The Math Behind Beauty

A plastic surgeon computes the perfect face.

By Bruno Maddox|Friday, June 01, 2007

“What is Beauty?” 

Marylin Monroe (1957) by Milton H. Greene
Image courtesy of Marquardt Beauty Analysis

Very little has surprised me more, in my years as a public intellectual, than how often I get collared on the street by some desperate pedestrian demanding an answer to this most fundamental question. Almost never. It hardly ever happens. 

Which is odd, because people still care about Beauty—quite a lot in fact—especially here in Southern California, if I can be the first to make that observation. Last night in my room at the Sunset Marquis I reached out for what I assumed was the room-service menu and passed a few fleeting surreal moments trying to imagine what “Upper Leg with Bikini” might taste like, for a mere $100. It turned out that I had grabbed the Beauty Menu by mistake and that for $240 someone was prepared to come to my room and give my skin a “Firming Renovateur.” 

But while people may care about being beautiful as much as they ever did, it seems they have largely stopped trying to figure out what Beauty actually is.

It wasn’t always thus. The ancient Greeks, for their part, were convinced that an explanation of, and definition for, Beauty was as concrete and discoverable as the answer to why the days got shorter in winter or why your toga weighed more after you’d gone swimming in it. Indeed, no less a thinker than Pythagoras, he of hypotenuse fame, logged some impressive early results. In music, Pythagoras showed that the notes of the musical scale were not arbitrary but reflected the tones produced by a lute string—or any string—when its length was subdivided precisely into such simple ratios as 2:1 or 3:2. In architecture and design, similarly, he managed to show that the shapes people found most pleasing were those whose sides were related by the so-called golden ratio. 

The golden ratio, briefly, is the proportional relationship between two lines a and b such that (a + b) is to a as a is to b; in other words, the ratio between the whole and one of its parts is the same as the ratio between its two parts. This doesn’t sound like much in algebra form (a/b = (a + b)/a) and still less when expressed as a decimal (1:1.61814). But draw a rectangle—or build a Parthenon—with sides of a and b, and the sheer cosmic rightness of the thing leaps out at you. If you were to be stranded on a desert island with one particular rectangle, that’s the one you’d go with. Palpably, it’s the first rectangle that occurred to God when he realized he needed another four-sided, right-angled shape to complement his juvenile masterpiece, the square.

This was good enough for Plato, the 800-pound gorilla of ancient Greek intellectual life, to include Beauty as one of his famous forms: those transcendent, invisible archetypes of which this reality is nothing but a set of blurry ramshackle imitations. Beauty was not in the eye of the beholder. On the contrary, to borrow Plato’s legendary cave metaphor, the beholder had his back to Beauty, able to see only its flickering shadows on the grimy cave wall of reality.

In short, the Science of Beauty was inaugurated by the two classical thinkers upon whose shoulders the science of pretty much everything else would eventually come to rest. Among historians of science, that’s what is known as a rollicking and auspicious start. 

Imagine the surprise, therefore, of one Dr. Stephen Marquardt, a plastic surgeon working in Southern California at the tail end of the 20th century, who checked in on the progress of the Science of Beauty since Pythagoras and found that very little had been made.

As Los Angeles plastic surgeons go, Marquardt (now retired from clinical practice) was the serious, unsleazy sort. His patients weren’t the standard Valley girls and divorcées whose breasts a doctor could breezily augment to the tinkle of a Japanese water feature before checking his teeth in the shine of his scalpel and heading off for cocktails at Skybar. His patients were deformed. They were people who were born without chins or who had taken a speedboat turbine to the face. And they came to him with dreams not of gorgeousness or superstardom but of one day being able to ingest food orally. 

Yet herein lay a paradox. The fact that aesthetic perfection was the last thing on his patients’ minds meant that Marquardt had to think about it all the time, far more than if he’d been just another surgeon slinging collagen up in Beverly Hills. People didn’t come to him wanting a cleft in their chin; they came to him wanting a chin, and they generally left it up to Marquardt to decide what the thing was actually going to look like. 

