Phidias numbers as a basis for Fibonacci analogues

P. S. Kosobutskyy
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 26, 2020, Number 1, Pages 172-178
DOI: 10.7546/nntdm.2020.26.1.172-178
Download full paper: PDF, 452 Kb

Details

Authors and affiliations

P. S. Kosobutskyy
Department of Computer-Aided Design
Lviv Polytechnic National University
S. Bandery 12 St., Lviv, 79646, Ukraine

Abstract

In this paper it is shown that there is a plurality of irrational values of the roots of a quadratic equation with equal modulus coefficients |p| = |q| ≠ 1 having properties of the numbers of Phidias φ = 0.61803… and Ф = 1.61803… It is shown that it is also possible to construct a set of sequences possessing the basic properties of the Fibonacci and Lucas sequences.

Keywords

  • Golden ratio
  • Quadratic irrationality
  • Roots of quadratic equation

2010 Mathematics Subject Classification

  • 11B37
  • 11B39

References

  1. Affleck, I. (2010). Solid-state physics: golden ratio seen in a magnet. Nature, 464 (7287), 362–363.
  2. Bergman, G. (1957). A number system with an irrational base. Math. Magazine, 31, 98–110.
  3. Bradley, S. (2000). A geometric connection between generalized Fibonacci sequences nearly golden section. The Fibonacci Quart., 38 (2), 174–179.
  4. Dunlap, R. (1997). The Golden Ratio and Fibonacci Numbers. World Scientific
    Publishing Co. Pte. Ltd.
  5. Finch, S. Mathematical Constants. Cambridge University Press, 2003.
  6. Franz, M. (2009). A Fractal Made Of Golden Sets, Mathematics Magazine, 82, 243–254.
  7. Horadam, A., & Shannon, A. G. (1988). Fibonacci and Lucas Curves, Fibonacci Quart., 26 (1), 3–13.
  8. Koshy, T. (2001). Fibonacci and Lucas Numbers with Application, A Wiley-Interscience Publication: New York.
  9. Kosobutskyy, P., & Karkulovska, M. (2019). Mathematical methods for CAD: The
    method of proportional division. Bulletin of the Lviv Polytechnic National University. Collection of scientific works. Scientific publication. Series: Computer Design Systems. Theory and practice. 908, 75–83.
  10. Kosobutskyy, P. S. (2018). On the Possibility of Constructing a Set of Numbers with Golden Section Properties. International Conference Algebra and Analysis with Application. July 1-4 2018, Ohrid, Republic of Macedonia.
  11. Krcadinac, V. (2006). A new generalization of the golden ratio. Web Resource:
    https://www.fq.math.ca/Papers1/44-4/quartkrcadinac04_2006.pdf.
  12. Leyendekkers, J. V., & Shannon, A. G. (2016). Some Golden Ratio generalized Fibonacci and Lucas sequences. Notes on Number Theory and Discrete Mathematics, 22 (1), 33–41.
  13. Leyendekkers, J. V., & Shannon, A. G. (2018). Generalized golden ratios and associated Pell sequences. Notes on Number Theory and Discrete Mathematics, 24 (3), 103–110.
  14. Shechtman, D., Blech, I., Gratias, D., & Cahn, J. W. (1984). Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett., 53, 1951–1953.
  15. Shneider, R. (2016). Fibonacci numbers and the golden ratio. Web Resource: arXiv:1611.07384v1 [math.HO] 22 Nov 2016.
  16. Shneider, R. (2016). Fibonacci numbers and the golden ratio. Website.com, Available online at: arXiv:1611.07384v1 [math.HO] 22 Nov 2016.
  17. Szakacs, T. (2017). k-order linear recursive sequences and the golden ratio. The Fibonacci Quart., 55 (5), 186–191.
  18. Vajda, S. (1989). Fibonacci & Lucas Numbers, and the Golden Section. Theory and Applications. Ellis Horwood limited.
  19. Vernon, R. P. (2018). Relationships between Fibonacci-type sequences and Golden-type ratios. Notes on Number Theory and Discrete Mathematics, 24 (2), 85–89.
  20. Yiu, P. (2003). Recreational Mathematics. Department of Mathematics Florida Atlantic University. Web Resource: http://math.fau.edu/yiu/RecreationalMathematics 2003.pdf.
  21. Yu, D., Xue D., & Ratajczak, H. (2006). Golden ratio and bond-valence parameters of hydrogen bonds of hydrated borates. Journal of Molecular Structure, 783 (1–3), 210–214.

Related papers

Cite this paper

Kosobutskyy, P. S. (2020). Phidias numbers as a basis for Fibonacci analogues. Notes on Number Theory and Discrete Mathematics, 26(1), 172-178, doi: 10.7546/nntdm.2020.26.1.172-178.

Comments are closed.