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

Babies can spot pretty face right from birth

By Mark Henderson, Science Correspondent
Times Online, Sept 6, 2004


BEAUTY is not in the eye of the beholder: British scientists have discovered that human infants are born with an innate concept of what makes an attractive face.

The findings suggest that ideals of facial beauty are not determined by culture alone, but also rely on universal standards that have been hard-wired, or imprinted, in our genes.

“Attractiveness is not simply in the eye of the beholder, it is in the brains of newborn infants, right from the moment of birth,” said Alan Slater, a developmental psychologist who led the study. “This view contradicts views arguing that the newborn infant enters the world as a tabula rasa — a blank slate on which experience will write.”

Dr Slater will present the results this week at the British Assocation Festival of Science at Exeter University.

Though infants cannot tell scientists that they prefer this face or that, a large volume of research has shown that their attention is much more easily captured by images they find pleasing or interesting.

In the new study, Dr Slater’s team used this effect to test whether newborns with little or no experience of the world shared their elders’ assumptions about facial beauty. The researchers took hundreds of pictures of female members of the public, and asked adult volunteers to rate their attractiveness on a scale of one to five.

Dr Slater then paired particularly beautiful faces, with an average score of close to five, with particularly unattractive ones scoring close to one. Care was taken to match qualities such as hair colour and length that might otherwise interfere with the experiment. Almost 100 newborn babies, with an average age of two days, were then shown these paired images.

About 80 per cent of the time the babies looked exclusively or mainly at the face judged “prettier”. The effect was also seen when the experiment was repeated with male faces and faces from many different ethnic groups.

Dr Slater said: “A lot of it is hard-wired, and you can’t get away from that hard-wiring.”

There are two probable evolutionary explanations for the phenomenon. First, facial symmetry — which is strongly linked to beauty across cultures — may be an “honest” signal of good genes, good health and an absence of parasitic diseases.

We may be primed to recognise this as a way of selecting sexual partners with the best breeding prospects. Second, the preference could be a by-product of an evolved mental capacity to recognise faces from birth. It is thought that babies fit faces to some sort of average facial template.

Beauty Is in the Brain of the Beholder

Ideals of facial beauty are not determined by culture alone, but also rely on universal standards that have been hard-wired, or imprinted, in our brains. Even babies can spot pretty face right from birth.

Live Science, 6 July 2011 - Whether you're admiring a painting or enjoying a song, all the works of art you favor lead to activity in the same region of the brain, a new study shows. The findings go some way to support the view that beauty is in the perception of the beholder rather than in the object.

Researchers asked 21 volunteers from different cultures and ethnic backgrounds to rate a series of paintings or pieces of music as beautiful, indifferent or ugly. Participants then looked at the pictures or listened to the music while lying in a functional magnetic resonance imaging (fMRI) scanner, which measures brain activity.


That the same area of the brain was activated for both visual and auditory beauty implies that beauty exists as an abstract concept within the brain, researchers from the University College London's Wellcome Laboratory of Neurobiology say.

The finding showed that the medial orbitofrontal cortex, which is part of the brain's pleasure and reward center, was more active in participants when they listened to music or viewed a picture that they had rated as beautiful. However, no particular region of the brain correlated generally with artwork rated as ugly.

Activity in another region, the caudate nucleus, located near the center of the brain, increased in proportion to the relative visual beauty of a painting. The region has been previously reported to correlate with romantic love, suggesting a neural association for the relationship between beauty and love.


The findings were published today (July 6) in the journal PLoS One.

Remi Melina, Live Science Staff Writer 

Why Do Men Find Women with Larger Eyes Attractive?

Zidbits, Science, June 2, 2011 - According to studies done, research has shown that men generally find larger eyes on women more attractive. This has lead to a growing trend (especially in Asia) for many women to use cosmetics and contact lenses with larger limbal rings to give the illusion of larger eyes. Some have even done this permanently through cosmetic surgery.

But why do men find larger eyes attractive?

There seems to be two separate reasons why larger eyes on a female are more attractive. The first is that larger eyes (along with fuller lips, bigger breasts and smaller chins) is a sign of higher levels of estrogen. Women that have higher levels of estrogen will be more successful and have an easier time conceiving than women with lower levels. This means that women with larger eyes will be seen as a better ‘mate’.

The second reason is that large eyes are a neotenous feature.

A neotenous feature is a characteristic of youth shared by babies and children that humans evolved to retain so they could attract quality mates with “protective and nurturing impulses” which would be more likely to help raise their offspring and raise reproductive success. Since human eyes do not grow in size during development as much as the rest of the body, the size of the eye relative to the face decreases as we grow. This means that babies have relatively large eyes compared to older children and adults. As a consequence, people who have large eyes are often perceived to be younger than they really are. Since perceived youth of a mate is also a sign of fertility, men tend to prefer women with neotenous features, like large eyes.

However, this isn’t saying that men want to mate with babies and small children – that would be incredibly maladaptive because they are not fecund. So, then, why do males usually prefer women who, in essence, look like babies?

It’s thanks to something that happens in evolution called Fisherian runaway selection. Fisherian selection suggests that, when one sex prefers mates with certain genetic traits, then, through the process of sexual selection, the other sex will come to possess the trait in increasingly exaggerated forms. 

The process is called “runaway” because over time, it would cause the development of greater preference and more pronounced traits, until the costs of producing the trait balance the reproductive benefit of possessing it. The peacock’s tail is an example of this. Its tail requires a great deal of energy to grow and maintain, it reduces the bird’s agility, and increases the animal’s visibility to predators. Yet it has evolved which indicates that peacocks with longer tails have some advantage.