Which was harder than it sounds. Often Marquardt would walk out of surgery thinking he’d gotten someone’s chin exactly right, only to find weeks later, when the bandages came off, that the thing just didn’t work on an aesthetic level. 

The solution, Marquardt decided, was to ramp up the degree of proportional precision. But he could find nothing useful in the literature. After Pythagoras with his golden ratio and Plato with his forms, the mathematics of Beauty went largely untouched until Leonardo da Vinci. Da Vinci’s Vitruvian Man, that famous sepia sketch of a nude, spread-eagled person touching a square and a circle with his extremities, asserted the eerie proportional coincidences of the ideal human form (arm span = height; height = hand length x 10) but said nothing about the face.

So Marquardt went it alone. He collected photographs of faces the world deemed beautiful and began measuring their dimensions. Whereupon something peculiar and thrilling presented itself: the golden ratio. Beautiful people’s mouths were 1.618 times wider than their noses, it seemed, their noses 1.618 times wider than the tip of their noses. As his data set expanded, Marquardt found indeed that the perfect face was lousy with golden ratios. Even the triangle formed by the nose and the mouth was a perfect acute golden triangle. 

Marquardt went public, making a splash with his unveiling of the Golden Mask, his understandably grandiose name for what was, if he was right, nothing less than a blueprint for the perfect face—and more than enough reason, you would think, for this reporter, passing through Los Angeles, to check in with Marquardt to see where his work has gone from there.

So I did, and I have to say I left Marquardt’s comfortable home in Huntington Beach not entirely convinced. Gunning my rented Ford Escape back to Los Angeles, I couldn’t help but think that the good doctor was overreaching—perhaps quite a lot—with this whole Golden Mask thing. 

society and culture may call us ugly, but that’s only because society hasn’t yet gone to the trouble of comparing our faces to the golden mask.

The iris, in particular, gave me pause. Marquardt contends that the golden ratio can be detected in the iris, the colored part of the eye. Take 10 golden triangles, arrange them with their sharp points touching, and you have a golden decagon, fitting perfectly within the iris of the eye, vertices neatly touching the rim. But surely, so would a square, if you sized it right. Or an equilateral triangle. Or a bull’s-eye. 

Then there was the way the Mask did not quite fit supposedly beautiful faces as well as Marquardt told me it did, while he helpfully talked me through the images on his Web site (beautyanalysis.com). As well as the way it seemed to fit supposedly ugly faces much better than you’d expect. Marquardt conceded this last point and hailed it as proof that the human race has evolved to the point that—hooray!—most of us, in objective terms, are actually rather attractive. Society and culture may call us ugly, but that’s only because society hasn’t yet gone to the trouble of comparing our faces to the Golden Mask, which was derived by studying faces that society deems beautiful . . . which would seem to me to invalidate the whole ball of wax.

It was only later that I changed my mind—a gradual, nay, ineffable process I should probably describe, for the sake of Beauty, as an epiphany at the end of a pier in Santa Monica while watching the sun go down through my Ray-Bans.

So what if Marquardt’s overreaching? I suddenly realized. If he’s right only in his assertion that the most pleasing faces have mouths that relate to the noses above them by the ancient and mysterious golden ratio, that’s not nothing. That’s a lot. And if he’s also right, as he once told The Washington Post, that the width of the front two teeth in a supermodel’s smile is 1.618 times the height of each tooth, then he is actually really onto something.

Maybe Plato was right as well: that nothing in this world is perfect, be it a table, a face, or the life’s work of a California scientist, until you tune out the noise and break through to what is true—and even a whiff of mathematical insight into Beauty gets the job done. For as John Keats once said, frantically overachieving en route to his glamorous early grave, “Beauty is truth, truth beauty—that is all ye know on earth, and all ye need to know.”

Other scholars can debate whether Keats, at 24, dying, working in the anything-goes medium of poetry, actually knew what he was talking about when he wrote those words. But I think perhaps that I, peering through the faux-deep shallows of Southern California to its faux-shallow depths, finally do. 

Discover Magazine - June 2007 Issue