WitCH 80: External Forcing

Yep, it’s Golden Ratio Day here at Mathematical CrapLand.

This is a weird one. It might be trivial, or very hard, or non-existent. We’re not 100% sure. (OK, we are sure, but it took a while.) Anyway, it’s a research paper appearing in a (not the) journal published by Nature, and so can tolerate a little extra scrutiny. The paper, titled The golden number seen in a mechanical oscillator, appeared a couple weeks ago in the open access Nature journal, Scientific Reports; the paper can be downloaded here. Here is the begining:


A seemingly ubiquitous irrational number often appearing in nature and in man-made things like structures, paintings, and physical systems, is the golden number. Here, we show that this astonishing number appears in the periodic solutions of an underactuated mass-spring oscillator driven by a nonlinear self-excitation. Specifically, by using the two-time scale perturbation method, it is analytically demonstrated that the golden number appears in the ratio of amplitudes, as well as in the oscillation frequency of the periodic solution, which is referred to as golden solution and, by applying the Poincaré method, it is demonstrated that this solution is asymptotically stable. Additionally, the analytic results are illustrated by means of numerical simulations and also, an experimental study is conducted.


Around 300 BC the Greek mathematician Euclid, introduced to the world the division in extreme and mean ratio in which, a given line is divided in two segments such that the ratio of the whole line to the larger segment is equal to the ratio of the larger segment to the shorter segment. Later, in the nineteenth century, this ratio was renamed as the golden section or golden ratio and, since then, its study has attracted the attention of mathematicians, historians, architects, among others1.

The numerical value of the golden ratio, which is strongly related to the Fibonacci sequence, see e.g. Ref.2 , can be obtained as follows. Consider a line of length x and divide the line such that the largest segment equals 1. Then, the segments are said to satisfy the extreme and mean ratio mentioned above if the following is satisfied


Rewriting this equation yields to the second order polynomial in x

    \[\color{blue}\boldsymbol{x^2 - x - 1 = 0,}\]

which has the roots

    \[\color{blue}\boldsymbol{\phi:=x_+ = \frac{1+\sqrt5}{2}=1.61803\dots, \quad x_- = -\phi^{-1}=-0.61803\dots}\]

The positive root φ is referred to as the golden number.

Interestingly, it has been found that the golden ratio frequently appears in nature. For example, in the structure of galaxies, in the arrangement of the petals of a rose, in the apple’s seeds, in the spiral shells of mollusks, and in the ratio of masses of two quasiparticles in cobalt niobate3–5, just for mentioning a few. For further examples, ranging from biology to architecture and music—including some controvertible examples on ancient buildings and paintings—the interested reader is referred to3,6–9.

Regarding physical systems, the golden number has appeared in, for example, a simple harmonic oscillator10, in the solutions of a spherical pendulum11, and in a network of resistors12.

It should be noted that, besides its intriguing geometric and mathematical interpretation, there exist examples where the golden ratio is of paramount importance. For example, a recent study13 has shown that the golden ratio distribution observed in the veins of tree leaves allows maximizing the bending stiffness and also, with this golden distribution, the area of the leaf exposed to the sun rays is larger, which ultimately is beneficial during the photosynthesis process. Furthermore, the golden ratio also finds interesting applications, like in the design of antennas for improving the bandwidth and increasing the antenna’s gain14,15, and in magnetic resonance imaging, for reducing the occurrence of eddy current related artifacts16, among others. In this paper, it is demonstrated that the golden ratio appears in the periodic solution of an underactuated mechanical oscillator, which is composed of two interconnected masses. Only one of the masses is excited by a weakly nonlinear term in order to have self-sustained oscillations in the system. Specifically, it is shown that the system has a stable periodic solution, which has the following properties: (1) the ratio of amplitudes corresponding to the actuated and non actu- ated oscillators is equal to the golden ratio, (2) the oscillation frequency of the periodic solution is equal to the natural frequency of the uncoupled oscillators scaled by the golden number. It is important to stress the fact that the aforementioned properties are independent of the intrinsic parameters of the oscillators. Furthermore, the occurrence of the golden number in the periodic solution is analytically demonstrated by using a perturbation method namely, the two-timescale method and also, analytic conditions for the stability of the periodic solution are derived by using the Poincaré method of perturbation. The obtained theoretical results are numerically illustrated and also, an experimental study is conducted in order to validate the analytic results. The experiments are performed using an electro-mechanical device.

