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The Mathematical Formula For The Perfect Face

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> The Mathematical Formula For The Perfect Face
post Aug 15 2007, 10:33 AM
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It is difficult to define perfect beauty as the parameters for a perfect face may vary according to individual preferences.

However, scientists have narrowed down to a simple mathematical ratio of 1:1.618, otherwise known as phi, or divine proportion, to set standards of beauty.

Only one formula has been consistently and repeatedly present in all things beautiful, be it art, architecture or nature, but most importantly in facial beauty.
Ideal facial proportions are universal regardless of race, sex and age, and are based on 'divine proportions'.

According to the formula, if the width of the face from cheek to cheek is 10 inches (25 centimetres), then the length of the face from the top of the head to the bottom of the chin should be 16.18 inches to be in ideal proportion. If you're keen to see how you measure up, keep in mind that the ratio of phi also applies to:

+ The width of the mouth to the width of the cheek.

+ The width of the nose to the width of the cheek.

+ The width of the nose to the width of the mouth.

The origins of the divine proportion

In the Elements, the most influential mathematics textbook ever written, Euclid of Alexandria (ca. 300 BC) defines a proportion derived from a division of a line into what he calls its "extreme and mean ratio." Euclid's definition reads:

A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.
In other words, in the diagram below, point C divides the line in such a way that the ratio of AC to CB is equal to the ratio of AB to AC. Some elementary algebra shows that in this case the ratio of AC to CB is equal to the irrational number 1.618 (precisely half the sum of 1 and the square root of 5).

Attached Image
C divides the line segment AB according to the Golden Ratio

Who could have guessed that this innocent-looking line division would have implications for numerous natural phenomena ranging from the leaf and seed arrangements of plants to the structure of the crystals of some aluminium alloys, and from the arts to the stock market?

Attached Image
Sunflower seeds

In fact, with the increasing realization of the astonishing properties of this number over the centuries since Euclid's definition, the number was given the honorifics "Divine Proportion" and "Golden Ratio."

Here I shall concentrate only on one of the surprising (claimed) attributes of the Golden Ratio - its presumed association with aesthetics, since it provides a wonderful example of an attempt to mingle mathematics with the arts.

Visualizing the Golden Ratio in 1-Dimension

Phi is only a number - and we don't really see numbers when we look at things.

However, there is a visual manifestation of Phi and the Golden Ratio - something that we can actually look at and see. It is this manifestation of that Golden Ratio which has been reported to be present in many things that are seen as beautiful.

This is a line which has been divided into two segments, the larger of which has a (magnitude) ratio to the smaller of 1.618:1

Attached Image

Where a=1.618 and b=1

A line which has been segmented into two parts having this 1.618:1 ratio is called:

A golden cut line
A golden sectioned line
A golden divided line
A Phi cut line
A Phi sectioned line
A Fibonacci sectioned line

This division of a line in such a manner is referred to variously as:

The golden cut
The golden section
The golden division
The Phi cut
The Phi section
The Fibonacci section

Creating the Golden Sectioned Line:

Create a line from a point
We create a line of any length.

Section that line
There are an infinite number or places that we can divide that line into two segments, and we can section (or cut) that line at any point we desire.

"The Golden Section"
However there is one place (and only one place - a unique place) where that line can be divided or "sectioned" so that the ratio of the smaller segment of the sectioned line to the larger segment of the sectioned line is 1:1.618

This ratio of 1:1.618 is called "The Golden Ratio".

The interesting and remarkable thing about this sectioning of the line into the golden ratio is that not only is the ratio of the smaller segment or the line to the larger segment of the line equal to 1:1.618


the ratio of the larger segment of the line to the whole line is also equal to 1:1.618.

And this particular division, or sectioning, of a line into segments with a ratio of 1:1.618 is called:
"The Golden Section"

The "Golden Sectioned Line"
And the line which is cut into the Golden Section is called:
"The Golden Sectioned Line"

The "Golden Section Point"
This place or point on the line where this golden sectioning occurs is called:
"The Golden Section Point"

The Repeating Phi Ratio & Phi Ratio Duplication (Growth):

Intriguingly, if we duplicate that golden sectioned line to form a new longer line, consisting of the smaller and larger segments of the original line plus a duplicate of the original line, then the ratio of the larger segment of the original line to the duplicate of the original line is also = 1:1.618.
If we delete the smaller segment of the original line, we are left with a new line consisting of the larger segment of the original line and the duplicate of the original line.

If we duplicate this new line, then the ratio of the larger segment of this new line to the whole new duplicate is also = 1:1.618.

