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Fractality in resistive circuits: the Fibonacci resistor networks

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  • Petrus H. R. Anjos

    (Universidade Federal de Catalão)

  • Fernando A. Oliveira

    (Universidade Federal Fluminense
    Universidade de Brasília)

  • David L. Azevedo

    (Universidade de Brasília)

Abstract

We propose two new kinds of infinite resistor networks based on the Fibonacci sequence: a serial association of resistor sets connected in parallel (type 1) or a parallel association of resistor sets connected in series (type 2). We show that the sequence of the network’s equivalent resistance converges uniformly in the parameter $$\alpha =\frac{r_2}{r_1} \in [0,+\infty )$$ α = r 2 r 1 ∈ [ 0 , + ∞ ) , where $$r_1$$ r 1 and $$r_2$$ r 2 are the first and second resistors in the network. We also show that these networks exhibit self-similarity and scale invariance, which mimics a self-similar fractal. We also provide some generalizations, including resistor networks based on high-order Fibonacci sequences and other recursive combinatorial sequences. Graphical abstract

Suggested Citation

  • Petrus H. R. Anjos & Fernando A. Oliveira & David L. Azevedo, 2024. "Fractality in resistive circuits: the Fibonacci resistor networks," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 97(8), pages 1-10, August.
    • Handle: RePEc:spr:eurphb:v:97:y:2024:i:8:d:10.1140_epjb_s10051-024-00750-z
    DOI: 10.1140/epjb/s10051-024-00750-z
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    References listed on IDEAS

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    1. J. H. Asad & A. Sakaji & R. S. Hijjawi & J. M. Khalifeh, 2006. "On the resistance of an infinite square network of identical resistors – Theoretical and experimental comparison," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 52(3), pages 365-370, August.
    2. Costa, I.V.L. & Vainstein, M.H. & Lapas, L.C. & Batista, A.A. & Oliveira, F.A., 2006. "Mixing, ergodicity and slow relaxation phenomena," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 371(1), pages 130-134.
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