Control Systems and Computers

https://doi.org/10.15407/csc.2022.01.032

Control Systems and Computers, 2022, Issue 1 (297), pp. 32-43

UDC 519.816

Tymofijeva N.К., Doctor of Sciences (Eng.), Chief researcher, International Research and Training Center for Information Technologies and Systems of the NAS and MES of Ukraine, Glushkov ave., 40, Kyiv, 03187, Ukraine, tymnad@gmail.com

SIGN COMBINATORIAL SPACES, FINITE SEQUENCES AND LOGARITHMIC SPIRALS

Introduction. Sign combinatorial spaces that exist in two states: convolute (tranquility) and deployed (dynamics), are considered. Spaces, in particular biological, physical, informational and some others, for which the axioms of sign combinatorial spaces, are valid, have a combinatorial nature. When they are deployed, combinatorial numbers (Fibonacci numbers) are formed, through which logarithmic spirals appear in living nature. These spirals are formed due to the finite sequences that take place during the deployment of the agreed spaces and which are presented geometrically using polar coordinates.

Formulation of the problem. The logarithmic spiral is geometrically represented through a “golden rectangle” in which one side is 1,618 times longer (“golden” number or golden section). The presence of the golden ratio in nature is manifested through Fibonacci numbers, which are formed from an arithmetic triangle from elements of finite sequences formed by the deployment of sign combinatorial spaces. But this spiral is transmitted through the “golden rectangle” indirectly. The problem is to trace its formation in nature through constructed sequences, the elements of which are represented by polar coordinates.

The approach proposed. Using the finite sequences that are formed during the unfolding of sign combinatorial spaces and the representation of their elements in polar coordinates, we can trace the dynamics of the formation of logarithmic spirals in nature.

Conclusion. Representation of biological space as a sign combinatorial space can explain various phenomena in nature. When unfolding these spaces from the convolute spaces finite sequences are formed, the sums of the members of which determine the number of combinatorial configurations in a subset of isomorphic combinatorial configurations and form an arithmetic triangle (Pascal’s triangle). Fibonacci numbers and, accordingly, a golden number are formed from an arithmetic triangle. The logarithmic spiral fits into a golden rectangle. The dynamics of the formation of the logarithmic spiral is traced
due to the finite sequences formed as a result of the deployment of the sign combinatorial spaces, the elements of which are presented in polar coordinates.

Keywords: sign combinatorial spaces, logarithmic spiral, polar coordinates, combinatorial configuration, finite sequences, fractals.

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1. Tymofijeva N.K. (2015) Znakovi kombinatorni prostory ta shtuchy’j intelekt Shtuchny`j intelekt, 67-68 (1–2). 180–189.
2. Sergienko I. V., Каspthitzkaja M. F. Modeli i metodu reshenija na EVM kombinatornux zadath optimizatzii, Kiev: Nauk Dumka, 1981. 281 p.
3. Burduk V.Ja. Diskretnoje metritheskoje prostranstvo, DGU, 1982. 99 p.
4. Skordev. Recursion theory on iterative combinatory spaces. Bull. Acad. Polon. Sci., Sér Sci. Math. Astronom. Phys. 1976. 24, № 1. P. 23–31.
5. Stojan Ju.G. Ob odnom otobrajenii kombinatornyx mnojestv v evklidovo prostranstvo. Xarkov, 1982. 33 p. (Preprint AN USSR. In-t probl. mashinostrteija; 173).
6. Tymofijeva N.K. Teoretyko-thyslovi metody rozvjazannja zadath kombinatornoij optymizatsiji. – Dysertatsija na zdobuttja naukovogo stupenja doktora texxnithnyx nauk za spetsialnistju 01.05.02 – matematythne modeljuvannja ta obthysljuvalni metody. Rukpys. IK im. V.M. Glushkova NAN Ukraijny, K. 2007. 374 p.
7. Tymofijeva N.K. Vykoystannja vlastyvosti periodythnosti dlja generuvannja kombinatornyx konfiguratsi’j. Systemy keruvannja ta kompjutery (USiM, Control systems & computers. 2021, №1(291). P. 15–28. DOI https://doi.org/10.15407/csc.2021.01.015.
8. Mandelbrot B. Fraktalnaja geometrija prirody. Ishevsk: NITS «Reguljartaja i xaotuthesraja dinamika», 2010. 656 p.
9. Depman I.JA. Istorija arifmetiki. M.: Gosud. uthebno-pedagogith. iz-vo Minist. prosvets. RSFSR, 1959. 423 p.
10. Mir matematiki: v 40 t. T 1: Fepnando Korbalan. Zolotoe sothenie. Matematitheski’j jazyk krasoty. Per. s angl. М.: De Agostini, 2014. 160 p.
11. Virthnko N. O., Ljashko I.I. Grafiky elementarnyx ta spetsialnyx funkshi’j: Dovidnyk. К.: Nauk. dumka, 1996. 584 p.
12. Pothemu prostye thisla obrazujut spirali. https://www.youtube.com/watch?v=DxntHp7-wbg (data obrashenija: 5.05.2021).

Received 02.06.2022