Mathematics is objective: two plus two equals four, regardless of your gender, race, orientation or aesthetic tastes.

Beauty is subjective: something (or someone) that may appear beautiful to one viewer may be considered a monstrosity by another.

So how can these two seemingly unrelated concepts tie together? Can we actually determine what a “beautiful” face should look like through mathematics versus subjectivity?

## The Golden Ratio

Some experts believe that innate conceptions of beauty relate to what is known as the “Golden Ratio”. The Golden Ratio (also called the “Divine Proportion”) symbolizes what occurs when the ratio of the longer side of a rectangle (L) to its smaller side (S) equals the ratio between the sum of both sides (L+S) and the larger side (L).

In mathematical terms, the Golden Ratio occurs when L/S = (L+S)/L. In nearly every instance, this ratio equals approximately 1.618.

## Ideas of Beauty: in Ancient Times

One of the more remarkable aspects of art from ancient cultures is that much of it reflects ideas of beauty from those times that still carry over today. Although ideas of what constituted “beauty” in those times are much different from today, the proportions of the eyes, nose, lips and chin found in Greek and Roman statues are still considered beautiful thousands of years later. Examples such as the Venus de Milo and of Athena at the Parthenon show how the Golden Ratio have affected ideas of beauty, even before the number had such a name.

## Ideas of Beauty: Renaissance

The presence of the Golden Ratio in art became much more evident during the Renaissance. Works such as Leonardo da Vinci’s “La Giaconda” (better known as “the Mona Lisa”) show how the Golden Ratio was used to determine where to place the facial features and how to capture the viewer’s eye. More than five hundred years after its creation, the picture of the unknown woman still draws thousands of visitors from around the world and the appreciation of millions in prints, replicas and parodies.

## Ideas of Beauty: Modern Times

Today, our ideas of beauty come not from statues or paintings, but from movie and TV screens. Researchers have applied the Golden Ratio to measure the facial contours of many of Hollywood’s most beautiful women, including Angelina Jolie, Jessica Alba and Halle Berry. In 2003, model Saira Mohan was featured in a Newsweek cover article titled, “The Perfect Face”. Her mixture of Irish, French and Indian ancestries has blended her facial features to such an extent that she retains the best proportions of all those ethnicities.

## Golden Ratio in Human Anatomy: Arms and Hands

In his famous drawing, Vitruvian Man, da Vinci showed that the ratio between a man’s height and the distance from his shoulder to the opposite middle fingertip is close to the Golden Ratio. Also, the typical distance from the elbow to the wrist is approximately 1.6 times that of the distance from the wrist to the middle fingertip. The Golden Ratio can also represent the ratio of the distances between the successive joints of each finger.

## Golden Ratio in Human Anatomy: Torso, Chest and Waist

For many decades, both sexes have been taught that the perfect “hourglass” female figure had measurements of “36-24-36”. These proportions represented the ideal of a large bust and wide hips with a small waist. In order to apply the Golden Ratio to these “figures”, a woman with a 36-inch bust and hips would need a waist of only 22.25 inches (36/1.618). A woman with a 24-inch waist would need bust and hip measurements of nearly 39 inches (24 x 1.618).

## Golden Ratio in Human Anatomy: Facial Structure

Beauty analysts and Cosmetic Surgeons have also developed a number of measurements that can, when compared to the Golden Ratio, determine the inherent attractiveness of the human face. In a conventionally attractive face, the ratios of specific facial measurements will be close to the Golden Ratio of 1.618.

The further the ratios of facial measurements are from the Golden Ratio, the less perceived attractiveness of the face. For instance, if the distance between the outside corners of the eyes is more than 1.6 times that of the distance from the hairline to the pupil, the eyes are too far apart and out of proportion to the rest of the face.

