Why Phi? – ‘Fractals seen in throbs of pulsating golden stars’

Posted: January 28, 2015 by oldbrew in Astronomy, Astrophysics, Fibonacci, Phi
Tags: ,

Kepler Space Telescope [NASA]

Kepler Space Telescope [NASA]

A very interesting report of a new science paper has appeared in the New Scientist:

‘William Ditto and his colleagues at the University of Hawaii, Manoa, compared the two strongest oscillations, or tones, made by the variable star KIC 5520878, using observations by NASA’s Kepler space telescope. They noticed that dividing the frequency of the secondary note by that of the primary, or lowest, note gives a value near the “golden ratio” – a number that shows up often in art and nature and is close to 1.618′

So is it real or did they perhaps just imagine it?
Let’s start with the abstract :

‘The unprecedented light curves of the Kepler space telescope document how the brightness of some stars pulsates at primary and secondary frequencies whose ratios are near the golden mean, the most irrational number.’

(continued) ‘A nonlinear dynamical system driven by an irrational ratio of frequencies generically exhibits a strange but nonchaotic attractor. For Kepler’s “golden” stars, we present evidence of the first observation of strange nonchaotic dynamics in nature outside the laboratory. This discovery could aid the classification and detailed modeling of variable stars.’

From the science paper itself:

‘We expanded this analysis to five other multi-frequency variable stars in Kepler’s field of view. The three additional golden RRc Lyrae stars KIC 4064484, KIC 8832417 and KIC 9453114, whose frequency ratios well approximate the golden mean, all exhibit signatures of strange nonchaotic dynamics, and their spectral scalings nearly collapse onto each other when the axes are proportionally scaled’

Another quote: ‘In contrast, two non-golden RRab Lyrae stars KIC 4484128 and KIC 7505345, whose frequency ratios well approximate the simple fraction 3/2, exhibit qualitatively different dynamics’

3:2 ratio = Fibonacci first order resonance like Neptune:Pluto

All this looks exactly along the lines we’ve been putting forward for the last two years here at the Talkshop.
The paper concludes:

‘Strange nonchaotic attractors have been observed in laboratory experiments involving magneto-elastic ribbons, electrochemical cells, electronic circuits, and a neon glow discharge, but never before in non-experiments in nature.
The pulsating star KIC 5520878 may be the first strange nonchaotic dynamical system observed in the wild.’

More about the Kepler Space Telescope here

Also reported here

  1. Joe Public says:

    I’m glad the image you selected had a description.

    At first glance, I thought it was the design for a new Dyson vac!

  2. oldbrew says:

    Maybe a space background is what it needs.

    [NASA image]


  3. oldbrew says:

    The paper does drop the ball in one place when they say:

    ‘A few dozen such stars have been documented
    [14] for mysterious behavior involving frequency
    ratios near 0.62, apparently without connection to the
    golden ratio ϕ ≈ 1/0.62 or the possibility of strange nonchaotic
    behavior. We refer to the former stars as golden
    stars and to the latter stars as strange nonchaotic stars,
    and we recognize that these classes may overlap.’

    Why say ‘apparently without connection to the golden ratio’?
    Obviously 0.62 is close to 0.618 which is the inverse of the golden ratio:


    They could also refer to Richard Merrick’s post on this website: link above at ‘Related’.

    Merrick also notes: ‘Standing wave reflection simply cannot be sustained when one of the three dimensions in a container is at or even near a golden ratio to another.’

    This paper: ‘As the most irrational number, the golden ratio can have significant dynamical consequences. For example, by the KAM theorem, dynamics with two frequencies in the golden ratio maximally resist perturbations.’

    Diagram from the Merrick post:

  4. Bill Ditto says:

    Hi this is Bill Ditto (one of the paper’s authors). When we started this work several months ago we just thought the ratio a coincidence (when it was brought to our attention) and that if the star cooincidently had two frequencies near the GM ratio we might see this Strange NonChaotic Dynamics. Thus the paper. However the more we looked the more we found that the ratio approx phi was noted but not linked to phi but just thought curious. Indeed it was brought to our attention by somebody who thought it was 1/Sqrt(2) and that that was curious. A preprint on the archives shows a paper that went back and looked at ratios of know variables and found hundreds of them with ratios very near this ratio!! We now think that we are on the track of a universal nonlinear dynamical mechanism why this would be the case. We are writing that work up and and once we can verify it we will submit it for publication. Thanks for your interest in our paper
    Bill Ditto
    University of Hawaii at Manoa

  5. tallbloke says:

    Hi Bill, thanks for dropping by, and welcome to the talkshop.
    Do take a look at the link Stuart (Oldbrew) posted
    We have been finding all manner of interesting solar system dynamics relating to Phi here over the last two years.
    Please stick around and kick a few ideas about with us.
    I got a couple of papers published by Copernicus (and then unpublished) along with some prestigious co-authors last year on this stuff

  6. tchannon says:

    We had this years ago for carbon emission control

    S’pos it’s a space scuttle.

    [reply: groan…]

  7. oldbrew says:

    Great to hear from one of the authors. Could we ask a favour and be informed when this next paper is published, or be advised where to look for it?

    ‘We now think that we are on the track of a universal nonlinear dynamical mechanism why this would be the case. We are writing that work up and and once we can verify it we will submit it for publication.’

    Looking forward to that.

  8. Wayne Job says:

    Very interesting, that most things least understood are controlled by non-linear equations that gravitate to a strange attractor. That these attractors are fibonacci numbers or golden numbers is no surprise as chaos is all of creation, and all chaos attractors lead to harmony. The only chaos that does not lead to harmony is politics.

