Yeah, we’ve written about golden ratio twaddle a few times, although we tend not to bother. Sir Theodore Cook said it all a century ago. But, on occasion, we have cause to give the phi nonsense another whack. And, as it happens, we’ve been working on one such post when more nonsense came rolling in.

Classy media outlets, such as *The Australian*, are currently trumpeting the news that Amber Heard, whatever her other failings, also has the most beautiful face in the world. Because golden ratio. This stunning news coming courtesy of some cosmetic surgery clowns.

This nonsense will never end, of course. *The Australian *report, which took the angle of seeking to abolish “the beauty standard”, and seemingly any beauty standard, was notable in that it quoted UNSW mathematician Thomas Britz at length:

*The ratio itself is very simple, and was initially seen thousands of years ago in mathematics.*

*Then it was extrapolated to be seen in nature, **and then those beautiful mathematical proportions were used to judge beauty.*

*A lot of people hold (the ratio) as important to them, and they aim towards it. *

The reporter then declares that the golden ratio “is embedded in our society”, quoting Britz to the effect that, during the Renaissance, “[m]illions of identical statues and portraits were produced and the beauty standard was born”. Then, Britz again:

*The Golden Ratio gave a bias to society that means you then seek that ratio out in order to find beauty. That still exists today. *

Uh, thanks very much Tom, but you’re not helping.

I think I love the misuse of the Golden Ratio more than I love the golden ratio itself.

I went to a Jeffrey Smart exhibition in Canberra earlier this year (it was great) – and here is the blurb on the wall for a painting called “On the Roof”, and the painting itself (warning – nudity)

(and did anyone notice that: the first letter of Amber is A which is the 1st letter in the alphabet; the first letter of Heard is H, which is the 8th letter of the alphabet; that there are 2 words in her name; that there are 5 letters in each part of her name; and that there are a total of 3 syllables in her name? All these numbers (1, 2, 3, 5, and 8) are part of the Fibonacci series – therefore golden ratio! It all makes cosmic sense!)

Hi there! I’m the mathematician mentioned in the article, and I agree with you. It was a happy and chaotic interview, right before the editorial deadline. I was not as precise as I should have been and some things got lost in editorial translation. This was more my fault than the reporter’s – sorry! Although it was just a tiny entertainment piece, it prompted me to write a proper article on the myths of the Golden Ratio. It appears here: https://theconversation.com/does-amber-heard-really-have-the-worlds-most-beautiful-face-an-expert-explains-why-the-golden-ratio-test-is-bogus-187018 . There was some editorial tweaking and one or two tiny errors but it should hopefully help dispel some misunderstandings, including those I inadvertently promoted in that smaller article.

Thanks very much, Thomas. An admirably collegial response to a not so collegial slap. It’s also magnanimous of you to mostly excuse the

Australianreporter for the nonsense in her article, although I don’t buy it. The reporter clearly had a pre-conceived angle on notions of beauty, it was an obtuse angle (ha ha), and an angle for which the golden ratio twaddle was irrelevant and confusing. Her shoehorning φ in where it didn’t belong, and couldn’t belong, was her doing.I mostly liked your Conversation article, even if the Amber Heard ship has sailed. And this stuff is basically unkillable. But/and, since you’re here, I’m gonna make three nitpicks, in increasing order of grumpiness.

1) The mostly debunking and (unfortunately paywalled) Archaeology article to which you linked was very interesting. I was not totally convinced, however, even by the 4/N (with N huge) purported examples of Ancient Greek use of φ in architecture. The Modon example had the most plausibility, and I’d have to think about it more, but it’s still simply number-juggling; no theory is evident, or even suggested, as to why whatever ratio was incorporated.

2) You state that Fibonacci and/or φ “appear in … the whirling arms of certain galaxies”. Name one.