This same process occurs with men’s preference for women with neotenous features.

GUIDELINE ON AESTHETIC MEDICAL PRACTICE II

Prerequisites of the Practitioner practising Aesthetic Medicine

The registered medical practitioner practising aesthetic medicine:

1. May be a general (family) practitioner or a specialist in any recognized field;
2. Must possess experience through recognized practical training courses conducted by bona fide professional bodies specialising in aesthetic medicine;
3. Shall provide documentary evidence that he has undergone such training and practical/written examination in a bona fide professional body. The Ministry of Health reserves the right to examine documents so produced and either permit or reject the application by a registered medical practitioner to be registered in the Aesthetic Medicine Practitioners Register.
4. Must exercise strict patient selection criteria, must communicate to the potential client/patient the risks involved, the possible outcome, obtain valid consent for the aesthetic procedure planned, and generally observe all aspects of the Code of Professional Conduct of the Malaysian Medical Council;
5. Must place client/patient safety as the primary concern and should provide aesthetic medicine services in an approved healthcare facility as required by the Private Healthcare Facilities and Services Act 1998 and Regulations 2006.
6. Is subject to all relevant parts of the Medical Act 1971, the Regulations 1974, the Code of Professional Conduct and Guidelines of the Malaysian Medical Council.

GUIDELINE ON AESTHETIC MEDICAL PRACTICE - Malaysian Medical Council

History and Background

Aesthetic medicine was introduced as a scientific movement by JJ Lengrand, a
Parisian Endocrinologist in France in 1973. It originates from the belief that the
well-being of an individual is about being satisfied with one’s health, job and the
environment he or she lives in. The goal is the creation of a harmonious physical
and psychological balance as desired by clients, by administering medical
treatment which is both preventive and corrective.

Although aesthetic medicine has not been recognized as a medical specialty in
Malaysia, the practice of aesthetic medicine is currently inevitably on the rise and
is gaining popular demand nationwide.

Definition

For purposes of this Guideline, Aesthetic Medicine is defined as an area of
multidisciplinary medical practice carried out by registered medical practitioners to provide treatment which is evidence-based, with the objective of addressing the aesthetic desires of clients.

Aesthetic Medicine is not a spin-off of traditional or complementary medicine, and
is recognised to be scientific in its approach and practice. Such practice may be
through non-invasive and minimally invasive modalities.

Double Suture and Twist (DST) - The Durable Non-Cutting Double Eyelid Procedure for Asians

Double-eyelid procedure is the most popular cosmetic procedure in Asia. Whereas most surgeons in the United States perform cutting-type double eyelid surgery, Japanese women prefer non-incisional or suture methods.

Publications in the early Japanese medical literature favoured the suture method. The first description of this method, by Mikamo was published in 1896. Between 1896 and 1950, 11 articles relating to the suture methods were published in the Japanese medical literature.

The configuration of a double eyelid consists of four important parameters, namely, height, shape, continuity and permanence. As the suture method has been criticized for its high breakage rate and poorly define crease which fade over time, permanency is the major challenge the suture method is facing along its evolution.

In 2001, an improved non-cutting method to create double eyelids in Asian patients is reported. Dr Katsuhiro Kure, and Dr Akihiro Minami described an improved procedure that uses double sutures and twists (DST) to create double eyelids in Asian patients. Dr. Kure says the non-incisional procedure is “simple, durable, and virtually scar free.” The method uses only two sutures on each upper eyelid, with the durability explained by the areas of ties and the central twists that result in more secure and therefore longer-lasting results.

The average operative time ranged from 20 minutes (one side) to 60 minutes (bilateral). Most patients did not require sedation, and the surgeries were performed with only local anesthesia. Recovery time was short (3-7 days), and complications were minor and rare.

The published report for breakage or loss of fold is approximately one percent per year (ten year follow up). An extraordinarily low failure rate for this method has therefore been scientifically established, comparable to an incision method. There is a notion that the DST technique frequently fails or disappears over time. In many anecdotal comments, the most common source of misinformation has been in confusing the DST technique with traditional suture techniques that are unreliable. It is therefore important to verify that the technique used is DST before you decide to undergo such procedure.

The credential of the doctor is of course first and foremost. The registration of a doctor can be verified at the Malaysian Medical Council website.

This method is suitable for younger candidates with eyelids which do not have lots of loose skin or fat, or any obvious eyelid deformities.

Some examples of double eyelids created using DST:

The types of eyelids which are not suitable for this non-cutting technique:

Obvious ptosis (droopy eyelids)

Obvious asymmetrical eyelids

 Dermatochalasis (Eyelid with lots of sagging skin)

Thick eyelids with lots of fat

These cases require incisional blepharoplasty (Cutting double eyelid surgery), a procedure which requires a highly meticulous and precise work of a well-trained and talented oculoplastic surgeon as it involves highly accurate measurements, skin removal, cutting into deeper structures and the ability and experience to manage possible complications promptly. Such invasive procedure if carried out by unscrupulous cosmetic practitioner is at risk of suffering a permanent deformity due to excessive fat removal, ptosis (droopy eyelid) due to damage to the muscle and tendon which opens the upper eyelid (levator muscle and aponeurosis), lagophthalmos (lid retraction or unable to close the eyelid fully) due to excessive skin removal and worst of all, blindness due to retrobulbar haematoma (accumulation of blood behind the eyeball) resulting in compression and damage to the optic nerve.

A thorough assessment is therefore crucial before deciding on which kind of double eyelid procedure is suitable for an individual. Appropriate referral to a talented and experienced oculoplastic surgeon has to be made where DST is not suitable to ensure safety and best outcome for the individual.