Ultimately, the obtained analytical, numerical, and experimental results presented here give clear evidence that the underactuated mechanical oscillator with weakly nonlinear self-excitation has “golden solutions”: the golden number appears in the amplitude and frequency of the stable periodic solution.

1. Herz-Fischler, R. A Mathematical History of the Golden Number (Dover Publications, 1998).

2. Dunlap, R. A. The Golden Ratio and Fibonacci Numbers (World Scientific, 1997).

3. Livio, M. The Golden Ratio: The story of Phi, the World’s most astonishing number (Broadway Books, 2003).

4. Coldea, R. et al. Quantum criticality in an Ising chain: Experimental evidence for emergent E8 symmetry. Science 327, 177–180. https://doi.org/10.1126/science.1180085 (2010).

5. Affleck, I. Golden ratio seen in a magnet. Nature 464, 362–363. https://doi.org/10.1038/464362a (2010).

6. Van Hooydonk, G. Euclid’s golden number and particle interactions. J. Mol. Struct. THEOCHEM 153, 275–287. https://doi.org/10.1016/0166-1280(87)80010-6 (1987).

7. Watson, A.R.  The Golden Relationships: An Exploration of Fibonacci Numbers and Phi. Master’s thesis, Duke University (2017).

8. van Gend, R. The Fibonacci sequence and the golden ratio in music. Notes Number Theory Discrete Math.20, 72–77 (2014).

9. Cartwright, J. H. E., González, D. L., Piro, O. & Stanzial, D. Aesthetics, dynamics, and musical scales: A golden connection. J. N. Music Res. 31, 51–58. https://doi.org/10.1076/jnmr. (2002).

10. Moorman, C. M. & Goff, J. E. Golden ratio in a coupled-oscillator problem. Eur. J. Phys. 28, 897–902. https://doi.org/10.1088/0143-0807/28/5/013 (2007).

11. Essén, H.& Apazidis, N. Turning points of the spherical pendulum and the golden ratio .Eur. J. Phys. 30, 427–432. https://doi.org/10.1088/0143-0807/30/2/021 (2009).

12. Srinivasan, T. P. Fibonacci sequence, golden ratio, and a network of resistors. Am. J. Phys. 60, 461–462. https://doi.org/10.1119/1.16849 (1992).

13. Sun, Z. et al. The mechanical principles behind the golden ratio distribution of veins in plant leaves. Sci. Rep. https://doi.org/10.1038/s41598-018-31763-1 (2018).

14. Gupta, S., Arora, T., Singh, D. & Singh, K. K. Nature inspired golden spiral super-ultra wideband microstrip antenna. In 2018 Asia-Pacific Microwave Conference (APMC), 1603–1605, https://doi.org/10.23919/APMC.2018.8617550 (2018).

15. Rawat, S., & Kanad, R. Compact design of modified pentagon shaped monopole antenna for UWB applications. Int. J. Electr. Electron. Eng. Telecommun. 7, 66–69. https://doi.org/10.18178/ijeetc.7.2.66-69. (2018).

16. Wundrak, S. et al. Golden ratio sparse MRI using tiny golden angles. Magn. Reson. Med. 75, 2372–2378. https://doi.org/10.1002/mrm.25831 (2016).

3 Replies to “WitCH 80: External Forcing”

  1. This is a case of mathematical pareidolia. The “ubiquity” of the golden ratio has been conclusively debunked by so many historians and mathematicians over the years that having any belief to the contrary is akin to believing in astrology or homeopathy. I’ve had a brief glance through the article, and it seems like a lot of work to end up with the polynomial x^2 – x – 1. It seems to be to be one of those things which if you look for, you’ll find. Like finding the golden ratio somehow hidden in the Old Testament, or the American Constitution, or the fugues of Bach.

    Somewhere I have an article in which the author found a “formula” for the circumference of the ellipse, under the mistaken impression that the arc-length was linear in the angle subtended at the centre. How do these things get published?

    1. Huh. I learned a new word.

      Thanks, Alasdair. I’m not sure the thing shouldn’t have been published. I’d be surprised if the mathematical arguments are unsound, and I can’t judge the merits. But it’s obviously over-egged, and it’s difficult to believe it would have been published in a Nature journal without the over-egging.

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