A scientific answer to the question, ‘What is the most beautiful equation?’

April 17, 2017

Richard A. Friedman in the Sunday New York Times wonders what it is about math that makes it so elegant and aesthetically appealing. He asks, “Is it the internal logic? The unique mix of simplicity and explanatory power? Or perhaps just its pure intellectual beauty?”

He adds, “We all know that art, music, and nature are beautiful. They command the senses and incite emotion. Their impact is swift and visceral. How can a mathematical idea inspire the same feelings?”

He suggests that in a time of “alternative facts” it can be appealing to know that the Pythagorean theorem still holds and that p continues to describe perfect circles.

You might think that the concepts of beauty and math have little to do with each other. “But,” Friedman writes, “our brains also appear to respond to mathematical beauty as they do to other beautiful experiences.”

He cites a 2014 study by Semir Zeki, a neuroscientist at University College London, and other researchers, who write, “The beauty of mathematical formulations lies in abstracting, in simple equations, truths that have universal validity. Many—among them the mathematicians Bertrand Russell and Hermann Weyl, the physicist Paul Dirac, and the art critic Clive Bell—have written of the importance of beauty in mathematical formulations and have compared the experience of mathematical beauty to that derived from the greatest art. Their descriptions suggest that the experience of mathematical beauty has much in common with that derived from other sources….”

The researchers used fMRI scanners to observe the brains of 15 mathematicians while they were thinking of various equations. The researchers found a strong correlation between a subject’s perception of an equation’s beauty and activation of the medial frontal cortex—“…the same area that has been shown to light up when people find music or art beautiful, so it seems to be a common neural signature of aesthetic experience,” Friedman explains.

Friedman notes that mathematicians tended to find formulas they understood well to be the most beautiful, but the correlation wasn’t perfect, and the researchers were able to separate the experience of beauty from meaning. Friedman explains that “…there were equations that subjects understood completely yet found ugly.”

I don’t recall previously hearing about the 2014 study, but as I was reading Friedman’s column I had a good idea which equation would come out on top, and I was right—Euler’s identity:

e + 1 = 0

The ugliest? Srinivasa Ramanujan’s infinite series for the inverse of π. I had not been aware of this equation but have to agree that, with all due respect to Ramanujan, it is not particularly aesthetically pleasing. I won’t try to reproduce it here in WordPress, but you can see it in the original study.

The researchers provide this quote from Dirac: “…the mathematician plays a game in which he himself invents the rules while the physicist plays a game in which the rules are provided by Nature, but as time goes on it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which Nature has chosen….” Consequently, “One should be influenced very much…by considerations of mathematical beauty.”

The researchers conclude, “Hence the work we report here, as well as our previous work, highlights further the extent to which even future mathematical formulations may, by being based on beauty, reveal something about our brain on the one hand, and about the extent to which our brain organization reveals something about our universe on the other.”

Friedman writes that he got his interest in math from his father, an electrical engineer. Friedman didn’t follow in his father’s footsteps, however, and is now a professor of clinical psychiatry and the director of psychopharmacology clinic at the Weill Cornell Medical College.

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