Strange Stars Pulse to the Golden Mean

Nature has revealed peculiar mathematical objects that connect order and chaos.

KIC 5520878, a pulsating star in the constellation Lyra, generates a unique fractal pattern that hints at unknown stellar processes.

Olena Shmahalo/Quanta Magazine

A pulsating star in the constellation Lyra generates a unique fractal pattern that hints at unknown stellar processes.

What struck John Learned about the blinking of KIC 5520878, a bluish-white star 16,000 light-years away, was how artificial it seemed.

Learned, a neutrino physicist at the University of Hawaii, Mānoa, has a pet theory that super-advanced alien civilizations might send messages by tickling stars with neutrino beams, eliciting Morse code-like pulses. “It’s the sort of thing tenured senior professors can get away with,” he said. The pulsations of KIC 5520878, recorded recently by NASA’s Kepler telescope, suggested that the star might be so employed.

A “variable” star, KIC 5520878 brightens and dims in a six-hour cycle, seesawing between cool-and-clear and hot-and-opaque. Overlaying this rhythm is a second, subtler variation of unknown origin; this frequency interplays with the first to make some of the star’s pulses brighter than others. In the fluctuations, Learned had identified interesting and, he thought, possibly intelligent sequences, such as prime numbers (which have been floated as a conceivable basis of extraterrestrial communication). He then found hints that the star’s pulses were chaotic.

Courtesy of the University of Hawaii, Mānoa

John Learned, professor of physics at the University of Hawaii, Mānoa.

But when Learned mentioned his investigations to a colleague, William Ditto, last summer, Ditto was struck by the ratio of the two frequencies driving the star’s pulsations.

“I said, ‘Wait a minute, that’s the golden mean.’”

This irrational number, which begins 1.618, is found in certain spirals, golden rectangles and now the relative speeds of two mysterious stellar processes. It meant that the blinking of KIC 5520878 wasn’t an extraterrestrial signal, Ditto realized, but something else that had never before been found in nature: a mathematical curiosity caught halfway between order and chaos called a “strange nonchaotic attractor.”

Dynamical systems — such as pendulums, the weather and variable stars — tend to fall into circumscribed patterns of behavior that are a subset of all the ways they could possibly behave. A pendulum wants to swing from side to side, for example, and the weather stays within a general realm of possibility (it will never be zero degrees in summer). Plotting these patterns creates a shape called an “attractor.”

Mathematicians in the 1970s used attractors to model the behavior of chaotic systems like the weather, and they found that the future path of such a system through its attractor is extremely dependent on its exact starting point. This sensitivity to initial conditions, known as the butterfly effect, makes the behavior of chaotic systems unpredictable; you can’t tell the forecast very far in advance if the flap of a butterfly’s wings today can make the difference, two weeks from now, between sunshine and a hurricane. The infinitely detailed paths that most chaotic systems take through their attractors are called “fractals.” Zoom in on a fractal, and new variations keep appearing, just as new outcrops appear whenever you zoom in on the craggy coastline of Great Britain. Attractors with this fractal structure are called “strange attractors.”

Then in 1984, mathematicians led by Celso Grebogi, Edward Ott and James Yorke of the University of Maryland in College Park discovered an unexpected new category of objects — strange attractors shaped not by chaos but by irrationality. These shapes formed from the paths of a system driven at two frequencies with no common multiple — that is, frequencies whose ratio was an irrational number, like pi or the golden mean. Unlike other strange attractors, these special “nonchaotic” ones did not exhibit a butterfly effect; a small change to a system’s initial state had a proportionally small effect on its resulting fractal journey through its attractor, making its evolution relatively stable and predictable.

Olena Shmahalo/Quanta Magazine. Source: Nicolas Desprez

The future path of a chaotic system through its attractor depends extremely sensitively on its exact initial state, such that paths from nearby starting points quickly diverge (top). In a strange nonchaotic system, however, paths with nearby starting points stay correlated (bottom).

“It was quite surprising to find these fractal structures in something that was totally nonchaotic,” said Grebogi, a Brazilian chaos theorist who is now a professor at the University of Aberdeen in Scotland.

Though no example could be positively identified, scientists speculated that strange nonchaotic attractors might be everywhere around and within us. It seemed possible that the climate, with its variable yet stable patterns, could be such a system. The human brain might be another.

