Mario Livio Ph.D.

Why?

On Symmetry and Aesthetics

What does symmetry mean and does it play a role in aesthetic experience?

Posted Oct 05, 2020

To most people, “symmetry” means the bilateral symmetry exhibited by, say, a butterfly, or the human face. That is, if you take a picture of a butterfly and draw a straight line down the middle of the figure, flipping the page over while keeping the line in place results in perfect overlapping. This is the type of symmetry mathematicians refer to as symmetry under reflection, which means that if you reflect one half of the shape, you obtain something that is identical to the other half. But symmetry is more generally defined as immunity to a possible change. For example, the phrase “Madam I’m Adam” reads the same if read back-to-front letter by letter. The same is true for the palindrome “A Man, A Plan, A Canal, Panama.” In such cases, we would say that the phrase is symmetric under the operation of back-to-front reading.

Many other symmetries exist. For example, most wallpaper designs are symmetric under translation. If you move by a certain distance in a certain direction, you see again the same pattern. The same is true for the string of characters XYZXYZXYZXYZXYZ. Snowflakes, on the other hand, are symmetric under rotations (about an axis passing perpendicularly through their center) by 60 degrees (also by 120, 180, 240, 300, and 360 degrees).

We generally tend to detect symmetric patterns quite easily and to remember them much better than we remember random shapes. Psychologists Emanuel Leeuwenberg, Wendell Garner, and Stephen Palmer suggested that these facts are related to the “information load” associated with shape representations. For example, to completely specify a perfect square, which is symmetric under rotation by 90 degrees about an axis perpendicular to its area through its center; and also symmetric under reflection about a line passing perpendicularly through the middle of two sides (or about the square’s diagonal), all we need to know is one length (the side of the square) and one angle (90 degrees).

On the other hand, for an arbitrary quadrilateral figure (which does not possess any symmetry) we would need to know four lengths and three angles (the fourth completes the sum of the angles to 360 degrees). In other words, the information load measures the complexity of the “code” (perceptual description) needed to generate the figure.

Using functional Magnetic Resonance Imaging (fMRI), visual neuroscientist Christopher Tyler and collaborators even identified an area of the brain which seems to encode the presence of symmetry in the visual field. Tyler presented subjects with a variety of patterns characterized by translation or reflection symmetry, and discovered that these stimuli activate a region around the middle occipital gyrus, a region of lateral occipital cortex whose function was otherwise unknown.

Symmetry plays an important role in perception. It helps in discriminating living organisms from inanimate objects, and (as studies in the 1990s have shown) even in the selection of desirable mates (where it is probably taken as an indicator of health). Here I would like, however, to mention an old, intriguing work, relating symmetry to a speculative attempt to formulate a mathematical theory of aesthetics.

Harvard’s influential mathematician George David Birkhoff spent half a year in 1928  traveling in Europe and the Far East and examining many art forms, ranging from poetry and music to the visual arts. This came on the heels of interest in aesthetics which occupied Birkhoff since his undergraduate days, and it culminated with him publishing a book entitled Aesthetic Measure in 1933. In that book, Birkhoff attempted to analyze that part of the sensation evoked by art which “is clearly separable from sensuous, emotional, moral, or intellectual feeling.”

He divided what he regarded as the aesthetic experience from a work of art into three phases: The first amounted to the effort invested in the attention necessary for perception; the second involved grasping that the work is characterized by a certain order, and the last was the appreciation of the value that rewards the entire mental exercise. He then suggested that the effort involved in the first stage is proportional to the complexity (which he denoted by C) of the work of art. He labeled the order characterizing the object by O. Clearly symmetries played a key role in O. Finally, Birkhoff denoted the feeling of value (which he called the “aesthetic measure”) by M.

The essence of Birkhoff’s theory of aesthetics could then be summarized as follows: within any particular class of aesthetic objects, be it paintings, vases for flowers, poems, or pieces of music, one can define a complexity C and an order O. Then the aesthetic measure is given by the simple formula M = O/C. That is, for any given degree of complexity, the aesthetic measure is higher the more order the object possesses. Similarly, for any level of order, the aesthetic measure increases when the object is less complex.

Birkhoff himself recognized that the precise definition of his parameters O, C, and M was tricky. Nevertheless, being a mathematician, he did offer detailed prescriptions for the calculation of these variables for entire classes of art objects, starting with simple geometrical shapes, continuing with Chinese vases and musical harmonies, and concluding with poems by distinguished poets.

If you think that it would be naive to expect that one can reduce all the intricacies of aesthetic pleasure to a simple formula you would be right, and Birkhoff would have probably agreed with you. Nevertheless, he pointed out that: “In the inevitable analytic accompaniment of the creative process, the theory of aesthetic measure is capable of performing a double service: it gives a simple, unified account of the aesthetic experience, and it provides means for the systematic analysis of typical aesthetic fields.”

In the next post, I’ll briefly describe the role of symmetry in the formulation of what we call the “Laws of Nature.”

References

Livio, M. (2005). The Equation That Couldn't Be Solved. New York: Simon & Schuster.