### How to Measure: By Hand

For those interested in taking the key measurements by hand, use a tape measure to record the following distances:

• From the top of the head to the chin
• From the top of the head to the pupil
• From the pupil to the nasal tip
• From the pupil to the lip line
• From the hairline to the pupil
• From the nasal tip to the chin
• From the nasal tip to the lip line
• From the lip line to the chin
• Between the outside corners of the eyes
• Width of the nose
• Length of the lip line

In an attractive face, the following ratios will be at or near 1.618:

• (From the top of the head to the chin)/(Width of the head)
• (From the top of the head to the pupil)/(From the pupil to the lip line)
• (From the nasal tip to the chin)/(From the lip line to the chin)
• (From the nasal tip to the chin)/(From the pupil to the nasal tip)
• (Width of the nose)/(From the nasal tip to the lip line)
• (Between the outside corners of the eyes)/(From the hairline to the pupil)
• (Length of the lip line)/(Width of the nose)

## How to Measure: By Photo

For those who do not have a convenient tape measure, or for those who want to save time and labor on measurements by hand, a reliable photo editing software package can be used to measure each element in pixels, rather than inches or centimeters. The computer’s measurements are often much more accurate than those compiled by hand. Users also place a proportional Golden Ratio rectangle over the features to be measure and determine who close each set of measurements come closest to the Divine Proportion.

## How to Measure: The Marquardt Mask

Dr. Stephen Marquardt developed a “mask” that assembles many of the previously mentioned ratios, plus a few more, into an outline of the “ideal” human face. The lines in the mask can be adjusted for sex, age, and ethnic background. The principle behind the mask is that the ratios it represents can quantify the facial structure that everyone, regardless of age, race or gender, can find appealing.

## Ethnic Features: African-American

Each ethnic group typically exhibits some variation from the “perfect” lines found in the Marquardt Mask. The most noticeable variations in African-Americans features from the mask’s frontal outline are the fuller lips and wider nostrils. Also, the brow lines are higher and the face is narrower in an attractive African American than they are on the lines featured on the mask. In the profile views, the nasal bridge and tip are slightly receded from the mask’s lines.

## Ethnic Features: Asian

The facial features of those patients with East Asian ancestry also show some significant deviations from the Marquardt Mask. The most noticeable of these differences are the medial and lateral epicanthic folds around the eyes. As with the African-American features, Asian features show wider nostrils and slightly higher brow lines. Where the African-American features display a slightly narrower face, the East Asian features have a noticeably wider face.

## Ethnic Features: Caucasian

Although the mask most closely resembles the features found in attractive Caucasians, some slight differences are still apparent. The most noticeable differences are the wider and longer nose found in many European ethnicities. Many Caucasian features are also much more flat and narrow than those found on the mask, including flat eyebrows, narrow eyes and thin lips. The typical Caucasian face also exhibits a longer, less protrusive chin than displayed on the mask.

## Research and Technology: Golden Ratio Study

A study at the University of Toronto used color photos of women’s faces and altered the proportions of the features, but not the features themselves, with photo editing software. In most instances, the researchers altered the photos so that the distance between the woman’s eye line and her mouth was 36 to 40 percent of the distance from the hairline to the chin. These numbers are related to the Golden Ratio of 1.618 by this equation:

[1 – (1.618 – 1)] = [1 – 0.618] = 0.382

The researchers then asked volunteers to judge between the altered and unaltered photos as to which one was more attractive. Most of the respondents found that the altered photos were more attractive.

## Research and Technology: Phi Matrix

A photo analysis software can determine how well a user’s features fit within the Golden Ratio. The software, known as Phi Matrix, measures the proportions of the eyes, nose, lips, chin, and other facial features to determine how closely they fit within the Golden Ratio proportions. With a few clicks of the mouse, the software forms a grid around the major facial features, measures the distances between those features and calculates the ratios.

What does math have to do with beauty? Based on extensive research, Dr. Slupchynskyj believes the two have more in common than previously thought. Although the saying “beauty doth lie in the eye of the beholder” still holds true, the eye also appreciates symmetry, proportion and structure in every aspect of life, including the human face.