The first laboratory demonstration of strange nonchaotic dynamics occurred in 1990, spearheaded by Ott and none other than William Ditto. Working at the Naval Surface Warfare Center in Silver Spring, Maryland, Ditto, Ott and several collaborators induced a magnetic field inside a metallic strip of tinsel called a “magnetoelastic ribbon” and varied the field’s strength at two different frequencies related by the golden ratio. The ribbon stiffened and relaxed in a strange nonchaotic pattern, bringing to life the mathematical discovery from six years earlier. “We were the first people to see this thing; we were pleased with that,” Ditto said. “Then I forgot about it for 20 years.”

In a very different field of academia, astronomers since ancient times have contemplated stars that throb. In 1912, Henrietta Leavitt discovered that the frequency at which variable stars pulsate corresponds to their intrinsic brightness, a feature that later allowed them to be used as “standard candles” for gauging cosmic distances. The conventional explanation for the stars’ behavior comes from Sir Arthur Eddington, who proposed in the 1930s that variable stars grow and shrink in a continual cycle: First, a star shines brightly, but as it gradually shrinks and heats up, its outer layer turns opaque, dimming the starlight. As the energy trapped inside the star builds up, the star expands under the pressure, cools, becomes transparent, and shines brightly once again.

The study of variable stars entered boom times in 2009 with the launch of the Kepler telescope, which looked for small aberrations in starlight as a sign of distant planets. The telescope gathered a trove of unprecedented data on the pulsations of variable stars throughout the galaxy. Other, ground-based surveys have added further riches.

The data revealed subtle variations in many of the stars’ pulsations that hinted at stellar processes beyond those described by Eddington. The pulses of starlight could be separated into two main frequencies: a faster one like the beat of a snare drum and a slower one like a gong, with the two rhythms played out of sync. And in more than 100 of these variable stars — including those, like KIC 5520878, belonging to a subclass called “RRc” — the ratios defining the duration of one frequency relative to the other inexplicably fell between 1.58 and 1.64.

“This is an entirely new ratio, and we don’t know what it’s about,” said Katrien Kolenberg, a variable star specialist with joint appointments at the Harvard-Smithsonian Center for Astrophysics and two universities in Belgium.

Within this window lies the golden mean. Upon seeing the number last summer, Ditto instantly suspected that KIC 5520878 and stars like it might be exhibiting strange nonchaotic dynamics. Indeed, when he and his co-authors tracked the brightness of the stars over time, they found signs of fractal yet nonchaotic variation in all four of the RRc stars in the Kepler data set, KIC 5520878 among them.

Lindner et al.

The evolution of the brightness of star KIC 5520878 forms an object called a “strange nonchaotic attractor,” shown here from two angles. The 3-D plot is created by locating the brightness of the star at each point in time with three coordinates: the brightness at that point in time along the x axis, the brightness at the previous point in time along the y axis, and the brightness two time increments earlier along the z axis.

As the star’s brightness waxes and wanes through time, it appears to wend a fractal path around and around a doughnut-shaped attractor. Zoom in or out, and the path looks similarly erratic at every scale of resolution. Like a craggy coastline, the peaks in the star’s brightness are variegated by smaller peaks occurring on shorter time scales, and those peaks are studded with even smaller and quicker peaks. But because the star is nonchaotic, there is no butterfly effect; paths that converge stay correlated. Imagine you are the star moving around its attractor. “You drop a bread crumb in a spot every half-hour,” Ditto said. “Two years later I come to that spot and start dropping bread crumbs. I fundamentally come back to your bread crumbs every few hours or days, no matter what I do. It’s crazy. Any two [nearby] points on the doughnut behave like that.”

The discovery, reported in February in Physical Review Letters, has some astronomers turning to arcane mathematics papers for new clues about the inner workings of variable stars.

“It’s really interesting because it’s a new kind of dynamical behavior,” said Róbert Szabó, a variable-star expert at Konkoly Observatory in Budapest, Hungary. “But it might be only a coincidence,” unrelated to the fundamental physics at work in the stars. Adding to the mystery are the hundreds of other variable stars that are also driven through their attractor by two frequencies whose ratio is near the golden mean, though not always near enough to give rise to such special dynamical behavior.

“All this begs the question,” Ditto said, “what is fundamentally going on with these stars that they end up with a ratio near the golden mean?”

