Phi (φ) of the Beholder

“Without mathematics there is no art.”
–Luca Pacioli

A commentary by Uliana Solovieva

Figure 1

Look at these two rectangles. Which one appeals to you?

Most people would readily agree that the left choice is somehow better. A new field of Neuroaesthetics, pioneered by Semir Zeki, has been busy understanding the mechanics of our gut feeling. It seems that inside the complex system, we call our brain, exists an autonomous math calculator. It sends signals to the brain when the art we view matches with intrinsic “pleasing” criteria. Such system is guided by the reward function of the internal model of the brain (Kawabata 2004:1700). This theory seems to downgrade us to mechanic calculating robots, but it can give us a new framework to understanding aesthetics. What I am arguing is that unnoticeably, mathematics guide our aesthetic appreciation of art, and the process by which artists create art.

The first obvious source of artistic appreciation is in the defined proportions/symmetry we see all around us. The internal OCD filter is satisfied when harmony is achieved. This phenomenon can be looked at through the lens of the golden ratio, famously utilized in the Fibonacci sequence. Under close observation, the golden ratio is found in the realistic art of Leonardo Da Vinci’s Vitruvian Man as well as surrealistic Salvador Dali’s Melting Clocks. Despite the obvious differences, both works of art are widely acknowledged as innately pleasing. By closely probing at these masterworks, math’s explanatory power can be traced in art as the source of aesthetic appreciation.

In discussion of art, one controversial issue has been the validity of art as a subjective experience. Some may readily challenge my stance by insisting a commonly accepted notion that “art is in the eye of the beholder.” However, new evidence suggests the underlying universality of art appreciation is based on math – symmetry and ratios. Mario Livio, in his article “Why Math Works” argues, “The universe has regularities, known as symmetries, that let physicists describe it mathematically. And no one knows why.” (Livio 2011:83). Most of us readily accept math’s power to explain fundamental properties of the universe, while art is left in the wishy-washy grey area of the human experience. Could these two seemingly incomparable subjects be treated as one? Let’s do so by returning to the rectangles from (Figure 1), Leonardo Da Vinci’s Vitruvian Man (Figure 2), and Salvador Dali’s Melting Clocks (Figure 3).

The key to unlocking this puzzle lies in mathematics of the golden ratio. It seems we are naturally predisposed to appreciate a ratio closest to 1.618033988749895… (Dense 2013:39) which is represented by phi (φ), a never ending and never repeating number. It has boggled the minds of people ever since Pythagoras discovered it in secrecy (Livio 2008:41). Look at the first rectangle in Figure 1 with sides 1.62 (length) to 1 (width). You are looking at the world’s most beautiful rectangle, called the golden rectangle. It needs no beauty pageants to be voted best, rather, it needs our brain circuitry to become so. Without the brain evolution handed down to us, there would be no aestheticians, no art critics, and no mathematicians. When the brain recognizes the golden mean, it activates reward pathways and we unnoticeably feel good and unconsciously ascribe the dopamine surge to the gut feeling “I like!”

Da Vinci’s Vitruvian Man (Figure 2) is the picture-perfect of all the perfect perfections of a perfect man. Da Vinci is said to have been inspired by the writing of the architect Vitruvius who once wondered what would a perfectly proportional human look like. Vitruvius suggested human proportions as a symbiosis between each constituent part of the body (Livio 2008:134). Ask any artist how to “properly” draw a human figure and you’ll get math – “Draw 8 heads horizontally, that’s your proportionate height. Legs will start on the “5th head” and it is also where hands end….” my art instructor would repeat time and time again. It becomes evident that the body’s proportionate relationship of each part to the other generates realistic proportions that we call beautiful. Moreover, just as our beautiful rectangle, humans are also quantified as beautiful within the frame of reference in accord to symmetry. Once again, it is all the working of golden ratio. Da Vinci shamelessly utilized these seemingly magical principles into the intricate layouts of his own artwork.

Leonardo Da Vinci’s Vitruvian Man
Figure 2

To understand the highly encompassing impact of φ in aesthetics, we must understand where it came from. Patterns are generated from sequences, and the Fibonacci pattern goes like this
1, 1, 2, 3, 5, 8, 13, 21, 34, 55…
This sequence is life’s way of generating φ! Adding up the previous two numbers together forms each consecutive number; then, by dividing current number by previous, we get fractions that inch up closer and closer to φ but never quite reach it 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13… (Dense 2013:39).

