Me, Myself and Math, a six-part series by Steven Strogatz, looks at us through the lens of math.

No other number attracts such a fevered following as the golden ratio. Approximately equal to 1.618 and denoted by the Greek letter phi, it’s been canonized as the “Divine Proportion.” Its devotees will tell you it’s ubiquitous in nature, art and architecture. And there are plastic surgeons and financial mavens who will tell you it’s the secret to pretty faces and handsome returns.

Not bad for the second-most famous irrational number. In your face, pi!

It even made a cameo appearance in “The Da Vinci Code.” While trying to decipher the clues left at the murder scene in the Louvre that opens the novel, the hero, Robert Langdon, “felt himself suddenly reeling back to Harvard, standing in front of his ‘Symbolism in Art’ class, writing his favorite number on the chalkboard. 1.618.”

Langdon tells his class that, among other astonishing things, da Vinci “was the first to show that the human body is literally made of building blocks whose proportional ratios always equal phi.”

“Don’t believe me?” Langdon challenged. “Next time you’re in the shower, take a tape measure.” A couple of football players snickered. “Not just you insecure jocks,” Langdon prompted. “All of you. Guys and girls. Try it. Measure the distance from the tip of your head to the floor. Then divide that by the distance from your belly button to the floor. Guess what number you get.” “Not phi!” one of the jocks blurted out in disbelief. “Yes, phi,” Langdon replied. “One-point-six-one-eight. […] My friends, each of you is a walking tribute to the Divine Proportion.”

I tried it. I’m 6-foot-1, and my belly button is 44 inches from the floor. So my ratio is 73 inches divided by 44 inches, which is about 1.66. That’s about 2.5 percent bigger than 1.618. But then again, nobody ever mistook me for Apollo.



The golden ratio originated in the ideal world of geometry. The Pythagoreans discovered it in their studies of regular pentagons, pentagrams and other geometric figures. A few hundred years later, Euclid gave the first written description of the golden ratio in connection with the problem of dividing a line segment into two unequal parts, such that the whole is to the long part as the long is to the short.

To make this problem more vivid and tangible, let’s think of it as a carpentry job. You’re working for Euclid, a notoriously fussy customer. He hands you two boards, each 60 inches long. Your job is to cut one of the boards into a long piece and a short piece, while leaving the other board whole. Sounds easy, but then Euclid says, “Not so fast, pal. The whole board and the long piece have to be in exactly the same proportion as the long piece and the short piece.”

You have no idea what the cranky old man means by that, so you simply saw one of the boards in half. Euclid screams when you hand him your work. “What’s this? Both these pieces are 30 inches long. That’s a ratio of 30 to 30, or 1 to 1. But the whole board is twice as long as either of them — a ratio of 60 to 30, or 2 to 1. Arrrgh! I told you, the whole to long and the long to short ratios are supposed to match. Try again!” and he hands you two new 60-inch boards.

So this time you cut the board much more unevenly into a 40-inch piece and a 20-inch piece, hoping to bring the ratios closer together. Now the ratio of whole to long is 60 to 40, or 1.5 to 1, whereas the long to short ratio is 40 to 20, or 2 to 1. Not bad, you think: 2 is pretty close to 1.5. But Euclid is not pleased, and he sends you back again.

At this point you have a few options. You could continue by trial and error, experimenting with various cuts between 30 and 40 inches long, progressing in steps of, say, one inch. That strategy would produce a fine result when you cut the board into pieces that are 37 and 23 inches long. The resulting ratios would then be whole/long = 60/37 = 1.62 (rounded to the nearest hundredth) and long/short = 37/23 = 1.61. Still not a perfect match, but much closer than before.

Better yet, you could remember what your high school algebra teacher taught you. If you set up Euclid’s demands as a word problem with x as the length of the long piece and 60 – x as the length of the short piece, then the whole/long ratio is 60/x and the long/short ratio is x/(60 – x). These ratios match if and only if

This equation can be rearranged and then solved for x, with the help of the quadratic formula from algebra class. (See, it is good for something!)

The solution comes out to be

inches, which is about 37.1 inches, rounded to the nearest tenth of an inch. That’s very close to what we found earlier by trial and error, but with the distinct advantage that it’s the only answer that will get Euclid off your back. For this cut, and only this cut, the whole to long ratio and the long to short ratio match exactly. Those ratios are both given by the golden ratio, a number whose value can be proved to equal

But how did Euclid ever come up with that cockamamie demand about ratios of whole to long and long to short? It’s actually a very natural demand to impose — if you’re fascinated by the geometry of pentagons, as Euclid and his predecessors were.

To see what pentagons have to do with dividing a line according to Euclid’s prescription, connect any two of the non-neighboring corners of a regular pentagon.

(Here, “regular” means that the pentagon is as symmetrical as possible. In particular, all its sides have the same length.) Clearly the red line is longer than any of the black sides, but how much longer? It turns out to be exactly 1.618… times longer. The ratio of red to black is golden.

Now if we take this ratio of pentagonal lengths as an alternative definition of the golden ratio, we can prove it amounts to the same thing as Euclid’s enigmatic recipe about how to cut a line into a long part and a short part. (That’s probably how Euclid concocted his recipe in the first place.) To see the equivalence of these two definitions, let’s go back to the pentagon and draw another red line.

Two kinds of isosceles triangles suddenly appear. One is tall and narrow, the other short and squat:

Our next move is irresistible. Connect all the remaining corners with red lines:

The resulting shape is known variously as a regular pentagram, pentacle, or five-sided star. In a coincidence that must have delighted the ancients, it carries a new, baby pentagon inside itself. By connecting its lines, that baby pentagon can give birth to its own baby stars and pentagons, ad infinitum. No wonder these five-sided shapes have been revered as symbols of fertility, rebirth and infinity.