The new paper offers one hypothesis. The KAM theorem, named for Andrey Kolmogorov, Vladimir Arnold and Jürgen Moser, holds that systems driven by frequencies in an irrational ratio tend to be the most stable; that is, they can’t easily be knocked off-kilter into a new state of motion. In that case, it might be the fate of unstable stars to evolve until they arrive at a number like the golden mean. “It’s the most robust number to perturbations, which means these stars may select it out,” Ditto explained.

Variable star specialists plan to develop new stellar physics models that incorporate the newfound secondary frequencies. This will allow them to examine the frequencies’ possible physical origins and the relevance of the KAM theorem. “If we have a good model, we can start to see what’s going on,” Szabó said, “what kind of pulsations and oscillations can be excited in these stars.”

As for Learned, who reported with his colleagues in The Astrophysical Journal that the blinking of KIC 5520878 is almost definitely natural after all, the events of the past year are a prime example of “science in action,” he said. “We started out looking for something kind of wacky, and we found something that may turn out to be of big importance in understanding stellar dynamics.”


Correction on March 11, 2015: An earlier version of this article asserted that the golden ratio is found in nautilus shells, which is a common misperception. Nautilus shells are closer to the ratio 4/3, or 1.33. 

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  • At first I was completely astonished that yet another (seemingly) natural phenomenon is related to the Golden Mean. It was a major epiphany in my mid-20’s how nature follows the rule of irrationality, which still follows a pattern, and not simply rational numbers. It’s sacred geometry in action.

    The more I read, the more I realised that this is ‘just another’ verification of what I realized about nature and our universe. It’s everywhere.. the design of snail shells, seashells, seed patterns in Sunflowers and so many other plant formations, spiral galaxy formations (Fibonacci spiral!), and now star illumination patterns. It’s everywhere. It’s a key to helping understand the nature of reality. Thanks for posting this!

  • This article risks promoting absurd myths surrounding the golden mean.

    Quoting the article:

    “Ditto was struck by the ratio of the two frequencies driving the star’s pulsations.
    [identified as the golden mean]
    This irrational number, which begins 1.618, is found in nautilus shells, golden rectangles and now the relative speeds of two mysterious stellar processes.”

    The spiraling coefficient for nautilus shells is plainly not the golden mean. The article talks about a ratio of two numbers then goes on to state that the golden ratio is an irrational number which, by definition, cannot be written as a ratio of two rational numbers. So their ratio of the two frequencies can’t possibly be the irrational golden mean (unless at least one frequency is irrational, which it can’t be if it was measured).

    “…the ratios defining the duration of one frequency relative to the other inexplicably fell between 1.58 and 1.64”

    So the ratios aren’t the golden mean then, they happen to be close to the number 1.62….Why is that so inexplicable? Why the ridiculous title?

    Please see this comic for a critical look at the golden mean and it’s appearance in nature:

    I’m sure the Simons Foundation is interested in mathematical literacy, this article doesn’t not help this aspect of its mission.

  • @mittimithai

    I’m not sure I understood your statement here: “The article talks about a ratio of two numbers then goes on to state that the golden ratio is an irrational number which, by definition, cannot be written as a ratio of two rational numbers. So their ratio of the two frequencies can’t possibly be the irrational golden mean (unless at least one frequency is irrational, which it can’t be if it was measured).”

    While it’s true that you can’t measure something exactly that has an irrational value, that doesn’t mean the thing doesn’t have an irrational value. ie, Frequencies can be irrational numbers, you just can’t measure it perfectly. For a geometric example, take a right triangle with sides of length 1 and 1, yielding a hypotenuse of length square root of 2 (irrational number). The length of the hypotenuse is an irrational number. If you try to measure that length with ruler you can’t get the value exactly, but you can still try to measure it and get close. So things with irrational values do exist, which means that, at least in theory, either or both of those frequencies could be an irrational number, so the ratio of those two frequencies could be an irrational number, and therefore could be the Golden Mean. The article doesn’t try to claim that they could measure it perfectly and indeed their value for the ratio is only fuzzily close to the Golden Mean

    For the record, I am not trying to support the idea that this really is the Golden Mean that they have found. I just wanted to clarify the mathematical argument.

  • I see what you are saying, but we quickly wind up in a philosophical debate that has been discussed ad nausem on internet forums:

    All interesting thoughts, but perhaps we can’t get very far discussing the physical “existence” of irrational numbers without getting very careful about what existence means etc.