With this pattern, a logarithmic spiral is generated in Figure 4, notice that parameters of each “box” making up the rectangle is in correspondence to the Fibonacci sequence numbers. The box in the center is 1:1, the next is 1:1, then 2:2, 3:3 and so on to create the good old golden rectangle.

Figure 4

With this groundwork, it becomes easy to see a connection between math in realistic art. However, what if I want to steer clear of obvious shapes and sizes, as in surrealism? That art style is known to disregard the boundaries of possibility and rely on the feelings, “pathos”, emotions of each viewer. Salvador Dali’s art demonstrates lack of required proportions, shadows, or concrete meaning that Da Vinci’s work exhibited. Did we finally hit the sweet spot of art containing no math backbone?

Figure 3

Not quite. Even though Dali, a master surrealist, seems unconcerned with the proper rules of symmetry and reality, in actuality tells us a hidden story of math through his Melting Clocks piece (Figure 3). Unintentional implicit use of the golden ratio in the creative process in Dali’s mind can be noticed by inserting a Fibonacci spiral over his masterpiece. It evidently lines up! Our eyes start from the brightest spot in the center and move to the left following the course of the tree and extending down to the clocks and completing the golden spiral in the anti-clock-wise fashion. Strangely once more, math seems to be underlying not only our aesthetic appreciation, but the way a painter paints. Could these algorithmic internalized calculations be present in all of us, secretly staging our actions since the day we can hold a paintbrush (pen/spatula/cup). Our brains are known to precisely calculate each movement when we take on even the simplest of tasks such as grabbing a cup of tea. Oh the intricacy of trajectories needed to move every muscle in perfect synchrony in order do it right. Could same or similar mechanisms lie in our understanding of art and the creation of art as a two-fold system, leading to the sense of aesthetic pleasure spiked with dopaminergic pathways and feel good vibes? It surely seems so.

Mario Livio’s remark of math having explanatory and predictive power stands clear to identify two opposite works of art – one realistic and one surrealistic – under the same umbrella of universal mathematical laws. These laws narrow the subjective ambiguity of human “liking” into a well-defined boundary. Dictated by the golden ratio φ in all things math – numbers, patterns and sequences act as the unconscious factors that choose what we do and do not like. This idea, can help you and any other folk looking to predict likability of a product. From artists to scientists, the mystery of the mystical aesthetic sense is no further than a math equation away,
as H.E. Huntley adds:

The description of this proportion as Golden or Divine is fitting perhaps because it is seen by many to open the door to a deeper understanding of beauty and spirituality in life. That’s an incredible role for one number to play, but then again this one number has played an incredible role in human history and the universe at large. (Huntley 1970:63)

Indeed, the golden ratio is referred to as a divine proportion found far beyond the properties of aesthetics: physicists find it in cosmic pulsars; botanists find it in growth patterns of leaves and biologists in the structure of DNA. Even though artworks of Dali and Da Vinci might not paint (no pun intended) the whole picture of how φ is found in every type of art. Nevertheless, they illustrate a microcosmic example of the role mathematics play in our internalized sense of beauty.

Livio, Mario. The golden ratio: The story of phi, the world’s most astonishing number. Broadway Books, 2008: 1-268.

Livio*, Mario. “Why math works.” Scientific American 305.2 (2011): 80-83.

Dence, Thomas. “SOME OLD AND NEW RESULTS FOR THE WORLD’S MOST FAMOUS SEQUENCE OF NUMBERS.” Journal of Applied Global Research 6.16 (2013): 38-43.

Kawabata, Hideaki, and Semir Zeki. “Neural correlates of beauty.” Journal of neurophysiology 91.4 (2004): 1699-1705.

Bielski, Carolyn. Beginning Art 1 Class. Lake Forest High School. Lake Forest, IL. September 2011. Keynote Address.

Huntley, H. E. The Divine Proportion: A Study in Mathematical Beauty. New York: Dover Publications, 1970. Print. 50-69.

Images found in:
“Leonardo’s Vitruvian Man.” The Vitruvian Man. Web. 28 Mar. 2016.

“15 Things You Didn’t Know About ‘The Persistence Of Memory'” Mental Floss. Web. 28 Mar. 2016.

“Lesson Plans Based on Movies & Film Clips!” Donald in Mathmagic Land. Web. 28 Mar. 2016.