Beauty in Mathematics
"Euclid alone has looked on Beauty bare," wrote the poet Edna St. Vincent Millay. And ever since Euclid's time, lovers of mathematics have marveled at its beauty, even if they have glimpsed it, as St. Vincent Millay wrote later in her sonnet, "once only and then from far away."
One of the beauties of mathematics is that it promises us eternal truths that do not depend on opinion or fashion. One plus one will never be three. But mathematics is not just about assertions of fact; the deeper truth usually lies in the explanation of the fact. A good definition of beauty in mathematics would be: simplicity that leads to insight. "Proof is beautiful," wrote the late Harvard mathematician Gian-Carlo Rota, "when it gives away the secret of the theorem, when it leads us to perceive the actual and not the logical inevitability of the statement that is proved."
Though the most beautiful solution is almost invariably the simplest one, it is not necessarily the most obvious. This counterintuitive kind of simplicity is hard won. It may come in a flash, but that flash is usually preceded by hours--or years--of hard mental labor. It reveals hidden patterns and deep meanings, and it often gives its discoverers the feeling that it was not the product of human intellect but must have been there all along. Surely this same sense of wonderment was felt by the twelfth-century Indian mathematician Bhaskara, who presented a famous proof of the Pythagorean theorem consisting of a single diagram with a one-word inscription: "Behold!"
Mathematical works, like symphonies or paintings, can be profoundly original. Often, this originality takes the form of a connection between two different problems or areas of mathematics that had not previously seemed to be related. Leonhard Euler, in his proof of the identity 1 + 1/22 + 1/32 + 1/42 + ... = pi2/6, daringly treated the sine function (from the world of trigonometry) as if it were a polynomial (from the world of algebra) and thereby discovered a connection that had never occurred to anyone. Quite apart from the beauty of Euler's proof, one could argue that the identity itself is beautiful, as it links two of the most organic concepts in mathematics: the series of integers 1, 2, 3, 4, ... and the irrational number pi. It also contains a strong whiff of the unexpected: how did pi get there, and where did that 6 come from? To number theorists, Euler's formula is now "elementary," but it will never stop being elegant.
Visual attractiveness plays a role in mathematics and is increasingly revealed as graphic technology improves. Any mathematical portrait gallery should include the infinite regresses and labyrinths of fractals such as the Mandelbrot set; the holistic unity of the icosidodecahedron, which represents the molecular structure of carbon-60 or buckminsterfullerene; and the graceful sweep of minimal surfaces such as the helicoid, which is shaped like a DNA molecule. Most mathematicians would probably agree that the beauty of these objects was already immanent in the mathematics even before technology caught up to the task of portraying them. They are beautiful because they are symmetric, and mathematics places a high value on symmetry. (It is the motivating idea behind group theory and an important theme in many other branches of mathematics.)
The examples of buckminsterfullerene and DNA mentioned above bring up the point that mathematics and nature are inextricably intertwined. The ancient Pythagoreans taught that "all is number" and discovered the relationship between numbers and musical harmony: a lute string that is twice as long as another will sound one octave lower. The Fibonacci numbers can be found in the elegant spirals of a pine cone, a pineapple, or a sunflower. Physicists have used mathematical beauty as a compass in constructing new theories of the universe, such as the "eightfold way" of Murray Gell-Mann that led to the prediction of quarks. Galileo wrote, "The universe ... cannot be understood unless one first learns to comprehend the language and interpret the alphabet in which it is written. It is written in the language of mathematics...." This connection between mathematics and the secrets and harmony of the universe may be, for many people, the best route to appreciating its beauty.
Nearly every subject in mathematics has been described as beautiful by someone at some time: public key encryption, differential equations, the central limit theorem, Maxwell's equations, fluid mechanics, graph theory. In mathematics, as in art, beauty is in the eye of the beholder. However, mathematicians--unlike some modern artists--have never turned their backs on beauty as a standard of merit. For them, the highest accolade that a new discovery can receive is not that it is logical or useful, but that it is elegant. "Beauty is the first test," wrote number theorist G. H. Hardy. "There is no permanent place in the world for ugly mathematics."
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