Is Beauty in the Eye of the Mathematical Beholder?

By Shawn Radcliffe


“Mathematics, rightly viewed, possesses not only truth, but supreme beauty.” ~ Bertrand Russell

If beauty is in the eye of the beholder, then mathematicians must have a curious eye indeed. In a study earlier this year in Frontiers in Human Neuroscience, researchers found that when 15 mathematicians viewed “beautiful” equations, the same area of the brain was activated as when people listen to music or look at works of art that are considered beautiful. The stronger the experience, the more activation in the brain.

Is this the result of mathematicians’ love of abstract concepts, or is there more at work here? In the study, researchers found some connection between how much the mathematicians understood an equation and how beautiful they found it. But not always. Certain equations were ranked ugly regardless of understanding.

Comprehension, however, is not always necessary for the experience of mathematical beauty, just as not knowing how a piece of music or art “works” doesn’t limit your appreciation of it. The researchers, led by Semir Zeki of University College London, also asked nonmathematicians to view the equations while their brain was scanned with a functional MRI. Again, certain equations caused the medial orbitofrontal cortex to light up—an area of the brain that’s also involved in integrating emotion, sensory experience and decision-making.

So which equations are mathematicians most likely to put on a pedestal and stare at for hours? Ones like Leonhard Euler’s identity:


And among the “ugly” equations is Srinivasa Ramanujan’s infinite series for 1/π, a particular cumbersome one:


Based upon the appearance of these equations, mathematical beauty may hinge on simplicity. Traditionally, simplicity has been an important component in mathematics, but some mathematicians view it as separate from beauty. Theoretical physicist Paul Dirac said, “It often happens that the requirements of simplicity and of beauty are the same, but where they clash the latter must take precedence.”

At its heart, though, beauty in mathematics may be more than just a personal preference. In fact, striving for beauty in mathematics can often lead to deeper insights, which also moves us closer to the ultimate answers about the universe.

This, no doubt, is what Edward Frenkel, a professor of mathematics at the University of California, Berkeley, and author of Love and Math, means when he writes that “Mathematics directs the flow of the universe, lurks behind its shapes and curves, holds the reins of everything from tiny atoms to the biggest stars.”

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