Methods of Constructing Equations for Objects of Fractal Geometry and R-Function Method

Abstract

AbstractThe paper discusses methods for constructing equations for objects of fractal geometry and method of R-functions. Basic concepts of the theory of fractals, areas of application and their types have been presented. The basic methods of constructing fractals are taken into account: L-system method, system of iterating functions, set theory method, and the R-function method. Equations of complex structures of fractal geometry have been developed based on the R-functions method. Using the of straight-line equation, the equation of a circle and constructive means of the method of R-functions R0: R-conjunctions and R-disjunctions are constructed various kinds of fractals, equations of fractals consisting of intersections of lines, tangencies of circles. Based on these equations, various prefractals were generated by specifying the number of iterations n and the angle of inclination. Equations are constructed for fractal antennas based on the “Cayley tree”, fractal ring monopolies and the Sierpinski curve that are used in antenna design. These fractals are very beautiful, which can be used in the creation of computer landscapes, in various illustrations, in telecommunications, in the textile industry, in drawing patterns in ceramic and porcelain products, as well as in the development of patterns for the modern design of Uzbek national carpets, fabrics, costumes, etc.KeywordsFractalSelf-similarityL-systemsCantor setIFSR-function methodR-conjunctionsR-disjunctionsPre-fractalSierpinski curve