  9. oldbrew says:

    Richard Merrick links to a research paper:

    ‘Abstract. Incursion is suggested to be a fundamental physical principle of the universe [1]. We take steps in this direction by analyzing recursion in the Schrödinger equation. The Golden Ratio Phi is an extraordinary and ubiquitous irrational number of value 1.618033…Phi’s presence may be seen in both biological and astronomical realms and recently in the quantum mechanical and physical realms. In the biological realm, the number Phi can be seen in both Phyllotaxis and DNA. In the astronomical realm, its presence is found in the spiral structures of galaxies. In physics, Phi can now be related to the g-factors of the electron, proton and neutron. This paper will demonstrate that Phi is also intimately related to the quantum realm by virtue of its presence in the quantum mechanical wave function ? (x, y, z, t). The basis for the compact incorporation of F into the wavefunction will be derived by solving the Schrödinger Wave Equation and the use of the Phi recursive heterodyning set of wavelengths ?n. Solutions to the Schrödinger Wave Equation based on these recursive wavelengths and Phi will be derived in both Cartesian and Polar coordinates.’


  10. tallbloke says:

    Anyone understand this abstract?
    Period-doubling cascades and strange attractors in the triple-well phi^6-Van der Pol oscillator
    “Duffing–Van der Pol equation with the fifth nonlinear-restoring force is investigated. The bifurcation structure and chaotic motion under the periodic perturbation are obtained by numerical simulations. Numerical simulations, including bifurcation diagrams, Lyapunov exponents, phase portraits and Poincaré maps, exhibit some new complex dynamical behaviors of the system. Different routes to chaos, such as period doubling and quasi-periodic routes, and various kinds of strange attractors are also demonstrated.”

  11. call me dumb but I see no practical application of this.

  12. Golden Stars Hum the #Asynsis tune. Coincidence? So do our Planets, another Stellar Cymatic + Celestial Mechanics QED for the AsynsisConstructal geometric-thermodynamics paradigm.
    http://asynsis.wordpress.com http://about.me/asynsis

  13. tallbloke says:

    Hi Salvatore. I won’t call you dumb, but I will call you short-sighted. Our research has found that phi relationships pervade the cosmos from the sub-atomic to the galactic scale, and that these are related to resonance and energy transfer in chaotic systems containing strange attractors.

    It’s time for climate science to grow up and stop boring us with simplistic pronouncements about ‘control knobs’.

  14. oldbrew says:

    Nigel Reading links to this report: ‘Golden Ratio offers unity of science’ – Nov 27, 2014


  15. tallbloke says:

    Nigel’s link is a bit woolly IMO. We need to think carefully about phi both as an emergent phenomenon (fractals) and a convergent phenomenon (Fibonacci).

  16. oldbrew says:

    Time for another look at Miles Mathis perhaps: Φ² + Φ = 1


  17. tallbloke says:

    Φ² + Φ = 1
    0.382 + 0.618 = 1
    1/Φ = 1 + Φ
    1.618 = 1 + 0.618

    Ratio of Mercury’s semi major axis to Earth’s

    Ratio of Venus’ sidereal orbit period to Earth’s

    To within less than 0.5% error:
    Venus passes Earth 5 times as Earth orbits Sol 8 times as Venus orbits Sol 13 times as Mercury passes Venus 21 times as Mercury orbits Sol 34 times.

    5-8-13-21-34 is the Fibonacci series, the ratio of successive terms of which converge to Φ

    But the interesting quirk here is the Mercury – Earth and Venus – Earth ratios. One is for distance, and the other for period. Keplers third law shows there is a relationship between the distance and period of an orbit: period² is proportional to semi-major-axis3

    Newton reduced the constant of proportionality in Kepler’s 3rd law (for bodies with huge mass differences) to
    2 / GMSun.
    The more universal formulation is
    P2 = 4π2 a3 / G(m1 + m2)

    Mathis reminds us that the surface area of a sphere is
    SA = 4πr2
    And that it falls off by the square, like gravity does.

    It’s a good puzzle.

  18. Thanks. Math is a very weak subject for me Rog, I have something I will be sending to you that has built upon my earlier climatic theory when the correct topic comes out with questions. I will welcome commentary pro or con.

  19. oldbrew says:

    Yes, Mathis notes in the link above (page 3) that the curves in this diagram are falling off by the (Phi) square:

    ‘The length of the side of a larger square to the next smaller square is in the golden ratio.’ – Wikipedia

    Spiral galaxy:

    Hurricane Isabel (by NOAA):

    Tropical storm Bansi (by NASA):

  20. oldbrew says:

    Some interesting stuff in this 2010 report:

    ‘Golden ratio discovered in quantum world: Hidden symmetry observed for the first time in solid state matter’


    A few quotes:
    ‘Researchers from the Helmholtz-Zentrum Berlin für Materialien und Energie (HZB), in co-operation with colleagues from Oxford and Bristol Universities, as well as the Rutherford Appleton Laboratory, UK, have for the first time observed a nanoscale symmetry hidden in solid state matter. They have measured the signatures of a symmetry showing the same attributes as the golden ratio famous from art and architecture.’

    ‘When applying a magnetic field at right angles to an aligned spin the magnetic chain will transform into a new state called quantum critical, which can be thought of as a quantum version of a fractal pattern.’

    “Here the tension comes from the interaction between spins causing them to magnetically resonate. For these interactions we found a series (scale) of resonant notes: The first two notes show a perfect relationship with each other. Their frequencies (pitch) are in the ratio of 1.618…, which is the golden ratio famous from art and architecture.” – Dr. Radu Coldea from Oxford University

  21. oldbrew says:

    Physics World (‘the member magazine of the Institute of Physics’) wades in with the same story but a few extra details:

    ‘Golden stars’ pulsate in a strange, non-chaotic way