May 9, 2008

The Most Marvelous Theorem

"What makes a theorem great?"

To Dan Kalman of American University, a great theorem is one that surprises. "It is simple to understand, answers a natural question, and involves familiar mathematical concepts," he writes in the April Math Horizons.

Kalman's nominee for the "Most Marvelous Theorem in Mathematics" concerns a remarkable relationship between the roots of a polynomial and the roots of its derivative. Kalman calls it Marden's theorem because he first encountered it in the 1966 book Geometry of Polynomials by Morris Marden (1905–1991). Marden himself traced the theorem to an 1864 paper by Jörg Siebeck.

Suppose you have a third-degree polynomial—a cubic function. Marden's theorem specifies exactly where to find the roots of the derivative of this polynomial.

A cubic has three roots in the complex plane. Typically, these roots form the vertices of a triangle. Draw an ellipse inside the triangle so that the ellipse touches the triangle's sides at their midpoints. The two foci of the inscribed ellipse are the roots of the polynomial's derivative.


"How can it be that the connection between a polynomial and its derivative is somehow mirrored perfectly by the connection between an ellipse and its foci?" Kalman asks. "Seeing this result for the first time is like watching a magician pull a rabbit out of a hat."

In a publishing tour de force, Kalman's technical article describing an elementary proof of Marden's theorem appears in the April American Mathematical Monthly and his interactive, online presentation of the theorem is simultaneously available in the MAA's Journal of Online Mathematics and Its Applications.

What's your candidate for the most marvelous theorem in mathematics?

3 comments:

Anonymous said...

When I first encountered Sharkovskii's theorem I was entranced on three levels. First it seemed astonishing that it was true. (I had no experience in that area at all) Second I wondered how anyone could discover it. What on earth was he looking for? And third how does one go about proving such a thing. I never looked at the original proof but struggled through the proof in a book by Devaney. I'm still not sure i understand the proof.

maxwelldemon said...

This is a tricky one, I think it is easy to overlook the wonder in some classical results as they are so familiar. The proof that there are an infinity of primes or the fact that the diagonal of a square is not commeasurable to its edge (the existence of irrational numbers). The existence of more than one infinity and of space filling curves both made me marvel at university. However having waffled I will nominate a theorem that is not very well known at all, but is of personal relevence. That is de Bruijn's proof that the Penrose tiling can be constructed as a slice of a 5 dimensional lattice.

The Penrose tiling was created as a geometric object to give the first example of a set of only two shapes that would tile the plane, but not periodically. The construction used for this was a substitution rule, where tiles are first expanded and then replaced by a patch of tiling. De Bruijn showed that there was a second, simple algebraic construction.

The marvel for me comes in the powerful way that (yet again) algebra are geometry can be used to mutually illuminate and explain each other.

maxwelldemon said...

This is a tricky one, I think it is easy to overlook the wonder in some classical results as they are so familiar. The proof that there are an infinity of primes or the fact that the diagonal of a square is not commeasurable to its edge (the existence of irrational numbers). The existence of more than one infinity and of space filling curves both made me marvel at university. However having waffled I will nominate a theorem that is not very well known at all, but is of personal relevence. That is de Bruijn's proof that the Penrose tiling can be constructed as a slice of a 5 dimensional lattice.

The Penrose tiling was created as a geometric object to give the first example of a set of only two shapes that would tile the plane, but not periodically. The construction used for this was a substitution rule, where tiles are first expanded and then replaced by a patch of tiling. De Bruijn showed that there was a second, simple algebraic construction.

The marvel for me comes in the powerful way that (yet again) algebra are geometry can be used to mutually illuminate and explain each other.