Combinatorial configurations, fractals, fractal dimension of combinatorial sets

Fìz.-mat. model. ìnf. tehnol. 2021, 33:170-174

Abstract

Combinatorial configurations and their sets are considered. The definitions of these objects are given, recurrent combinatorial operators are introduced, with the help of which they are formed, and rules are formulated according to which their sets are ordered. The property of periodicity, which takes place in the generation of combinatorial configurations, is described. It follows from the recurrent way of their formation and ordering. The fractal structure of combinatorial sets is formed due to the described rules, in which the property of periodicity is used. Analysis of these structures shows that they are self-similar, both finite and infinite, which is characteristic of fractals. Their fractal dimension is introduced, which follows from the rules of generating combinatorial configurations and corresponds to the number of these objects in their set.

References

1. Lypskyi, V. (1988). Kombinatorika dlija prohrammistov. Per. s polsk. M.: Mir. (in Russian).
2. Sachkov, V. N. (1977). Kombinatornyje metody dickretno’j matematiki. М.: Nauka. (in Russian).
3. Tymofijeva, N. K. (2015). Znakovi kombinatorni prostory ta shtuchy’j intelekt Shtuchny`j intelekt, 1-2(67-68), 180–189.
4. Tymofijeva, N. K. (2007). Teoretyko-chyslovi metody rozviazannia zadach kombinatornoi optymizatsii: avtoref. dys. dokt. tekhn. nauk: 01.05.02. Kyiv: ІK іm. V.M. Gluchkova NАN Ukrainy.
5. Mandelbrot, B. (2010). Fraktalnaja geometrija prirody. Ishevsk: NITS «Reguljartaja i xaotuthesraja dinamika».