Beauty is in the $\Phi$ of the beholder: the mathematics of human attraction

Human attraction is often regarded as a mystery. How many times has one of your friends got a new partner and you’ve thought to yourself “Dear God, what do they see in them?”? Probably quite a lot – I know I have (sorry friends)!

The thing is, human attraction has many components. Is their body physically attractive? Do you find their personality attractive?  Do you find their facial features attractive? Is their intelligence or skillset attractive to you? Is their sense of humour attractive to you? The list goes on.

So, how can we even begin to look at human attraction from a mathematical perspective? It’s certainly not a straightforward task, but people have been trying to figure it out for centuries. In the first part of this series, we take a look at some of the most influential discoveries about the mathematical roots of human physical attraction…

The Golden Ratio

The Golden Ratio (also called divine proportion) has been a hot topic ever since our good old Ancient Greek friend Euclid mentioned it in his Elements, and since then many papers have been written on the links between beauty and mathematics.

A representation of the Golden Ratio in the lengths of two line segments.

Simply put, the Golden Ratio (denoted by the Greek letter “phi” or $\Phi$) exists when we have a line segment and cut it into two pieces so that the longer part (a) divided by the smaller part (b) is equal to the sum of (a) + (b) divided by (a), which both equal 1.618.[1]

This same idea can be extended to 2-D. For example, to make a “golden spiral”, we start with a “golden rectangle” – that is a rectangle whose sides are in the ratio 1.618… ($\Phi$): 1. We then split this rectangle into squares; these squares having dimensions according to the Fibonacci sequence (1,1,2,3,5,8,13…). By drawing an arc between the corners of each square, we can make a “golden spiral”.

A Golden rectangle with a Golden spiral inside it

Actually, it has been shown that the Golden Ratio extends much further than just lines and shapes. In fact, it has been proposed that it is the key to aesthetically pleasing architecture, creating eye-catching graphics, many patterns within nature, and even the attractiveness of a human.

Plato’s Forms and Jung’s Archetypes

The idea that there is a “golden”, “divine”, “perfect” version of the human form is not a new one. In fact, it’s a very old one.

A depiction of Plato.

Back in Ancient Greek times, someone you might have heard of called Plato began to discuss his ideas about “Forms”. He proposed that all objects have an ideal “Form” (or structure) that only exists in principal and never in reality. In the context of beauty, a thing was only thought to be beautiful if it approaches this ideal Form. [2]

A photograph of Karl Jung.

Plato’s teachings were then developed further by a man called Karl Jung who developed his own “Theory of Archetypes”. According to Jung, an archetype is “an unconscious idea, pattern of thought, image, etc., inherited from ancestors of the race and universally present in individual psyches”. We now believe that the “ideal” human face (and potentially the body too) is one of these “archetypes”; a subconscious image that we are born with and carry throughout our lives. [3]

So, this raises the question, what does the perfect human look like?

Da Vinci’s Vitruvian Man

Leonardo Da Vinci (yes, that guy who painted the Mona Lisa) was the first person to apply divine proportions to the human body. His famous drawing “Vitruvian Man” depicts the ‘perfect’ man, whose proportions are based on guidelines from “De Architectura” by a Roman engineer and architect called Vitruvius. [4]

In his notes accompanying the drawing, Da Vinci notes that these are the ideal measurements of the human body according to Vitruvius:

  • Four fingers equal one palm.
  • Four palms equal one foot.
  • Six palms make one cubit.
  • Four cubits equal a man’s height.
  • Four cubits equal one pace.
  • Twenty four palms equal one man.

In addition to this, it is specified that: “If you open your legs so much as to decrease your height 1/14 and spread and raise your arms till your middle fingers touch the level of the top of your head you must know that the centre of the outspread limbs will be in the navel and the space between the legs will be an equilateral triangle. The length of a man’s outspread arms is equal to his height.”

Leonardo then went onto interpret these rules to create his drawing and to write his own set of guidelines for the ‘perfect’ man:

  • “From the roots of the hair to the bottom of the chin is the tenth of a man’s height.”
  • “From the bottom of the chin to the top of his head is one eighth of his height.”
  •  “From the top of the breast to the top of his head will be one sixth of a man.”
  • “From the top of the breast to the roots of the hair will be the seventh part of the whole man.”
  • “From the nipples to the top of the head will be the fourth part of a man.”
  • “The greatest width of the shoulders contains in itself the fourth part of the man.”
  • “From the elbow to the tip of the hand will be the fifth part of a man.”
  • “From the elbow to the angle of the armpit will be the eighth part of the man.”
  • “The whole hand will be the tenth part of the man.”
  • “The beginning of the genitals marks the middle of the man.”
  • “The foot is the seventh part of the man.”
  • “From the sole of the foot to below the knee will be the fourth part of the man.”
  • “From below the knee to the beginning of the genitals will be the fourth part of the man.”
  • “The distance from the bottom of the chin to the nose and from the roots of the hair to the eyebrows is, in each case the same, and like the ear, a third of the face.” [5]
Da Vinci’s drawing of the ‘Vitruvian Man’.