    My main quibble is the overall tone (and title) that associates any number around 1.6 (which is close to other concise mathematical expressions) with an “inexplicable” resemblance to the golden mean. The article does quote Róbert Szabó appropriately, but it’d be nice if a science magazine could do a little more to avoid common pitfalls in reasoning here.

    FWIW I find the correction a bit of a cop out. The sentence now reads:

    “This irrational number, which begins 1.618, is found in certain spirals, golden rectangles and now the relative speeds of two mysterious stellar processes.”

    Well, which spirals exactly? What’s so special about them? The part about golden rectangles is a tautology (“the number 2 is found in rectangles with aspect ratios of 2”). There isn’t anything special about golden rectangles. The sentence is suggestive and only helps perpetuates common misconceptions.

  • mittmithai:

    If we stop and think about the nature of physical reality we must conclude that *all* measurements are approximate. There are no measurements that are exactly integers, decimals, or irrational numbers, because there is no such thing as “exactly” if we aren’t in the realm of pure mathematics or logic.

    The Golden Mean is interesting to be sure, but whether or not it has true bearing (aside from our penchant for finding patterns out of habit) depends upon whether or not it is an inevitable result of physical processes in nature expressing mathematical truths. That the Nautilus constructs such a lovely symmetric shell certainly looks like a statistical outlier in comparison with the number of cephalopods who construct completely different shells, thus leading us towards the suggestion that the nautilus shell isn’t any more golden than any other type of shell as far as nature is concerned.

    I’m more apt to lean towards the idea that nature selects the simpler way of doing things, for the reason that random chance favors the odds of the simpler over the complex. There are *massive* caveats to this idea, of course. I’m speaking to the complexity of given options for solving a type of problem. While we can marvel at the numerous ways that nature has expressed or devised a given solution we must also realize that there are infinite other ways that nature has *not* devised a solution, and that even the most unexpected results still fall within the range of plausible.

    That a mathematical ratio might interplay with physical reality in a way that expresses as more stable results is, as a result, a factor in a less complex solution. So the Golden Ratio might, indeed, be more than just human observational bias. Statistical stability, like measurement, is an exercise in approximation.. or at least in medians and averages. Certain mathematical relationships and geometric relationships can then be expected as a result of the fundamental forces of physical reality.

    Which is why stars and planets are spherical and not any other shape. That we accept a given result (spheres) from a given set of inputs (sufficient mass and gravity) as a matter of course indicates that these relationships aren’t all mystical belief. The pulsating star in Lyra might very well be a happy accident that comes out phi, or it might be that it is an expression of mathematical relationships upon reality cropping up in very macro form.

    Either way, to dismiss the possibility that the Golden Mean is somehow expressed here in some way — just because the way people use it annoy you — is acting in just as silly a fashion as people who see the Golden Mean everywhere.

    I will grant you that the writing of this article goes a bit hyperbolic: the word “inexplicable” is definitely incorrect, as the physicist in question has some theories to explicate it in the same damn paragraph.

  • Actually the golden mean comes up in many mathematical contexts. It is the
    simplest non-terminating continued fraction for example, but for that reason is (one of ) the hardest irrational numbers to approximate with rationals. So in some ways it is not at all surprising that it should come up in the context of a dynamical system that is midway between a ratio of simple harmonics and a chaotic system. There is a lot of nonsense out there about the golden ratio but this article seems perfectly OK.

  • I do not think the fact that “systems driven by frequencies in an irrational ratio tend to be the most stable” is a sufficient explanation for the golden mean showing up. There is, of course, an infinity of irrational numbers to choose from (there are more irrational numbers than rational ones), so why choose the golden mean? There must be some other physical explanation to why precisely the golden mean turns out to be the optimal one.

  • Separating the content of the article vs. the paper:
    I don’t know a lot about dynamical systems, but the paper mentions the golden ratio in terms of analytic results of KAM theory, this seems to be fairly straightforward and true ( I can see where the corresponding sentence in the article comes from and cannot fault the author for that.

    If one looks at the comic post earlier, you can see there is no reason why phi cannot show up in a physical model (afterall it is the solution to a simple quadratic equation), the KAM theory context here seems uncontroversial and completely valid.

