Devlin's Angle

May 2007

The Myth That Will Not Go Away

Part of the process of becoming a mathematics writer is, it appears, learning that you cannot refer to the golden ratio without following the first mention by a phrase that goes something like "which the ancient Greeks and others believed to have divine and mystical properties." Almost as compulsive is the urge to add a second factoid along the lines of "Leonardo Da Vinci believed that the human form displays the golden ratio."

There is not a shred of evidence to back up either claim, and every reason to assume they are both false. Yet both claims, along with various others in a similar vein, live on.

The latest math writer to fall victim to this peculiar compulsion is the author of an otherwise excellent article in Science News Online (week of May 5, 2007). The main focus of that article is the appearance of the golden ratio in nature, which is real, substantiated, and of considerable scientific interest. So much so, in fact, that one wonders why the writer felt the need to spice up her story with falsehoods in her third sentence - for the two golden ratio claims I gave above are direct quotes from her lead.

Before I go any further I should say that my purpose is not to attack a fellow math writer. Indeed, let me confess that for many years I too fell victim to the very same compulsion, as a fairly quick search through some of my earlier writings will testify.

My first suspicions that all was not right with some of the claims made about the aesthetic appeal of the golden ratio were aroused when I admitted to myself that I personally did not find the golden rectangle the most pleasing among all rectangles. My doubts grew when tests I performed on several classes of students revealed that few people, when presented with a page of rectangles of various aspect ratios, picked out as the one they found most pleasing the golden rectangle. (Actually, given that the aspect ratio of any actual rectangle you draw can be only an approximation to a theoretical ideal, a more accurate description of my experiment would be that few people picked the rectangle that most closely approximated the theoretical ideal of a golden rectangle.)

Then I read an excellent article by the University of Maine mathematician George Markowsky, titled "Misconceptions about the golden ratio", published in the College Mathematics Journal in January 1992. In his article, Markowsky subjected many of the common claims about the golden ratio to a fairly rigorous review, and found that quite a few of them come up decidedly short. Further evidence against many of the common claims you see made about the golden ratio were provided by writer Mario Livio in his 2002 book The Golden Ratio: The Story of PHI, the World's Most Astonishing Number.

In my June 2004 "Devlin's Angle" and in an article I wrote for the June 2004 issue of Discover magazine, I added my own contribution to Markowsky and Livio's valiant attempts to inject some journalistic ethics into the scene (to wit, checking facts before going into print), but by all appearances the three of us have had little success. Particularly when novelist Dan Brown repeated many of the most ridiculous golden ratio chestnuts in his huge bestseller The Da Vinci Code. (No, I'm not giving a live link to that!)

With so many wonderful things to say about the golden ratio that are true and may be substantiated, why oh why do those myths keep going the rounds? Why do we so want to believe that, say, the ancient Greeks designed the Parthenon based on the golden ratio? (For the record, they did not; which is to say, there is not a shred of evidence that they did any such thing, and good reason to believe they did not.)

Yet the myth has undeniable appeal, even to individuals who claim to be innumerate. Radio producers, for instance. I gave a "debunking the golden ratio myths" talk at a science cafe event in San Francisco recently (Ask a Scientist), and by chance the local NPR radio station KQED sent a team to record the event for their science series Quest. The format for my talk was that I first repeated - as earnestly and breathlessly as I could - the various claims you see made about the golden ratio, followed by a fact-based dissection of each claim. I should have guessed that the clip the radio folk would choose to include in their broadcast was where I repeated the "the ancient Greeks believed . . ." story, but left out my subsequent denouncement thereof. Sigh. In the process of my trying to inject some facts into the picture, while the 120 or so people in my audience got the real story, tens of thousands of radio listeners heard me saying . . . Well, you get the picture. I dare not repeat it yet again!

Undaunted, let me sally forth once again into this mire of misinformation and try to set the record straight.

Here we go again

In the unlikely event that someone reading this article does not know what the golden ratio is, let me give the standard mathematical definition.

If you try to determine how to divide a line segment into two pieces such that the ratio of the whole line to the longer part is equal to the ratio of the longer part to the shorter, you will rapidly find yourself faced with solving the quadratic equation

x2 - x - 1 = 0
The positive root is an irrational number whose decimal expansion begins 1.618. This is the number now called the golden ratio, sometimes denoted by the Greek letter PHI, which I will refer to using the HTML-friendly notation GR

Euclid, in his book Elements, described the above construction and showed how to calculate the ratio. But he made absolutely no claims about visual aesthetics, and in fact gave the answer the decidedly unromantic name "extreme and mean ratio". The term "Divine Proportion," which is often used to refer to GR, first appeared with the publication of the three volume work by that name by the 15th century mathematician Luca Pacioli. (He has a lot to answer for!) Calling GR "golden" is even more recent: 1835, in fact, in a book written by the mathematician Martin Ohm (whose physicist brother discovered Ohm's law).

There is no doubt that the GR has some interesting mathematical properties. It crops up in measurements of the pentagram (five-pointed star), the five Platonic solids, fractal geometry, certain crystal structures, and Penrose tilings.

The oft repeated claim (actually, all claims about GR are oft repeated) that the ratios of successive terms of the Fibonacci sequence tend to GR is also correct.