3) You conclude that “there is currently no evidence suggesting the Golden Ratio φ determines facial beauty – or any visual beauty for that matter”. That is correct and good, but it way, way undersells the intrinsic idiocy of the entire game. It is not simply that no evidence exists for φ as the key to beauty; it is that the very idea that one precise number, or even one imprecise number, might be the key to beauty is absurd. That is just not how beauty works. Obviously. Burkard and I hammered the one aspect here, and Theodore Cook, much more elegantly, hammered the other aspect here.

In brief, and with genuine respect for your debunking Conversation article, I think you’re being too considerate, in two senses of the word. First, I think you’re being way too lenient on purveyors of nonsense. Secondly, even acknowledging your pretty solid debunking, you still seem to be too willing to consider there might be something to these φ games.

The only reasonable first approximation is that

allthis φ stuff is crap. Then, sure, phyllotaxis for example makes us take a big step back. And, yes, we may improperly dismiss a Modon or two. But starting with the assumption that it’s all crap, and then demanding each time to be convinced, is the only way to converge to the truth on this.Hi Marty, thank you very much for your kind and well-argued reply!

Again, I agree with all your points.

With respect to the galaxies, my memory had been that a few certain galaxy formation processes had been argued to act in similar ways to phyllotaxis in plants, geometries formed from physical packings. I couldn’t imagine quite how that would work, given that galaxy formations are driven by very different forces or “objectives” than plant formations but I deferred to the experts. However, I’ve just now tried and failed to find the scientific article in mind, so the more natural conclusion is that I’ve more likely misremembered this, and probably read this in some non-scientific article or book and mentally filed it incorrectly. Thanks for correcting me on this!

The occurrence of Fibonacci numbers and phi in nature is very limited. I had stated this in the original draft of the article but it disrupted the article’s flow and was cut. I’m ok with that since the article still conveys the limited nature of phi’s (approximate) occurrence in nature.

With respect to the other points, I completely agree. To anyone with understanding of maths – or the scientific method for that matter – it makes no sense to go looking for phi in architecture or faces or any other physical object with lots of fuzzy data. phi is an infinitely precise number; it is not 1.5 or even 1.618. If you have a wealth of fuzzy data like a face or a photo of an old building, then you can go looking for – and find – countless numbers and ratios but doesn’t mean that any of them have particular relevance. That is just cherry picking and toying around with pattern recognition: spotting images in clouds or the stars. It’s like seeing how a particular horoscope matches your life, at least if you tweak facts to fit and ignore data that doesn’t fit. These “discoveries” of phi would be more credible if one were to search for *all* ratios in a given object and then apply statistics to see whether any of them appeared more prevalently than others, say. If numbers close enough to phi kept appearing, then that could be taken as weak evidence for the existence of phi. But this is not how people approach detecting phi; they go looking for it, which is methodologically wrong. The archaeology article makes this error but was mostly valid since it largely finds no evidence of phi.

I know that you know all this already but this is a good opportunity to state it clearly myself. I’m very happy with the article in The Conversation but it was bound by several natural constraints. One was a word limit that cut my initial draft down to a third. Another constraint was that it should be readable and appealing, which cut some of the more subtle points and side branches. And a third was that it should appeal to a broad audience of mostly non-mathematicians; that cut some beautiful maths and some mathematical observations, like the one above.

One of the cut parts was about the interesting suggestion by Adrian Bejan in the 2010 article

https://www.witpress.com/elibrary/dne-volumes/4/2/403

that the claimed preference for phi in visual images might be actually be a preference for the ratio 3:2 , or 1.5 . Through physical and mathematical arguments, Bejan claimed this to be the ratio providing quickest flow of information when we scan a a painting, page or screen with those dimensions.

