I watched the majority of a program on PBS yesterday (hosted by John Cleese of Monty Python fame) called "The Human Face with John Cleese," and the topic of this installment was "Beauty."
I stumbled onto the program 15 minutes or so into it, but it's clear that they were trying to demystify (or better yet: universalize, or perhaps most accurate: generalize) a concept of human beauty.
What made me stop as I flipped through was a photo array featuring faces ranging from the gorgeous to these hideously deformed faces you see in lepers and "elephant-man" types. But this was old hat to me—concepts of human beauty are limbically associated with physical symmetry, particularly of the face, because higher degrees of symmetry
suggest (again, this is on a limbic level) better health and by extension, better reproductive viability.
But what really piqued my interest was that some researchers (there was an evolutionary psychologist, a "reconstructive" surgeon, and other professional types) speculate on a mathematical foundation to "universal" concepts of beauty. And to my absolute shock, it was my favorite number, phi or φ or 1.6180339...
For those of you unfamiliar with φ, I'll try not to bore you and thus simply say it is a number which has come to be known as "the Golden Number" or "the Divine Ratio." It is a number found throughout nature, as well as pop culture manifestations in the movie
Pi (perhaps the most inappropriately named movie of all time, after
Final Destination: 2). If you want to know more about φ, the actual number, I'll offer a little more background at the end of this post.
What's important for the discussion of beauty and the assertion of a mathematical formula for human beauty is that this number is purported to be a "visual harmonic" (if I may be so fast and loose with the terminology). In the same way that a harmonic is achieved on guitar by plucking a string at a natural node according to the wavelength of the string, some purport that φ is some harmonic point resting at a "visual node" of some ocular wavelength (
not to be confused with the wavelengths of light itself).
What all that basically means is that objects displaying this ratio seem most visually appealing, most pleasing to the eye. (This assertion itself is the subject of much debate, and rightfully so as it's the result of Renaissance-era Neoplatonism.) The most common assertion of this is found in the "Golden Rectangle." (Some believe that the Parthenon in Athens is such a rectangle.) A golden rectangle is one whose sides form the ratio of 1.618:1. Below is a rectangle with a height of 1 and a width of 1.618...

What makes the rectangle so "golden" or "divine" is that these proportions can be used to create infinitely many congruent rectangles inside of it:
(The congruent rectangles can be made by removing the square from the main rectangle, the secondary rectangle, and so forth.)THE NITTY GRITTY: 1.618...The assertion of Dr. Stephen Marquardt (who, to my distress though not my surprise, is a SoCal native and "reconstructive" [read: plastic] surgeon) is that the faces of "beautiful people" follow certain geometric ratios. For instance: the width of the mouth to the width of the nose is 1.618:1. The width of the secondary incisor to the primary incisor is 1.618:1. An exaggerated and erroneous depiction of all these golden ratios that may be found in the human face follows:
(No idea what the cattle head is doing in that collage.)Furthermore, Dr. Marquardt stated that the ratio of φ exists throughout the human body as well. In "well-proportioned" people, the ratio of their overall height to the height of their belly button is 1.618:1. On the phalanges of the fingers: the ratio of the length of the first phalange to the second is 1.618:1, and the second to third, 1.618:1. (I presume this assertion carries across all the fingers, and it's not clear to me whether he meant that this is a universal human ratio, or only one found in aforementioned "well-proportioned" people.)
(Artist's rendering of the possibility of golden ratios on the human body.)And while the motivations of Dr. Marquardt obviously go beyond the narcissistic, obsessive-neurotic motivations of all academics (including yours truly), what is difficult to dismiss is that this φ is a number found throughout nature: φ is the reason why four-leafed clovers are relatively rare. See, 4 is not a Fibonacci number. (It is at this point that I am forced to delve into the background of φ and of course some mathematical specifics.
The Italian mathematician, Fibonacci, noticed that—in nature—numbers belonging to a specific sequence of numbers appear far more frequently than numbers not in this sequence. (I think he noticed this while breeding rabbits? Something along those lines... google it if you're curious.) This sequence of numbers has subsequently come to be known as "the Fibonacci sequence" or "Fibonacci numbers." The general formula for any given number in the sequence is to simply add the previous two numbers in the sequence. And the sequence starts with 0 and 1. Thus:
Fn = Fn-1 + Fn-2 (any given number in the sequence [Fn] is simply the sum of the previous two numbers [Fn-1 + Fn-2])
The first few numbers in the series are thus:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
Notice that 4 is not in the sequence. That is what makes 4-leafed clover so rare: 4 is not a "natural harmonic." (Again, I'm being fast and loose with the terminology.)
Now, if you divide any number in the sequence by the number before it, the result approaches the number, φ, 1.618...
1/0 = ∞
1/1 = 1
2/1 = 2
3/2 = 1.5
5/3 = 1.666...
8/5 = 1.6
13/8 = 1.625
21/13 = 1.615
Notice how the result of these divisions keeps getting closer to φ, to 1.618.... ?
What I find even
more interesting is that mathematicians claim that φ is the
most irrational number, even more irrational than the most famous irrational, π, 3.14... . And perhaps it makes it more difficult to belive (or easier, depending on your own philosophy) that the most irrational number stems from a most straightforward equation. To approximate φ, simply repeat this formula for as long as you have the stamina:
(Shoutout to all the ThinkPad peeps... whut whut!!)OR:
phi = 1 + 1
-----------
1 + 1
------
1 + 1
------
1 + etc.
And I believe it's at this point that I will end my analysis. Beauty and irrational numbers, physical attraction and φ's prominence throughout nature... this all feels like it's verging on some sort of limit of its own, or better yet: a divergence. As a recovering atheist, I can't allow myself to say it, but as an agnostic I'm confident leaving it at: There's something serious going on here—something foundational, something structural.