March 12, 2007

Euler's Beauties

Those who assert that the mathematical sciences say nothing of the beautiful are in error. The chief forms of beauty are order, commensurability, and precision.—Aristotle, Metaphysics, XIII 3.107b

In 1988, David Wells surveyed readers of the Mathematical Intelligencer to get a sense of what mathematicians consider to be beautiful in their field. He provided a ballot listing 24 famous theorems. The results were published in 1990. Leonhard Euler (1707–1783) was responsible for three of the top five choices.

The readers ranked Euler's relation linking e, pi, and i as the most beautiful equation in mathematics.



It was Euler himself who introduced the constant 2.718281828459 . . . to the mathematical world as the base of natural logarithms and designated it, "for the sake of brevity," e. Euler's relation follows from his discovery of the following remarkable identity (for any real x), when x = π.



Of course, Euler's relation can be rewritten as e + 1 = 0.
"As math professors are fond of observing, this equation assembles the five most important constants in mathematics," William Dunham points out in his book Euler: The Master of Us All.
  • 0—the additive identity.
  • 1—the multiplicative identity.
  • π—the circular constant.
  • e—the base of the natural logarithms.
  • i—the imaginary unit.
"That these five superstar numbers should be related in so simple a manner is truly astonishing," Dunham writes. "That Euler recognized such a relationship is a tribute to his mathematical power."

Of the world's most beautiful theorems, Euler's formula for a polyhedron, tying together the number vertices (V), edges (E), and faces (F), ranked second: V + F = E + 2.

Euler's paper on this relation, sometimes called the Euler-Descartes formula, played a central role in early combinatorial topology.

Coming in fifth was the sum of an infinite series.



This equation represents one of Euler's earlier triumphs. In the previous century, a number of mathematicians had tried and failed to determine the exact value of this infinite series. Numerical approximations had shown the sum to be around 8/5, but the exact answer proved elusive.

"Well into the next century the problem remained unsolved, and anyone capable of summing the series was certain to make a major splash," Dunham recounts. "When it happened in 1735, the splash was Euler's. The answer was not only a mathematical tour de force but a genuine surprise . . . . This highly non-intuitive result made the solution all the more spectacular and its solver all the more famous."

What two theorems kept Euler from capturing the top three spots in the survey? One was Euclid's theorem that the number of primes is infinite. The other was the existence of five regular polyhedra.

Euler is getting a lot of additional attention this year because 2007 is the 300th anniversary of his birth (on April 15, 1707), and there are various celebrations of his life and works occurring around the world. Dunham himself recently talked about Euler at public events at the Embassy of Switzerland and the MAA Carriage House in Washington, D.C. The MAA Web site and online journal Convergence have information about additional events.

References:

Dunham, W., ed. 2007. The Genius of Euler: Reflections on His Life and Work. Washington, DC: Mathematical Association of America.

______. 1999. Euler: The Master of Us All. Washington, DC: Mathematical Association of America.

Wells, D. 1990. Are these the most beautiful? Mathematical Intelligencer 12(No. 3):37-41.

______. 1988. Which is the most beautiful? Mathematical Intelligencer 10(No. 4):30-31.

Artist Justin Mullins presents his gallery of the world's most beautiful equations (along with some ugly expressions) at http://www.justinmullins.com/.

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