    Given the context of nonsense around phi, I think it is important to take special care in distinguishing claims and carefully assert the significance of phi when it comes to experiments. I have a fairly superficial understanding of the significance of phi in the context of KAM theory, it isn’t clear if to me if the authors of the paper are “reaching” with their interpretation of significance of phi. I don’t see much in the way of a statistical argument in the paper, so I can’t help but feel suspicious given the type of thinking we so often see with the golden ratio. The title (and some of the wording) of this summary article feels far to complicit with phi-ction.

  • Excellent article.

    Just adds to the literature identifying the relationships found between mathematics and reality. Honestly, it’s tough to not be a mathematical realist after viewing a lot of the articles on this website. Keep them coming…

  • Thanks for the interesting comments. I asked William Ditto to kindly respond to some of your queries about the golden ratio and the KAM theorem. He wrote the following:

    “[T]he strange non chaotic dynamics is really the strong argument that the number is not just near the golden mean but near enough to have golden mean consequences. One cannot get exactly the GM in nature (you can in mathematics). Same for fractals, physical fractals are fractals approximately till one hits resolution limitations of nature. However, the duck analogy holds here, we saw something that looks a duck and then looked to see if we observed that it walks like a duck, quacks like a duck, etc.”


    “Regarding KAM, that gets a bit technical but [collaborator] John Lindner gave a great physics colloquium (very accessible to undergrads and lay public) that was video taped and the video and his slides (in pdf) are also available on this website:

    Feel free to point them to this link as he goes over KAM and why Phi and KAM in better detail than I can in a reply. The astrophysical models are still a work in progress but we have at least one simple model (in talk) that shows why phi is the only ratio that remains and it is related to KAM theory.”

    -Natalie Wolchover

  • It was a nice video that broadly surveyed a few areas and I certainly learned a few things.

    I must say though, the significance of the golden ratio isn’t obvious to me from watching it. There seem to be two different uses of phi in the video: one where the value fit is within 2% of the golden ratio another where a plot is zoomed out to see a split around one over phi. I mean no insult to the authors, but this sort of thing feels a lot like “phi-shing” we often see…

    The video also claims that phi is “the painter’s most aesthetic ratio”, this is a common claim that makes no sense when examined critically.

  • mittimithai,

    I don’t get what’s so hard to understand regarding aesthetics and the golden ration. It’s been used for centuries by artists, so I don’t understand how it is a “common claim that makes no sense when examined critically.”

    Here’s some further reading for you:

    No one is saying its the end all and be all of beauty, but to say that it hasn’t been used extensively in the arts is completely false.

  • Pete, I encourage you to read the comic posted earlier.
    The links you provide say the same thing I do: no clear evidence that there is anything special about the Golden ratio in terms of aesthetics. The concluding paragraphs from the first link you gave:

    I will certainly not attempt to make the ultimate sense of sex appeal in an article on the Golden Ratio. I would like to point out, however, that the human face provides us with hundreds of lengths to choose from. If you have the patience to juggle and manipulate the numbers in various ways, you are bound to come up with some ratios that are equal to the Golden Ratio.

    Furthermore, I should note that the literature is bursting with false claims and misconceptions about the appearance of the Golden Ratio in the arts (e.g. in the works of Giotto, Seurat, Mondrian). The history of art has nevertheless shown that artists who have produced works of truly lasting value are precisely those who have departed from any formal canon for aesthetics. In spite of the Golden Ratio’s truly amazing mathematical properties, and its propensity to pop up where least expected in natural phenomena, I believe that we should abandon its application as some sort of universal standard for “beauty,” either in the human face or in the arts.

  • mittimithai,

    I’ve read your comic, and I’m well aware of the reservations within the first article when it comes to aesthetics, which is why I was sure to add “No one is saying its the end all and be all of beauty…” to my last comment. That being said, he also mentions how its included in a plethora of artistic works.

    It’s a little strange that you also mention the second article as having nothing to offer when it comes to explanation, when the whole thing is about how it might relate to the brain interpreting its surroundings in an advantageous manner.

    I think you have this conception that anyone reading this article and saying “oh cool” is some sort of new age numerology mystic. Again, it’s just really neat to see these mathematical patterns crop up in nature, and it allows for all kinds of mathematical/philosophical inquiry (I love philosophy of mathematics, and articles like this could help generate more discussion). Quanta has been great about printing articles with fascinating mathematical insights into our world, and I hope Natalie and others keep it coming.

