Both the Mathematician and the Artist Perceive Beauty in Symmetry

Most people tend to look upon the relationship between science and art in the same way the late Stephen J.Gould regarded the relationship between science and religion: as “non-overlapping magisteria”.

However this is (in both cases) a dubious proposition. As pattern-seeking animals we seek order, consistency and symmetry. I mentioned in my previous piece on form and function, that we have a natural dislike of the extraneous and vestigial. In this respect, scientists, particularly mathematicians, and artists have more in common than you might think. In fact, sometimes they are different manifestations of the same intent. The pattern-seeking element of our brains is something all humans have in common and has been integral in reaching this stage of our evolution, although the means by which we perceive or reproduce these patterns is extremely varied.

Parthenon Floor Plan

No doubt you have heard, as I have many times, friends discussing their talents or careers in the context of how they think; that they are just ‘wired’ that way. I know extremely intelligent mathematicians who have little or no skill at expressing themselves through writing or art, or writers (myself included) who are mystified when presented with complex mathematical equations. There are parallels here (though I am not clear on the neurological details of this observation) between this spectrum of information-processing and spatial synaesthesia, where someone would see numerical sequences as points in space; or certain kinds of autism whereby someone may have difficulty interpreting human expressions but is very gifted when it comes to processing and replicating patterns or sequences.  Some disciplines spanning these magisteria have more in common than others.
Right: Floor Plan of the Parthenon. Source: Wikimedia Commons

Mathematics and music, in particular, share something like a common language (my high school music teacher also taught mathematics, and spoke often about the relationship between the two). Others can be more difficult to reconcile, but the cerebral commonalities are still there.

Some of the most beautiful examples of the interrelation between art and mathematics are to be found in the architecture of antiquity. The Parthenon, one of the most exquisite pieces of architecture of the ancient world, is said to embody the ‘golden ratio’; a term often represented visually by a pattern resembling an ammonite shell, and used in both science and art to describe two values where the ratio of the sum of both values to the larger of the two is the same as the ratio of the larger to the smaller of the two. Similarly, analysis of the Mosque of Uqba in Tunisia has revealed application of the golden ratio in its design. Of course, it has been said in the case of the Parthenon that this is an unintentional coincidence; however, if it was by accident rather than design that it possesses these qualities, it seems to further underscore that art has a natural inclination towards the mathematically elegant.

Early examples of this inclination towards mathematical perfection in architecture, including the Parthenon, can be found in the societies that drove our early scientific progress. ‘Natural philosophy’, as it was referred to before the sciences had advanced enough to be considered separate fields of enquiry, encompassed philosophy, spirituality, the sciences and, very often, art. The scientific community of the time was therefore one of polymaths who made little or no distinction between the natural and the transcendent. In a way we have come full circle, being as we are now sufficiently advanced to find transcendence in the beauty of science itself.

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Read: Da Vinci, Polymaths and the Art-Science of Innovation

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This interrelation between science and aesthetics is reciprocal of course. Mathematical elegance emerges in great works of art, but staggeringly beautiful systems can also emerge from the pursuit of pure science and even from (relatively) simple mathematical formulae.  This is revealed in an absorbingly beautiful way in the study of fractals (patterns that reproduce ‘self-similarly’ at any magnification), the most famous of which are the Mandelbrot and Julia sets. These fractal models resemble a hallucinogenic infinite landscape that, when observing it through a series of magnifications, have an intensely hypnotic quality.

SymmetryAt its most macrocosmic, we find that theoretical physics can have grandiose aesthetic aspects to it. Lawrence Krauss, in his explanation of why the shape of the universe matters aesthetically, explains how general relativity describes a curved universe which allows it to have one of three geometries: open, closed or flat. A flat universe is one with a total energy of zero, which, according to Krauss, is the “only mathematically beautiful universe”. Similarly, Garrett Lisi in creating his ‘theory of everything’ portrays the fundamental interactions between all particles and forces in the universe in the form of a breath-taking, eight-dimensional model. A perfect example of how that which is elegantly holistic, from a mathematical point of view, is most pleasing to us aesthetically.

As with form and function, the idea that aesthetics and science occupy entirely separate realms is a fundamental misunderstanding. Probably the best illustration that I have come across of the folly of this dichotomy is Richard Feynman’s discourse on the beauty of a flower. He saw profound elegance at all scales of the universe, from the quantum to the cosmic, about which he spoke at length, and, in fact, later became an amateur artist himself. As Feynman points out, the beauty an artist sees is available to everyone. To take it one step further, the transcendent potential of science and the symmetry of mathematics are also available to everyone and quite often they are one and the same.

As the architects of the Parthenon show, it is not necessarily important to understand it, merely a matter of looking at the world with less restrictive eyes and allowing our pattern-seeking nature to take it all in.