Slightly different brain-process arguments could also be used to explain, or at least speculate, why numbers such as phi might be seen as attractive. They are definitely seen as beautiful in maths, and that’s because they hit a sweet spot between order and chaos, so to speak. They provide our brains with good pattern recognition challenges and they give us enough surprises for us to measure good randomness, both functions of which – pattern recognition and randomness-measuring – is something that our brains like to do, with dopamine rewards and emotions including that of beauty. We also know that that beauty emotion is much the same whether we experience it in maths, music, art or a pretty face, so it would be natural to wonder whether the beauty of phi in maths could translate to us feeling beauty when spotting phi in visual images. If phi were not too easy and not too hard to recognise as a ratio that some how “fit”, then that might support the notion of phi as lending beauty to visual images.

However, there is no actual empirical evidence to support that notion. On the other hand, the study

Click to access 12027.pdf

found that Koreans prefer rectangles with the proportions 1:1 (square), 4:3 and, especially, sqrt(2):1 (or 1.414:1). The study found that preferences rely on a number of factors, including cultural factors. In the same way, one could imagine that if phi is now used enough in art, photography and cosmetic surgery, then we might end up having a preference for ratios close to phi, being used to them. This would however still be a cultural preference, and the number phi would still not hold any intrinsic relevance to our sense of beauty, let alone any mystical or Platonic significance.

Thanks a lot for the opportunity to present some of article “off-cuts” here! I find these off-cuts to be among the more interesting parts of the story here – but they didn’t fit into the clear and accessible article format.

Thanks again, Thomas. Your off-cuts are very interesting, and it’ll take me some time to explore them.

Briefly on the galaxies, I guess it’s possible someone has argued for a phyllotaxis kind of thing, but I’ve hunted on occasion and I’ve never located anything like that. All I’ve ever found is the silly slide à la nautilus shells: logarithmic and therefore φ.

Hi again, Thomas. Just a couple more thoughts on the suggested arguments – cultural and physiological – for the preferences of certain ratios.

First, it seems plausible enough that the mechanical way we see and then process what we see would give rise to preferences for preferred window shapes (and sizes). It’d also be surprising if such preferences didn’t play a subconscious role in art and its appreciation. But again, this only goes so far, and not very far. The idea that the magic of a Da Vinci or a Rembrandt or an Anybody might be substantially explained by a few dumb rectangles is just nuts.

Secondly, it is clear that cultural preferences can be self-reinforcing. If a certain artistic structure is preferred in a given culture, that encourages the continued employment of that structure, which then becomes more familiar, and snowball on. This self-reinforcing might be less or more conscious. So, in the case of φ-art, one might argue that the conscious promotion has meant there is, now, a cultural preference for φ-rectangles and whatnot. But I don’t believe it. I don’t believe the perception of φ-ratios is accurate enough or intrinsically motivated enough for such ratios to create any resonance or memory. I’ve certainly seen no evidence that it is.

This all reminds me of the end of

The Enigma of Kaspar Hauser, when they finally “explain” the enigma.Hi Marty, thanks a lot for another interesting reply!

Nice film references too! 🙂

I agree with you on both points. For the second, it is not unlikely that some people, like photographers and painters, might nowadays be biased towards φ in images, and biased against images not following the φ rules that they have been taught to follow. However, my guess is that 50-70 years or so of these subcultural feedback loops is not enough to significantly influence the broad and culturally heterogeneous public, even if most of us are massively influenced by advertising images.

Even for artists or photographers who follow φ rules, you could say that they never actually use the number φ itself, only varying degrees of approximations thereof. This cultural worship of φ is not about an actual number – it is the worship of the idea of a magical number, an ideal. One might argue that this ideal does actually make art beautiful, regardless of whether the measurable expressions of that ideal are accurate, at least to those who worship that ideal. They might be happy to interpret numbers very broadly in order to show the presence of that ideal. It’s really the idea of the number, not any actual number, that makes such people feel beauty and related emotions, as well as the sense of belonging to the in-group who know the “secret” about φ. Evidence for this is that worshipers of φ are rarely mathematicians.

Your point about the absurdity of trying to describe masterful art by a few simple rectangles is really good!

