UDC 004.925.86
ALGEBRAIC FEATURES OF FRACTAL STRUCTURE CALCULATION ALGORITHMS
K. A. Maikov, Dr. Sc. (Tech.), full professor, Informatics and control systems Department, Bauman Moscow State Technical University, Moscow, Russia;
orcid.org/0000-0003-1864-2397, e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.
A. N. Pylkin, Dr. Sc. (Tech.), full professor, RSREU, Ryazan, Russia;
orcid.org/0000-0001-9925-2870, e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.
S. N. Kuzmenko, Ph.D. (Phys. and Math.), associate professor, Department of mathematics, physics and Informatics, Kerch state marine technological University, Kerch, Russia;
orcid.org/0000-0002-1039-1274, e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.
A. A. Teplov, post-graduate student, Informatics and control systems Department, Bauman Moscow State Technical University, Moscow, Russia;
orcid.org/0000-0002-1785-6089, e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.
The analysis of the methods to construct known geometric fractals allowed revealing algebraic features of fractal algorithms composition, namely fractal operators i f , which are the basic operations to construct fractals of a certain type. The result of using operators for geometric fractals is the structure with fractures of triangular, square, trapezoidal, etc. forms. Multiplication and addition operations are introduced for the operators, as well as the concept of unit, zero and inverse operator which allows us to define periodic and quasi-periodic fractal structures. In the product with an infinitely large number of cofactors-basis operators it is possible to obtain ultimate generalization of the concept of fractal as a structure without self-similarity, but with naturally decreasing scale. Fractal operators i f have complex switching properties. Giving stochastic character to basic operators i f allows calculating stochastic fractals and analyzing regularities of their structure. For periodic fractals, as well as the union of fractals, average or generalized dimension is introduced. An algorithm to form periodic and stochastic fractal structures is proposed, a distinctive feature of which is the implementation of probabilistic choice of basic geometric primitive on current iterative cycle. Software implementation of the algorithm proposed confirmed the validity of algebraic approach in study and modeling of fractals with complex structure.
Key words: geometric fractals, stochastic fractals, algorithms to construct fractals, algebra of fractal operators.