"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?