# Golden variables

An article on physicsworld.com reported the discovery of variable stars whose periodic dimming and brightening frequencies have a ratio at or very near the golden ratio.

The objects were found in data from the Kepler space telescope by looking for stars with two characteristic pulsation frequencies that have a “golden ratio” of approximately 1.62. The discovery could shed light on the physics that drives variable stars and also help astronomers come up with better classification systems for these objects.

The Golden mean is a ratio that can be found by dividing a line into two parts in such a way that the longer part divided by the smaller part is also equal to the whole length divided by the longer part.  More clearly, if the longer segment has length a and the shorter segment has length b, then:

a/b = (a+b)/a = 1.6180339887498948420 …=φ

The well-known Fibonacci sequence is nicely tied to this ratio. In the Fibonacci sequence, each term is the sum of the two previous terms.  The sequence begins like this:

1,1,2,3,5,8,13,21,….

The ratio of 3 to 5 is 1.666 and the ratio of 13 to 21 is 1.625. Further into the sequence, the ratio of 144 to 233 is 1.618.  These ratios get closer and closer to the irrational number that is the golden ratio, a number whose decimal expansion is infinite and non-repeating (like the decimal expansion of p).

A number of natural occurrences of this ratio, are often cited.  The growth of certain flower petals follow the Fibonacci sequence, as does the progression of tree branches.   A golden spiral gets wider (or further from its origin) by a factor of φ for every quarter turn it makes.   Snail shells and nautilus shells follow this pattern, as does the cochlea of the inner ear.  It can also be seen in the shape of certain spider’s webs.

Scientificamerican.com also reported on this discovery.  And they balanced their story with this mathematician’s judgment:

“Many claims about natural phenomena and the golden ratio are exaggerated,” says mathematician and computer scientist George Markowsky of the University of Maine, Orono. “I refuse to accept anything off by 2 percent or more as evidence of the golden ratio. After all, around any real number there are infinitely many other real numbers. People don’t seem to write papers about the mystic properties of .6 (which is very close to .618….).”

But whether the ratio is or approximates the golden ratio, it’s presence does seem to signify that this variable star has some distinguishing characteristics. It is a type of periodic variable star called an ‘RR Lyrae’ variable.  The Physics World piece tells us that the presence of the golden ratio in this dynamical system could indicate that the star behaves as a ‘strange non-chaotic attractor,’ or that the system is fractal and non-chaotic.  The golden ratio is an irrational number that is often understood in geometric or growth terms. I find it interesting that while this special ratio is being perceived here in terms of timed brightness, or some measure of duration,  the significance of its presence may still be the way it provides information about the star’s structure.

To study the dynamics of the star, Learned and Hippke joined forces with physicists at the University of Hawaii and the College of Wooster, including John Lindner. To verify that the star is indeed a strange non-chaotic attractor, the team did a “spectral scaling” analysis. First, the researchers did a Fourier transform of a time sequence of the brightness… creating a power spectrum with peaks at a large number of frequencies. Then, they counted the number of peaks above a threshold value, repeating the process over a wide range of threshold values. Finally, they plotted the number of peaks above the threshold as a function of the threshold.

They found that the number of peaks was pretty well constant until the threshold reached an inflection point (a point on a curve at which the sign of the curvature changes). When this occurred, the number dropped rapidly and obeyed a power law. According to the team, this behaviour is indicative of strange non-chaotic dynamics. Interestingly, Lindner points out that a similar analysis of the variability of the coastline of Norway yields the same power-law exponent of –1.5.