But where does Euclid’s cutting recipe appear in all this? Well, notice that the diagram above, with the five-sided star, also contains an assortment of baby isosceles triangles, with proportions identical to those we found earlier. For example, there’s a baby version of the squat triangle.

It’s “similar” to the original squat triangle, meaning that it has the same angles, the same shape and the same proportions; it’s just smaller. (A proof of this uses the facts that both triangles are isosceles and they have a base angle in common.) Hence the ratios of the lengths of their corresponding sides must be equal — in other words, blue is to black as black is to red.

That equality of ratios sounds a lot like Euclid’s requirement that the short part is to the long part as the long part is to the whole. Sure enough, that’s exactly what this equation says. The red line does play the role of the whole, with the black and blue lines serving as its long and short parts, respectively:

One red line cuts another in a “golden section,” forming two segments — one black, one blue — whose lengths are in the golden ratio.

Given the marvelous patterns in diagrams like these, it’s easy to see why they’ve inspired reverence and wonder.

Unfortunately, in the more than two millenniums since Euclid, the golden ratio has suffered from so much hype, numerology and wishful thinking that it’s become hard to separate the myth from the math. Many of its supposed occurrences in nature, anatomy, art and architecture don’t stand up to careful scrutiny. For example, you can find lots of books and Web sites claiming that the shell of the chambered nautilus obeys the golden ratio, but in reality, nautilus shells have average growth ratios between 1.24 and 1.43, quite far from 1.618.

So be skeptical the next time you see the golden ratio being used to sell blue jeans, stock tips or the perfect smile.

The upside is, if a nautilus can’t get its proportions golden, maybe I shouldn’t worry so much about mine.

Pass the nachos.

NOTES

1. The quoted material about human proportions appears on page 95 of D. Brown, “The Da Vinci Code” (Doubleday, 2003).

2. The two best sources of information about the history, math and science of the golden ratio are Mario Livio’s book “The Golden Ratio” (Broadway Books, 2002) and Ron Knott’s Web site Fibonacci numbers and the golden section. Livio’s book is especially strong on the cultural context of the golden ratio, with many wonderful stories about the fascination it exerted on such luminaries as Pythagoras, Kepler, Dali and Le Corbusier. Knott’s site includes clickable images that let you draw and count spirals on sunflower and coneflower heads, to see how patterns related to the golden ratio manifest themselves in these plants. Although ardent in their admiration for the golden ratio, both Livio and Knott maintain their scientific objectivity. They take pains to separate the bogus sightings of the golden ratio from the valid ones.

3. The golden ratio and its cousin, the Fibonacci sequence, are on magnificent display in a short film by Cristóbal Vila called “Nature by Numbers.” (As of this writing, it has been watched 2.4 million times on YouTube.) In his notes on the math and science behind the movie, Vila is refreshingly honest about the artistic liberties he took; he admits that he had to fudge the shape of the golden spiral to get it to match a real nautilus shell. He also links to articles that show that although nautilus shells do have the shape of logarithmic spirals, just like the golden spiral does, they are not actually based on the golden ratio; they use ratios that average around 1.33, as discussed by Akkana Peck and by Ivars Peterson, using Clement Falbo’s measurements of a nautilus shell collection.

The myth that the nautilus spiral obeys the golden ratio was previously refuted in J. Sharp, “Spirals and the Golden Section,” Nexus Network Journal, Vol. 4, No. 1 (Winter 2002) and R. Fonseca, “Shape and order in organic nature: The nautilus pompilius,” Leonardo, Vol. 26 (1993), pp. 201–204. This video on the “golden ratio nautilus” by George Hart demonstrates visually, rather than statistically, just how far from golden a real nautilus shell is.

4. The clearest examples of the golden ratio in the biological world are found in the spiraling seed patterns and growth patterns of certain flowers and plants. Our current theoretical understanding of why the golden ratio so often appears in this context is nicely summarized in Julie Rehmeyer’s “Mathematical lives of plants: why plants grow in geometrically curious patterns,” Science News, Vol. 172, No. 3 (2007), pp. 42–45. Explore the patterns for yourself at this excellent interactive site at Smith College.

5. For debunking of some of the most oft-repeated claims about the golden ratio, see these posts by Keith Devlin, Julie Rehmeyer (who reports that the golden ratio is being used to sell blue jeans, and who admits to falling prey to some of the more plausible myths herself) and Nick Seewald (written when he was a freshman at Notre Dame). Earlier trenchant analyses are presented in M. Gardner, “The cult of the golden ratio,” Skeptical Inquirer, Vol. 18 (1994), pp. 243–247; G. Markowsky, “Misconceptions about the golden ratio,” College Mathematics Journal Vol. 23, No. 2 (January 1992), pp. 2–19; and C. Falbo, “The golden ratio: A contrary viewpoint,” College Mathematics Journal, Vol. 36, No. 2 (Mar., 2005), pp. 123–134.

6. If you followed the argument about the pentagon closely, you might have noticed that we didn’t quite prove that one red line cuts another into two pieces whose lengths are in the golden ratio. The missing step was that we never established that the segment to the right of the red dashed line in the final figure deserved to be drawn in black — that is, we didn’t prove that it has the same length as a side of the pentagon. For the full proof, see Proposition 8 in Book 13 of “Euclid’s Elements,” translated by T. L. Heath, edited by D. Densmore (Green Lion Press, 2002).

Thanks to Margaret Nelson for preparing the illustrations, and Paul Ginsparg, Tim Novikoff, Andy Ruina and Carole Schiffman for their comments and suggestions.