The GR also has a particularly elegant expansion as a repeated fraction. (This is almost explicit when you rearrange the quadratic equation that defines it.)

Turning from mathematics to the natural world, once you discount the initial throwaway lines about Leonardo Da Vinci and the ancient Greeks that the writer of the Science News article lets slip in, everything she says there about the role played by the golden ratio in plant growth is, to the best of my knowledge, correct - and it's fascinating.

Nature does, it seem, favor the golden ratio. But not exclusively so. In my 2004 debunking article in "Devlin's Angle" I inadvertently let yet another falsehood slip in. I claimed there that you can find the golden ratio in the growth of the Nautilus shell. Not so. The Nautilus does grow its shell in a fashion that follows a logarithmic spiral, i.e., spiral that turns by a constant angle along its entire length, making it everywhere self-similar. But that constant angle is not the golden ratio. Pity, I know, but there it is.

It's when you leave the mathematical world and the natural world, however, that the falsehoods start to come thick and fast.

Numerous tests have failed to show up any one rectangle that most observers prefer, and preferences are easily influenced by other factors. As to the Parthenon, all it takes is more than a cursory glance at all the photos on the Web that purport to show the golden ratio in the structure, to see that they do nothing of the kind. (Look carefully at where and how the superimposed rectangle - usually red or yellow - is drawn and ask yourself: why put it exactly there and why make the lines so thick?)

Another spurious claim is that if you measure the distance from the tip of your head to the floor and divide that by the distance from your belly button to the floor, you get GR. But this nonsense. When you measure the human body, there is a lot of variation. True, the answers are always fairly close to 1.6. But there's nothing special about 1.6. Why not say the answer is 1.603? Besides, there's no reason to divide the human body by the navel. If you spend a half an hour or so taking measurements of various parts of the body and tabulating the results, you will find any number of pairs of figures whose ratio is close to 1.6, or 1.5, or whatever you want.

Then there is the claim that Leonardo Da Vinci believed the golden ratio is the ratio of the height to the width of a "perfect" human face and that he used GR in his Vitruvian Man painting. While there is no concrete evidence against this belief, there is no evidence for it either, so once again the only reason to believe it is that you want to. The same is also true for the common claims that Boticelli used GR to proportion Venus in his famous painting The Birth of Venus and that Georges Seurat based his painting The Parade of a Circus on GR.

Painters who definitely did make use of GR include Paul Serusier, Juan Gris, and Giro Severini, all in the early 19th century, and Salvador Dali in the 20th, but all four seem to have been experimenting with GR for its own sake rather than for some intrinsic aesthetic reason. Also, the Cubists did organize an exhibition called "Section d'Or" in Paris in 1912, but the name was just that; none of the art shown involved the golden ratio.

Then there are the claims that the Egyptian Pyramids and some Egyptian tombs were constructed using the golden ratio. There is no evidence to support these claims. Likewise there is no evidence to support the claim that some stone tablets show the Babylonians knew about the golden ratio, and in fact there is good reason to conclude that it's false.

Turning to more modern architecture, while it is true that the famous French architect Corbusier advocated and used the golden ratio in architecture, the claim that many modern buildings are based on the golden ratio, among them the General Secretariat building at the United Nations headquarters in New York, seems to have no foundation. By way of an aside, a small (and not at all scientific) survey I once carried out myself revealed that all architects I asked knew about the GR, and all believed that other architects used the GR in their work, but none of them had ever used it themselves. Make whatever inference you wish.

Music too is not without its GR fans. Among the many claims are: that some Gregorian chants are based on the golden ratio, that Mozart used the golden ratio in some of his music, and that Bartok used GR in some of his music. All those claims are without any concrete support. Less clear cut is whether Debussy used the Golden Ratio in some of his music. Here the experts don't agree on whether some GR suggestive patterns that can be discerned are intended or spurious.

I could go on, but you get the picture, I hope. I've done my bit once again, and attempted once more to right my own previous wrongs in inadvertently adding fuel to this curious forest-fire of myths.

Just as happened last time, I anticipate receiving some truly ANGRY emails from readers incensed that I should dare question their long cherished beliefs about this particular number. (Maybe that's why we say it is irrational?) And there we scratch the surface of what I think is a fascinating aspect of human nature. People, at least many people, seem to really WANT there to be numbers with mystical properties. So much so that they are prepared to put aside their otherwise wise insistence on evidence or proof. (Many of the emails I got last time demanded that I give proof that the golden ratio is not in the design of the Parthenon, for instance. Which is of course to get the scientific method completely backwards. The hypothesis in that case is that the GR did figure in the Greeks' design, and that is what needs justification - and does not get. Ditto all the other spurious GR claims.) This almost religious attachment to a number, or to numbers in general, has a long history, going back at least as far as the Pythagoreans. I have a theory as to why people have this deep attachment to numbers. But I dare not put it in print for fear that it will assume a life of its own, and once out of the bottle, no one will be able to contain it.

Devlin's Angle is updated at the beginning of each month.
Mathematician Keith Devlin (email: [email protected]) is the Executive Director of the Center for the Study of Language and Information at Stanford University and The Math Guy on NPR's Weekend Edition. Devlin's most recent book, THE MATH INSTINCT: Why You're a Mathematical Genius (along with Lobsters, Birds, Cats, and Dogs) was published in 2005 by Thunder's Mouth Press.