The sense of beauty and satisfaction that this reductionism can give is much the same as the sense of beauty and wonder in religions and in science, and in those contexts you also have the beautiful in-group feeling of knowing some “secrets” with some universal significance or explanatory powers. We’re hardwired to try to understand things through patterns and explanations, and our brains reward us with happy chemicals for doing that. The only danger here, though, is that we can get addicted to happy emotions and take pattern recognition and reductionism beyond the point where it is useful.

You might be interested in a recent article that I wrote about the Golden Ratio and its beauty… not in nature or art, or whatever, but in actual maths:

https://www.parabola.unsw.edu.au/2020-2029/volume-58-2022/issue-2/article/beauty-golden-ratio

After writing the article, it seems to me that it’s most beautiful in contexts of infinite limits, and sometimes not that pretty in a few of the more finite settings, though many of those are still quite pretty. It’s about that balance of order and chaos – pattern recognition and entropy-measuring, and the related dopamine-fueled beauty emotions. Here, the patterns occasionally get annoying since they break symmetries too much.

On further reflection, the beauty of the Golden Ratio maybe mostly lies at a higher meta-level of abstraction, namely the delight in seeing it appear in so many different contexts. This beauty of the Golden Ratio is not too different from the beauty of the Golden Ratio that non-mathematicians experience: the beauty of some magical number (that they would be hard-pressed to define) that, according to what everyone says, appears unexpectedly everywhere in nature and art. However, where mathematicians might say: “Yeah, the Golden Ratio reflects a very basic recurrence relation and equation, so you’d expect to see it appear in lots of places”, it would be natural for most non-mathematicians to extrapolate beyond reason about the Golden Ratio, just because there is a joy and beauty in finding deep and simple truths, even if they are wrong – for some, exactly because they make no sense.

Anyway, you might enjoy the article!

Thanks, Thomas. It’s a very nice article. A couple nitpicks (it’s what we do here …):

*) Although you are writing about very pretty mathematics, and in a mathematically solid manner, I’m not sure it’s not still playing into the phi-games. You’re writing about phi for a reason. What reason? Simply because that’s where the beautiful maths is?

*) I’m not sure the beginning history as you’ve written it is that solid, or should be presented in quite the manner you have. It’s a while since I’ve looked at it, but here’s my memory/understanding.

The pentagram and, more so, the golden rectangle indeed gives simple irrationality proofs, and the Pythagoreans *may* have done something like what you have suggested. However these proofs are better expressed and labelled as incommensurability proofs: the two labelled edges cannot both integer lengths (which is all the Greeks understood), since the shorter edges then produced by smalling the diagram would also be integers, and we would get an impossible infinite descent.

In particular, this proof doesn’t require the contemplation of, or even the creation of, any ratio of edge lengths, which is not a Greek kind of thing. So, even if the Pythagoreans discovered irrationality/incommensurability via the pentagram, that doesn’t imply, and I think it’s unlikely to be true, that they the Pythagoreans contemplated the *ratio* phi in any solid manner.

Thanks a lot, Marty; I’m happy that you liked the article! 🙂

I wrote it mostly because it is a rich and fascinating topic to write about (and hopefully to read about) and because I was already in the flow of thinking about this general topic: good and effective timing. It also completed my Fibonacci paper well, which feels nice. As editor of Parabola, I’m always after good articles; that was also a small factor. Speaking of which, please feel more than welcome to contribute anything to Parabola yourself! 🙂

Your points about the history section are presumably correct but my aim was to give very brief historical context, with interesting/entertaining anecdotes, balancing reasonably accuracy (up to historical probability) with clarity for a modern reader. Readers interested in more historical details were invited to read Livio’s book, which is a good starting point.

Does Livio’s book still have a nautilus shell on the cover …

More relevant, I realise the history wasn’t the purpose of your article. Nonetheless, I think the temptation to impose phi on a discussion, rather than phi being so prominent that it properly demands discussion, is always there. All the time people are throwing 5-things around and pronouncing “golden ratio”. I think the history is a more subtle version of this kind of thing.