Now, Da Vinci’s man was based on Vitruvius’ writings, not the Golden Ratio, but the drawing actually has a handful of dimensions that align with golden ratios…

In the distance from Da Vinci’s guide line drawn at the hairline to the guide line at the foot, the following are all at golden ratio points:

  • The navel, which is most often associated with the golden ratio of the total height and not the height of the hairline.
  • The guidelines for the pectoral nipples.
  • The guidelines for the collar bone.

In the distance from Da Vinci’s guide line drawn at the elbow to the guideline at the fingertips:

  • The base of the hand is at a golden ratio point. [6]

And so, accidentally or not, Da Vinci became the first person to relate the Golden Ratio to human body proportions.

The Marquardt Mask

Since Da Vinci’s drawing in the late 15th century, mathematical understanding has improved greatly and, along with the invention of computers, it has become easier to create visual representations of these ‘perfect’ human proportions.

The Marquardt Mask on Marilyn Monroe.

More recently, in 2001, a Californian surgeon named Dr Stephen R. Marquardt decided to investigate the mathematics of human facial beauty. The result was the Marquardt Beauty Mask (or the Marquardt Phi Mask) – a facial mask derived from multiple golden dodecagon matrices.

This mask seems to fit many well-known faces which are commonly deemed as beautiful and Marquardt claimed that the mask applies across history regardless of sex or ethnicity. [7]

The Marquardt mask on a variety of famous faces and people of colour

Two new Golden ratios?

After the ground-breaking publication of “Marquardt’s Mask”, researchers became interested in investigating the truth behind this ‘mathematically perfect’ face. Many were pretty sceptical that something as seemingly subjective as facial beauty could be simplified so easily.

One particular study, carried out by The University of California and The University of Toronto, claimed to disprove this idea that facial beauty was centred around the Golden ratio (as proposed by the Marquardt Mask).

The study focused on two types of facial measurement ratios: horizontal and vertical. It was discovered that individual attractiveness is optimised when the face’s vertical distance between the eyes and the mouth is approximately 36% of its length, and the horizontal distance between the eyes is approximately 46% of the face’s width. The researchers then claimed that these “preferred” ratios were significantly different from the classic Golden ratio.

However, this is where they were wrong… The “ideal” face presented in this study actually contains a number of Golden ratios within the proportions of its key facial markers! The picture to the left demonstrates a handful of Golden ratios present within both the vertical and horizontal “Golden” faces.

Furthermore, working in pixels, we can actually show that the dimensions of the two faces can be derived from the Golden number (phi) 1.618. [8]

  • 38 = Phi to the -2 power (e..g., $1/\Phi^2$ as a percentage).
  • 62 = Phi to the -1 power (e.g., $1/\Phi$ as a percentage or 0.618 multiplied by 100).
  • 100 = $\Phi$ Phi to the 0 power.
  • 162 = $\Phi$ Phi to the 1 power (e.g., $\Phi$ as a percentage or 1.618 multiplied by 100).
  • 200 = 162 + 38.
  • 224 = 162 + 62.
  • 262 = Phi to the 2 power (e.g., $\Phi^2$ as a percentage or 2.618 multiplied by 100).

The “new” horizontal golden ratio of 0.46 can then be computed from these Phi-based numbers as:

  • Position of left pupil, at the midpoint of first golden ratio line at 38 pixels and third golden ratio line at 100 pixels = (38+100)/2 = 69.
  • Position of right pupil, at the midpoint of fourth golden ratio line at 162 pixels and fifth golden ratio line at 224 pixels = (162+224)/2 = 193.
  • Distance between the pupils = 193-69 = 124.
  • “New” golden ratio defined as the distance between the pupils in relation to the width of the face = 124 / 262 = 0.47.

Participants could only choose between .46 and .48, so it’s quite possible that, if given the opportunity to choose this option, .47 may have been found to be the “horizontal Golden ratio”. Even so, the visual difference between .46 and .47 would not be significant.

The same kind of derivation can be done for the “vertical Golden ratio face” (click here for more info).

So, does maths make you attractive?

Well, as a mathematician myself… Just kidding. On a serious note, I think that maths has a lot more to do with human physical attraction than one would assume. I’m not saying that I think there’s a single-solution, straightforward, formula for being physically attractive; people have different preferences and beauty is always somewhat in the eye of the beholder. For example, maybe to one person, a strong jawline is important, but to another person, nicely shaped eyebrows are the key to attractiveness.

The thing is, whilst different features are important to different people, we all seem to have roughly the same idea of what is attractive – watch some Love Island for modern-day evidence of that one. As suggested throughout history, it seems extremely unlikely that there’s not some sort of mathematically-based ‘ideals’…

In the second part of this series, we’ll take a look at those aspects of attractiveness below the surface and the joys of online dating. So, see you next time for some swiping right!

Emilia is a UEA Mathematics graduate, Norwich local, and incoming Consulting Development Analyst at Accenture. Maths-wise, she is particularly interested in statistics and computing, as well as their respective applications to the finance industry. Outside of spending hours of her life puzzling over numbers, she also enjoys a good ol’ pub quiz, shopping, and traveling with her mates.