## Sunday, April 17, 2011

### the most beautiful equation

Etudes is an exercise in beauty; to me, its brilliance lies in its choreographic elegance and almost mathematical playfulness of the musicality. The strict counts and clean lines give this ballet a stunning, unexpected quality--in a way, Etudes is the most beautiful science put to music. This got me wondering: is there such a thing as "beautiful" math or science? And my research led me to what many scholars agree is the "most beautiful theorem in mathematics": Euler's Identity.

Named for Swiss-German mathematician Leonhard Euler, Euler's Identity is the equality in analytical mathematics:

where e is the base of natural algorithms (Euler's number); i is the imaginary unit--i² = −1; π is pi.

The reason Euler's identity is considered remarkable is because of its mathematical beauty. The three basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. The equality also connects five fundamental mathematical constants:

- The number 0, or the additive identity
- The number 1, or the multiplicative identity
- The number π, ever-present in trigonometry, the geometry of Euclidean space, and analytical mathematics
- The number e, which is the base of natural logarithms
- The number i, or the imaginary unit of the complex numbers

Euler's Identity is a special case of Euler's formula from complex analysis, which reads (for any real number x):

And notably,

since cos π = -1 and sin π =0, then it must be true that

This gives us Euler's Identity:

The simplistic elegance of this equation, in mathematical beauty standards, is stunning; many scholars have waxed poetic about this one equality. A poll of readers conducted by The Mathematical Intelligencer magazine named Euler's Identity as the "most beautiful theorem in mathematics"; in another poll of readers by Physics World magazine Euler's Identity tied with Maxwell equations (of electromagnetism) as the "greatest equation ever". There is an entire 400-page mathematics book written by Dr. Paul Nahin devoted to the identity: Dr. Euler's Fabulous Formula; the tome professes that Euler's Identity sets "the gold standard for mathematical beauty." After proving Euler's Identity during a lecture, Benjamin Peirce, the noted American philosopher/mathematician and a professor at Harvard University, said, "It is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth." Perhaps Stanford University mathematics professor Dr. Keith Devlin was most poetic: "Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's Equation reaches down into the very depths of existence."

Etudes has the same sort of clean beauty and technical impressiveness as Euler's Identity, albeit in a completely different way. Etudes takes the precise structure of the ballet class and emphasizes the beauty behind pure technique, much as this equality stresses the importance and beauty of the most basic numbers and functions in mathematics.