One of the strangest and most enjoyable presentations we’ve given was at a Brisbane conference on art and design. The conference organiser appeared from nowhere, requesting that we give an address on the golden ratio and art. This was a little puzzling, since our views on such matters were presumably known to them. We replied as such:

*“But it’s all bullshit.”*

*“Yes, we want you to come up and tell us it’s all bullshit.”*

Which we dutifully did.

Our talk was well received. The artists seemed relieved that they could stop trying to make sense of something that made no sense. And, the conference was a blast. (It turns out that artists and designers have a significantly better idea of fun than mathematicians. Who knew?)

Golden ratio garbage, linked to both art and nature, has been around for ages, but it has really piled up in the last century. As it was taking off, the art critic Sir Theodore Cook wrote a brilliant, scathing critique. About 25 years ago, mathematician George Markowsky published a careful and thorough debunking. There have been plenty of other critiques, and of course we gave the cult a whack as well. But none of it helps. There will always be another clown waiting in the wings, ready to bring out her nautilus shells or Parthenons or whatnot.

That is all by way of introduction to last month’s bumper crop of nonsense. We try to steer clear of the phi fetish; the systemic perversion of education and (thus) democracy matters a hell of a lot more than some dumb clickbait. On occasion, however, it’s too much to ignore. And some golden nonsense is significantly disturbing.

Most recently we had Bella Hadid declared to be the most beautiful woman in the world because of, you know, ‘science’. This astonishing theorem was announced by that august journal, *The Daily Mail*. The theorem was repeated all around the world, almost always without irony. Predictably, the source for the theorem was a plastic surgeon, who performed some computery gimcrack eyes divided by chin plus nose ratio thing.

Whatever. Just some clown wanting to sell his dubious wares. Who passes his stuff to the newspapers, run by clowns selling their dubious wares. But then there’s the serious science as well.

Earlier in the month, Dr. Rafael Tamargo and Dr. Jonathan Pindrik, two medical researchers at John Hopkins, published a research paper in *The Journal of Craniofacial Surgery*: *Mammalian Skull dimensions and the Golden Ratio (Φ)*. We heard about their paper because the media clowns grabbed onto John Hopkins’ catchy press release, with its very catchy title: *Golden Ratio Observed in Human Skulls*.

Tamargo and Pindrik observed nothing of the sort, of course. One simply cannot detect an irrational number in our fuzzy, approximate view of nature. What Tarmago and Pindrik did was *imagine* the golden ratio in human skulls, and their imaginations were pretty wild.

Tamargo and Pindrik were exploring the accepted idea that the human cranium evolved to accommodate an increasingly large and complex brain. They measured 100 human skulls (and 70 skulls of other mammals). For each skull they calculated the ratio of the ‘nasioiniac arc’ and ‘parieto-occipital arc’ (green divided by red in the diagram below), as well as the ratio of the parieto-occipital arc and the ‘frontal arc’ (red divided blue). With a little algebra one can show that that the two ratios are equal if and only if that common ratio equals the golden ratio, (1 + √5)/2 ≈ 1.618…

Tamargo and Pindrik found that the two ratios in their human skulls averaged to about 1.64 and 1.57, which, they write, “are essentially identical and closely approximate Φ”. Well, yeah, sort of. As they admit, however, the ratios also “closely approximate” 1.6. But then the title *The Ratio 8/5 observed in Human Skulls* isn’t quite as grabby, is it?

Still, maybe Tamargo and Pindrik are correct? After all, the golden ratio sometimes appears in nature, in approximate form. So, why not here? Because there is simply no reason to think so. Tamargo and Pindrik bear the burden of proof, and there is not a hint of a proof in their paper.

Any claim for the appearance of the golden ratio must be supported by a model, an argument *why* the golden ratio might be appearing. Without that, or without a *lot* of decimal places, the claim is just number-mongering; the claim is no stronger or weaker than the claim for 1.6 or 1.62, or whatever.