  • Peter, Not sure what else I can say that hasn’t been said in the cartoon, the use of phi in art is greatly exaggerated and the speaker in the video is plainly wrong when he suggests that it is “the painter’s most aesthetic ratio”.

    The second link isn’t very credible, the “golden-ness” of the Parthenon has been debunked in many places, no need to repeat the argument here. The second link also contradicts the information from the first link! Look at the quote about Mondrian in the first link:

    “false claims and misconceptions … (e.g. in the works of Giotto, Seurat, Mondrian)”

    and then look at the second link’s position on the same thing:

    “The Dutch painter Mondrian used it in his abstract compositions”

    The second link is also perpetuating the common misconception that the golden ratio is somehow involved in the Mona Lisa, it simply isn’t (links in my comic post).

    Dali is a different case, it is fairly clear he was directly referencing phi as it had been popularized (with many many misconceptions) by phi proponents.

    Phi deserves some special attention since so many of us get it wrong (see the correction on this very post). I encourage you to post some examples of the artwork where phi has been deliberately employed. I’d ask that you cite examples that demonstrative of your earlier statement that phi has “been used for centuries by artists”.


    The above might help as far as examples go. I really don’t have a stake in the aesthetics game. I’m not an artist, and I can’t say for sure, but I really don’ think that mentioning the golden ratio in relation to art is doing evil to people’s minds, especially if it’s consciously incorporated by artists that may be more mathematically inclined.

  • That site is a case study in pseudoscience around the golden mean (I’m sure I refer to it derisively somewhere in my comic post).

    All of those lines and rectangles are nothing more than tea leaf reading. We can all find those sorts of things if we look long enough…Very little evidence that phi has anything to do with the composition (except Dali, which I explained above). You are not alone in your misperception here, the reason why I made a comic about it.

  • The Phi comes up in nature quite often because of it’s ability to aid the subject displaying it’s ability to outperform the other options (as in plants). However, this isn’t likely a survivability instance. Magnetic forces can play a role in creating a golden number and I would suspect that some form of magnetism plays a role in this one. I would expect that this star or it’s enveloping system is richer than others in heavy elements.

  • Not every golden ratio phenomenon in physics is questionable.
    For example consider the Periodic Table. As we all know you can get as arbitrarily close to the golden ratio as you want by dividing consecutive Fibonacci numbers (or their generalized relatives such as Lucas numbers). Taking Fib numbers as ATOMIC numbers generates a very interesting pattern- up to 89 (the last Fib number that is also the atomic number of a known element), ALL of them are first members of orbital half rows. In fact the ODD Fibs map to the first half orbital and the EVEN Fibs map to the second. Up to and including 89 there are NO exceptions (though above 89 things get out of whack, but then again there aren’t likely to be any real elements that far up). The Lucas numbers have a similar patterning trend, without exception up to 18, but to last members of orbital half rows (including the odd/even split). Starting with 29 (Cu) and 47 (Ag) we see a mismatch in POSITION within periods, but BEHAVIORALLY because of anomalous electronic configurations that rob one s electron to give it to the nearly filled d orbital they still work. 76 (Os) doesn’t have the correct electronic configuration for this sort of ‘fix’, but still has a behavioral solution- by acting as if it were xenon in compounds.

    The Pascal Triangle generates Fibonacci series numbers by summing along the so-called ‘shallow’ diagonals. Atomic structures (both electronic and nuclear) show a good number of Pascal mathematical patterning. We all know that traditionally the periods of the periodic table end with the noble gases. But from a quantum mechanical perspective they should end with the s-block alkaline earth elements (but including He here). All s2. If one looks at the atomic numbers of the s2 elements one can see that EVERY OTHER number is a tetrahedral number- that is 4, 20, 56, *120. All the intermediate numbers 2, 12, 38, 88 are arithmetic means of pairs of the tetrahedral numbers (2 works if you pair 4 and 0), reflecting the fact that periods are paired for length when the table is constructed as above (the Janet Left-Step table, devised in the late 1920’s). Those period lengths are all half-/double-square numbers, intimately related to the tetrahedrals, 2, 8, 18, 32, 50…
    And in the nucleus, under the simple harmonic oscillator model, so called ‘magic’ numbers of nucleons are ALL doubled tetrahedral numbers 2, 8, 20, 40, 70, 112, 168… Obviously there are number theoretical motivations involved, but nobody in the professional realm is really giving any thought to this.

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