This self-duplication can continue on to infinity (i.e. forever) with each line segment in a ratio of 1:1.618 with its adjacent and succeeding line segment along the formed line.

This self-duplication never occurs at any other division, section or ratio of any line or line segments. This continuous self-duplication only occurs with the golden section.

Because of its unusual properties, the number "1.618" has been given its own name - that name is "Phi".

What is the Golden Ratio, and why is it important?

The Golden Ratio is approximately 1 : 1.618

Besides for possessing some remarkable and unique characteristics, the Golden Mean is found in ALL living creatures on Earth. Along with the Fibonacci Sequence (which is a whole-number system approximating the Golden Ratio, discovered by Leonardo Pisano Fibonacci), this ratio is found in plants and animal life wherever one looks. For example, this ratio can be found in fingers one's hand, amongst many other places, and it is prevalent in the skeletal structure of all creatures.

The Fibonacci Sequence is as follows: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, ...

The sequence is calculated as follows.

1 + 1 = 2

1 + 2 = 3

2 + 3 = 5

3 + 5 = 8

5 + 8 = 13

The Fibonacci sequence is an approximation of the Golden Ratio, and as one will see, the higher one goes in the Fibonacci Sequence the closer ones gets to the Golden Ratio.

8 / 5 = 1.6

13 / 8 = 1.625

21 / 13 = 1.615

233 / 144 = 1.618

As you can see, each consecutive number in the sequence is derived from the sum of the previous two. The Fibonacci series is a Fractal sequence (a fractal is a mathematically defined system that, when represented graphically, usually forms self-replicating or recurring patterns).

The Fibonacci sequence is the formula that plants use when deciding how many branches to grow next, but this series can be found everywhere in nature.

Attached Image

Geometry in Nature and the natural world:

Our reality is very structured, and indeed Life is even more structured. This is reflected though Nature in form of geometry. Geometry is the very basis of our reality, and hence we live in a coherent world governed by unseen laws. These are always manifested in the natural world. The Golden Mean governs the proportion of our world and it can be found even in the most seemingly proportion-less living forms.

Clear examples of geometry (and Golden Mean geometry) in Nature and matter:

All types of crystals, natural and cultured.

The hexagonal geometry of snowflakes.

Creatures exhibiting logarithmic spiral patterns: e.g. snails and various shell fish.

Birds and flying insects, exhibiting clear Golden Mean proportions in bodies & wings.

The way in which lightning forms branches.

The way in which rivers branch.

The geometric molecular and atomic patterns that all solid metals exhibit.

Another, less obvious, example of this special ratio can be found in Deoxyribonucleic Acid (DNA) - the foundation and guiding mechanism of all living organisms:

Attached Image
The geometry of DNA

The golden ratio in the arts

Many books claim that if you draw a rectangle around the face of Leonardo da Vinci's Mona Lisa, the ratio of the height to width of that rectangle is equal to the Golden Ratio. No documentation exists to indicate that Leonardo consciously used the Golden Ratio in the Mona Lisa's composition, nor to where precisely the rectangle should be drawn. Nevertheless, one has to acknowledge the fact that Leonardo was a close personal friend of Luca Pacioli, who published a three-volume treatise on the Golden Ratio in 1509 (entitled Divina Proportione).

Attached Image

Another painter, about whom there is very little doubt that he actually did deliberately include the Golden Ratio in his art, is the surrealist Salvador Dali. The ratio of the dimensions of Dali's painting Sacrament of the Last Supper is equal to the Golden Ratio. Dali also incorporated in the painting a huge dodecahedron (a twelve-faced Platonic solid in which each side is a pentagon) engulfing the supper table. The dodecahedron, which according to Plato is the solid "which the god used for embroidering the constellations on the whole heaven," is intimately related to the Golden Ratio - both the surface area and the volume of a dodecahedron of unit edge length are simple functions of the Golden Ratio.

These two examples are only the tip of the iceberg in terms of the appearances of the Golden Ratio in the arts. The famous Swiss-French architect and painter Le Corbusier, for example, designed an entire proportional system called the "Modulor," that was based on the Golden Ratio. The Modulor was supposed to provide a standardized system that would automatically confer harmonious proportions to everything, from door handles to high-rise buildings. But why would all of these artists (there are many more than mentioned above) even consider incorporating the Golden Ratio in their works? The attempts to answer this question have led to a long series of psychological experiments, designed to investigate a potential relationship between the human perception of "beauty" and mathematics.

The golden ratio and aesthetics
by Mario Livio

Is beauty in the eye of the beholder?