Predictably, the media tended to swallow the press release whole, in the lazy and gullible manner they’ve turned into an art form. We could only find one properly sceptical report, one that was willing to suggest that at least some anatomists considered the skull-phi thing to be “ridiculous”.

Of course Tamargo and Pindrik are by no means the first people to spot phi in the human body and, in proper scholarly fashion, they cite earlier research. They write: “In the clinical sciences, Φ has been found to underlie cardiac anatomy and physiology, gait mechanics, and the aesthetic dimensions of the face.” And no, they didn’t reference *The Daily Mirror* for that last one. Without checking, however, there’s no particular reason to believe that the literature they cite has any greater validity. It is reasonable and proper to question whether the majority of this research is anything more than silly number games.

Sir Cook’s brilliant critique was titled “A New Disease in Architecture”. Perhaps it’s time for someone to write a follow-up: “A New Disease in Medicine”.

Right! I could never comprehend the weird obsession people have with \phi, and how it always seems to be related to people’s faces. Presenting the “it is bullshit” talk to artists seems like community service!

Thanks, Glen. Not really a community service. Just entertaining myself pointing out idiocy while the ship slowly sinks.

In defence of Phi as a number (not as a golden ratio – that is just rubbish) I rather like the way the number appears in, say, infinite surds or infinite fractions, but my love of the number ends there. Any person who insists that Mathematics must correspond to something in the “real world”… well, my opinion on this is known well enough.

…back to the wine.

Here’s a fun fact about phi, unrelated to infinite surds or infinite fractions, for your pleasure, then: For any triangle ABC, sin(A) + sin(B)sin(C) <= phi. (This can in fact be proven without any calculus other than the knowledge that sin(x) <= 1.)

Just quickly then, under what circumstances does the equality hold?

The equality holds when B = C, and A = 2arctan(phi-1).

Cool. Thanks. I really like these unexpected trigonometric “delights”. Never before seen one involving phi, so thanks!

Thanks, ed. So it is perhaps more accurate to refer to this as a fun fact about √5?

I’m not too sure what would make it “more accurate” to be one than the other in this context, but the proof I came up with does involve getting a factor of √5/2 as an intermediate step. Make of that what you will.

Hi ed. You are right, that “more accurate” is perhaps misguided here, and it’s more a question of psychology than yes or no. My point is that if we say some certain special number X (pi or phi or whatever) occurs in this or that situation, then to an extent we choose to see that number.

For example, in the formula for the volume of a sphere, we usually say pi occurs, rather than 4pi/3 (or 8pi/3). Sometimes the proof/construction suggests choosing our special X over X + 1 or 3X, and sometimes it is suggested by convention.

So, if the number (1 + √3)/2 occurs in some calculation then it may well be more normal and natural to point to √3 occurring. But what if (1 + √5)/2 appears?

That seems often to be the thing with occurrences of phi in mathematics. Yes, phi is genuinely there in some sense, and it’s not wrong to say so. And it may be exactly right to say so, for ratios of fibonacci numbers and so on. But it may be more natural, except for the desire to spot phi, to declare, rather, that √5 is genuinely there.

Thanks, ed (?). I didn’t know that one. A nice one to puzzle over, in the spare time I’ll never get.

Thanks, RF. I definitely wasn’t disputing the coolness of phi as a number. This can also be overstated, since other irrationals can appear in essentially the same way. But sure, phi is a cool number.

Let us never forget the time that Mary Tai rediscovered the trapezium rule and got 75 citations for it.

Jesus, that’s hilarious. How did I not know of that? Will have to look carefully and do a post.

Hilarious in the same way that we sometimes get cursed to live in interesting times …. It is a total disgrace.

There has been a lot of on-line comment, and a blog of sorts on it here:

@edderiofer: I am in your debt for pointing this out. I will add this anecdote to all future calculus classes I teach.

Please do not google “fibonacci retracement”. You do not want to know how many books were written about it, and how many talks on that topic have been given inside honest universities.

Thanks, Franz. I see that Fibonacci support garbage come up a lot in the junky finance media. Does it appear in more academic contexts as well?