The Mathematics Of Beauty And The Golden Ratio

Beauty In The “PHI” of the Beholder

Is beauty just in the eye of the beholder or is there a method behind it? Actors, actresses and models exhibit features that many people like. When someone mentions that a dog looks adorable or a baby looks cute, these are taken as compliments. Natural beauty it seems, is not a random thing. Observed in nature, it seems to show a repeated pattern that has symmetry but not quite a simple explanation. In humans and animals, it appears to be a genetic code that is passed on from generations through DNA. It appears that there is something at work here that cannot be created at will, but only by nature.

Observing beauty in objects and people, early mathematicians detected what seemed to be a pattern. This is what is called the “Golden Ratio” symbolized by the Greek letter Φ which has a constant value of Phi=1.618. Leonardo da Vinci used the ratio to define a symmetry in structures, including the human body in which it is aesthetically pleasing to the human eye. In Da Vinci’s study it is the proportion of the measurements of a person’s body parts in relation to another body part. So let’s say you compare the measurements of the two body parts, like the nose denoted by a to the lips denoted by b. Then the ratio is computed as:(a + b)/a = a/b
= 1.6180339887498948420 ...
= Φ (phi)

To further explain this relation, the number (ratio) is found by dividing the object into 2 parts. It must be divided so that the longer part divided by the smaller part is also equal to the whole length divided by the longer part.

Fibonacci Numbers

What interests me here is something called the Fibonacci number sequences, a very fundamental lesson in mathematics. It is related to the golden ratio in a way that the ratio of any two Fibonacci numbers is close. The first two numbers in the Fibonacci sequence are either 1 and 1, or 0 and 1, depending on the chosen starting point of the sequence, and each subsequent number is the sum of the previous two. The sequence is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 …. where F is:


As the numbers in the sequence get higher or larger, they get even closer to 1.618 in terms of decimal precision. A lot of the ratio results in 1.6xxxxx, so there is a close relationship here. Using the formula b/a here are some examples.Let's start with the following Fibonacci numbers a=5 and b=8.Take the ratio: 8/5 = 1.6Next, take the Fibonacci numbers a=21 and b=34.Take the ratio: 34/21 = 1.61904762Now take these two Fibonacci numbers a=144 and b=233. Take the ratio: 233/144 = 1.618055556

Another way to look at Fibonacci numbers is through visualization. A fine example of that is what is called the Fibonacci Spiral.

This illustration shows the sequence of Fibonacci numbers as it spirals from bottom of the box. It follows the numbers 1,2,3,5,8,13,21,34 …. The formula (a + b)/a = a/b can be used for this particular spiral example.

Photography and Retouching

It seems to follow a pattern in the most beautiful and attractive people. There is a template called the Marquardt Beauty Mask, which was created by Dr. Stephen Marquardt. This is just a template that describes the idealized form of the human face. In no way is this a standard for beauty, so don’t get me wrong on that explanation. It is a geometric mask that is based on proportions that are calculated using the golden ratio. The template can be superimposed on top of an existing photograph of a face to show how well it fits those dimensions.

I have tried to fit the mask on some photos I have taken to see if there is a perfect match. Some of them do not appear to come close. In others, it appears to be a close fit. This reinforces the notion that beauty is computational, based on instructions coded in a person’s DNA. However, let us not accept this as the scientific evidence just yet. The construct of beauty in human beings is rather complicated because of the diversity in culture and the human gene pool. People come in different shapes, sizes and forms. Therefore, beauty itself cannot be just based on proportions but also on the perspective of people.

In the photograph’s face, you can come to close proximity to the values of the golden ratio. In my example the model had a lip measure of 2"(b) and nose width of 1.23"(a). The ratio is 1.6260162601…. The mask fit is not perfect but the concept states the closer the fit the closer to the ideal proportions of the golden ratio.

When applying concepts of the golden ratio to photography, there is actually one common technique that many are quite familiar with. It is called the Rule of Thirds. This basically places an image in such a way as to not be too centered on the frame. Instead you are composing your subject as if the image were superimposed with the Fibonacci spiral. A photo shot using the Rule of Thirds shows the symmetry that seems to fit with the Fibonacci spiral. This is more open to interpretation because not all photographers compose an image the same way.

We can use the golden ratio to divide the frame into a longer and shorter part. Then apply the Fibonacci spiral principle. This is of course not visible when composing the image, so it will depend on the photographers framing of the subject when creating the image.

Applying the Rule of Thirds in photography. Rather than centering the wine bottle and glass in the frame, visualize it in a way where there is symmetry with the subject to the frame. Either place the wine bottle at the small square where the spiral begins or place it within the longer length of the frame. This tends to be where the eyes glance when people first glimpse the image. It is a point of interest in your image.

Photo editing software like Photoshop and the ones that come with smartphone cameras allow features to make photographs appear more stunning. In portraits, it can alter the shape of the face and enhance it as well to make the person look better. The software can automatically use a setting that can apply the golden ratio, but for the most part it is the user who can determine the proportions. The liquify tool in Photoshop is an example of this. The user can use this tool to make adjustments until the face or body in the photograph appears in good proportions. This is customizable, so it really depends on the user and is not generated automatically by software.

Art, Architecture and Science

The golden ratio is not just observed in people, but is actually used in architecture and art work as well. It is also observed in flower petals, seashells and sunflower seeds. In the Great Pyramid of Giza, the length of one side of the base is 756 feet with a height of 481 feet. The ratio of the base to the height is roughly 1.5717 (756/481) which is close to the Golden ratio. It is also seen in art like the Mona Lisa painting by Leonardo da Vinci. Euclid even linked 1.618 to the construction of the pentagram.

Remarkably, it has also been observed in DNA sequences as well. For example, a DNA molecule measures 34 angstroms by 21 angstroms at each full cycle of the double helix spiral. If you take the ratio of 34 to 21, the result is 1.61904 which is close to 1.618 in precision to at least two decimal places to the right. It doesn’t seem to be just coincidence when these patterns have been observed to repeat, not just random.

The Eye or The PHI?

In digital image editing today, software use these ratios to morph faces and body parts to look more symmetrically appealing to the eye and printed in glossy magazines for commercial use. The proportions appear close, but not exactly always to the value of the golden ratio. This does not mean everyone with proportions close to this ratio have perfect faces. It is just another way to describe how beauty is not just random, but a symmetrical pattern nature has determined. This is the x factor, that has universal appeal. When it comes to interpretation however, beauty is still in the eye (not the PHI) of the beholder.