The pioneering (rather crude) experiments in this field were conducted by the German physicist and psychologist Gustav Theodor Fechner in the 1860s. Fechner's experiment was simple: ten rectangles varying in their length-to-width ratios were placed in front of a subject, who was asked to select the most pleasing one. The results showed that 76% of all choices centered on the three rectangles having ratios of 1.75, 1.62, and 1.50, with a peak at the "Golden Rectangle" (with ratio 1.62). Fechner went further and measured the dimensions of thousands of rectangular-shaped objects (windows, picture frames in the museums, books in the library), and claimed (in his book Vorschule der Aesthetik) to have found the average ratio to be close to the Golden Ratio.

Many psychologists have repeated similar experiments since then, and obtained rather conflicting results. Michael Godkewitsch of the University of Toronto, for example, pointed out that average group preferences often do not reflect the most preferred choice. For example, the brand of tea that everybody ranks second-best may on the average be rated best, but nobody will ever buy it. Godkewitsch therefore noted that first choices provide a more meaningful measure of preference than averages. Godkewitsch concluded from a study conducted in 1974 that the preference for the Golden Rectangle reported in the earlier experiments was an artifact of the rectangle's position in the range of rectangles presented to the subjects. He noted: "The basic question whether there is or is not, in the Western world, a reliable verbally expressed aesthetic preference for a particular ratio between length and width of rectangular shapes can probably be answered negatively."

Other experiments, however, gave different results. In particular, British psychologist Chris McManus concluded in 1980 that: "There is moderately good evidence for the phenomenon which Fechner championed." Nevertheless, McManus acknowledged that "whether the Golden Section [another name for the Golden Ratio] per se is important, as opposed to similar ratios (e.g. 1.5, 1.6 or even 1.75), is very unclear."

The entire topic received a new twist with a flurry of psychological attempts to determine the origin of facial attractiveness. For example, psychologist Judith Langlois of the University of Texas at Austin and her collaborators tested the idea that a facial configuration that is close to the population average is fundamental to attractiveness. Langlois digitized the faces of male and female students and mathematically averaged them, creating two-, four-, eight-, sixteen-, and thirty-two-face composites. College students were then asked to rate the individual and composite faces for attractiveness. Langlois found that the 16- and 32-averaged faces were rated significantly higher than individual faces. Langlois explained her findings as being broadly based on natural selection (physical characteristics close to the mean having been selected during the course of evolution), and on "prototype theory" (prototypes being preferred over non-prototypes).

Science writer Eric Haseltine claimed (in an article in Discover magazine in September 2002) to have found that the distance from the chin to the eyebrows in Langlois's 32-composite faces divides the face in a Golden Ratio. A similar claim was made in 1994 by orthodontist Mark Lowey, then at University College Hospital in London. Lowey made detailed measurements of fashion models' faces. He asserted that the reason we classify certain people as beautiful is because they come closer to Golden Ratio proportions in the face than the rest of the population.

Many disagree with both Langlois's and Lowey's conclusions. Psychologist David Perret of the University of St. Andrews, for example, published in 1994 the results of a study that showed that individual attractive faces were preferred to the composites. Furthermore, when computers were used to exaggerate the shape differences away from the average, those too were preferred. Perret claimed to have found that his beautiful faces did have something in common: higher cheek bones, a thinner jaw, and larger eyes relative to the size of the face.

An even larger departure from the "averageness" hypothesis was found in a study by Alfred Linney from the Maxillo Facial Unit at University College Hospital. Using lasers to make precise measurements of the faces of top models, Linney and his colleagues found that the facial features of the models were just as varied as those in the rest of the population.

I will certainly not attempt to make the ultimate sense of sex appeal in an article on the Golden Ratio. I would like to point out, however, that the human face provides us with hundreds of lengths to choose from. If you have the patience to juggle and manipulate the numbers in various ways, you are bound to come up with some ratios that are equal to the Golden Ratio.

Furthermore, I should note that the literature is bursting with false claims and misconceptions about the appearance of the Golden Ratio in the arts (e.g. in the works of Giotto, Seurat, Mondrian). The history of art has nevertheless shown that artists who have produced works of truly lasting value are precisely those who have departed from any formal canon for aesthetics. In spite of the Golden Ratio's truly amazing mathematical properties, and its propensity to pop up where least expected in natural phenomena, I believe that we should abandon its application as some sort of universal standard for "beauty," either in the human face or in the arts. wacko.gif

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post Sep 11 2007, 06:20 AM
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post Sep 13 2007, 07:31 PM
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If you found that hilarious, I am sure you wud enjoy this one too.

Beauty Is A Product Of Photoshop


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