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Symmetry: Culture and Science
Volume 33, Number 3, pages 209-220 (2022)
https://doi.org/10.26830/symmetry_2022_3_209

COMPLEX AND HYPERBOLIC FIBONACCI NUMBERS AND PHYLLOTAXIS

S. V. Petoukhov^1*, E. S. Petukhova², V. I. Svirin³

¹ Department of Biomechanical Systems, Mechanical Engineering Research Institute, Russian Academy of Sciences, Moscow, M. Kharitonievsky pereulok, 4, 101990, Russia.
Email: spetoukhov@gmail.com
Web: http://petoukhov.com
ORCID: 0000-0001-7355-1813

² Department of Biomechanical Systems, Mechanical Engineering Research Institute, Russian Academy of Sciences, Moscow, M. Kharitonievsky pereulok, 4, 101990, Russia.
Email: miraddams@gmail.com
ORCID: 0000-0002-5192-1462

³ Department of Biomechanical Systems, Mechanical Engineering Research Institute, Russian Academy of Sciences, Moscow, M. Kharitonievsky pereulok, 4, 101990, Russia.
Email: vitaly.i.svirin@gmail.com
ORCID: 0000-0001-9878-5640

^* corresponding author

Abstract: The article is devoted to studying the extensions of the additive Fibonacci sequence into the additive sequences of 2-dimensional complex and hyperbolic numbers having Fibonacci coordinates. This study is connected with the active use of complex and hyperbolic numbers by contemporary sciences. Special attention is paid to features of the matrix representations of complex and hyperbolic Fibonacci numbers. The authors believe that complex and hyperbolic Fibonacci numbers, having interesting features, will be useful in different scientific fields.

Keywords: Fibonacci numbers, phyllotaxis laws, complex numbers, hyperbolic numbers, matrix representation, eigenvalues.

References:
Aspray T. (2011, 13 August) Fibonacci analysis – Master the basics, Forbes. https://www.forbes.com/sites/tomaspray/2011/08/13/fibonacci-analysis-master-the-basics/#ac784a66f95b

Barker E., Barker W., Burr W., Polk W., Smid M. (2012) Recommendation for ket management, NIST Special Publication 800-57, NIST. https://doi.org/10.6028/NIST.SP.800-57p1r3

Bodnar O.Ya.(1992) Geometry of phyllotaxis, Reports of the Academy of Sciences of Ukraine, 9, 9-15.

Bodnar O.Ya. (1994) Golden Ratio and Non-Euclidean Geometry in Nature and Art, Lviv: Publishing House „Sweet”.

Brasch T. von, Byström J., Lystad L.P. (2012) Optimal control and the Fibonacci sequence, Journal of Optimization Theory and Applications, 154 (3), 857–78. https://doi.org/10.1007/s10957-012-0061-2

Cohn, M. (2005) Agile Estimating and Planning, Prentice Hall PTR, ISBN 978-0-13-147941-8.

Darvas G. (2007) Symmetry, Basel: Birkhäuser, xi+508. https://doi.org/10.1007/978-3-7643-7555-3

Darvas G., Koblyakov A.A., Petoukhov S.V., Stepanyan I. (2012) Symmetries in molecular-genetic systems and musical harmony, Symmetry: Culture and Science, 23, 3-4, 343-375. https://doi.org/10.26830/symmetry_2012_3-4_343

Harizanov V.S. (1995) Review of Yuri V. Matiyasevich, Hilbert's Tenth Problem, Modern Logic, 5, 3, 345–355.

Hu Z.B., Petoukhov S.V. (2017) Generalized crystallography, the genetic system and biochemical esthetics, Structural Chemistry, 28, 1, 239-247, https://doi.org/10.1007/s11224-016-0880-0

Jean R. (2006) Phyllotaxis, A Systemic Study in Plant Morphogenesis, Cambridge Univ. Press.

Kantor I.L., Solodovnikov A.S. (1989) Hypercomplex numbers, Berlin, New York: Springer-Verlag, ISBN 978-0-387-96980-0. https://doi.org/10.1007/978-1-4612-3650-4

Klavžar S. (2011) Structure of Fibonacci cubes: a survey, IMFM Preprint Series, Ljubljana, Slovenia: Institute of Mathematics, Physics and Mechanics, 49 (1150).

Krzywkowski M. (2010) New proofs of some Fibonacci identities, International Mathematical Forum, 5, 18, 869 – 874.

Olsen Sc. (2006) The Golden Section: Nature's Greatest Secret, Bloomsbury USA, ISBN-10: 0802715397, ISBN-13: 978-0802715395.

Penrose R. (2016) The Road to Reality: A complete guide to the laws of the universe (reprint ed.), Random House, pp. 72–73, ISBN 978-1-4464-1820-8.

Petoukhov S.V. (1981) Biomechanics, Bionics and Symmetry, Moscow: Nauka [in Russian].

Petoukhov S.V. (1999) Biosolitons, Moscow: GP Kimrskaya tipographia, 288p. http://petoukhov.com/?page_id=278.

Petoukhov S.V. (2008) Matrix Genetics, Algebras of Genetic Code, Noise Immunity, Moscow: RChD [in Russian], http://petoukhov.com/matrix-genetics-petoukhov-2008.pdf

Petoukhov S.V. (2011) Matrix genetics and algebraic properties of the multi-level system of genetic alphabets, Neuroquantology, 9, 4, 799-820. http://petoukhov.com/NEUROQUANTOLOGY_PAPER_PETOUKHOV_2011.pdf, https://doi.org/10.14704/nq.2011.9.4.501

Petoukhov S.V. (2016) The system-resonance approach in modelling genetic structures. Biosystems, 139, 1-11. https://doi.org/10.1016/j.biosystems.2015.11.001, PMid:26545937

Petoukhov S.V., He M. (2010) Symmetrical Analysis Techniques for Genetic Systems and Bioinformatics: Advanced Patterns and Applications, IGI Global, USA. http://www.evernote.com/l/AFnRwSleQRRMI7PGAKFzZgkUEdMP10Lv5J0/, https://doi.org/10.4018/978-1-60566-124-7

Rapoport D.L. (2016) Klein bottle logophysics, self-reference, heterarchies, genomic topologies, harmonics and evolution, Part I: Morphomechanics, space and time in biology & physics, cognition, non-linearity and the structure of uncertainty, Quantum Biosystems, 7, 1, 1-72.

Rapoport D.L., Perez J.C. (2018) Golden ratio and Klein bottle logophysics: the keys of the codes of life and cognition, Quantum Biosystems, November, 9, 2, 8-76.

Smolyaninov V.V. (2000) Spatio-temporal problems of locomotion control, Uspekhi Fizicheskikh Nauk, 170, N 10, 1063–1128. https://doi.org/10.3367/UFNr.0170.200010b.1063

Stakhov A.P. (2009) The Mathematics of Harmony. From Euclid to Contemporary Mathematics and Computer Science, New Jersey: World Scientific. https://doi.org/10.1142/6635