processes instead of the standard Picard iteration is presented. In the algorithm’s analysis, visualizations of the dynamics were used. The conducted experiments and the discussion regarding their results allow to understand the influence of tuning on the proposed algorithm. The understanding of the tuning mechanisms can be helpful in using other evolutionary algorithms. Moreover, the presented visualizations show intriguing patterns of potential artistic applications.

Abstract In this paper, we generalize the idea of star‐shaped set inversion fractals using iterations known from fixed point theory. We also extend the iterations from real parameters to so‐called q‐system numbers and proposed the use of switching processes. All the proposed generalizations allowed us to obtain new and diverse fractal patterns that can be used, e.g. as textile and ceramics patterns. Moreover, we show that in the chaos game for iterated function systems—which is similar to the inversion fractals generation algorithm—the proposed generalizations do not give interesting results.