Without a doubt, my favourite theorem in all of mathematics — in the sense that it quite literally made me gape with astonishment when I first read it — is Marden’s Theorem. The result is named after Morris Marden, who publicized it in 1945 but actually attributes the result to Jörg Siebeck (1864), and provides a wonderfully simple geometric relationship between the zeroes of a third-degree polynomial with complex coefficients and the zeroes of its derivative.

**Theorem.** Let *p*: ℂ → ℂ be any polynomial of degree three with complex coefficients such that the roots *z*_{1}, *z*_{2} and *z*_{3} of *p* form a non-trivial triangle Δ in the complex plane ℂ. Then

- there is a unique ellipse
*E*(known as the*Steiner inellipse*) that is tangent to all three sides of Δ at their midpoints; - and, furthermore, the foci of the Steiner inellipse
*E*are precisely the zeroes of the derivative*p*′ of*p*.

To me, that’s just incredible: you start with a cubic *p*, associate one obvious geometric object Δ with it by the “where are the zeroes?” operation, discover that there’s another unique geometric object *E* associated to Δ, and the two points (foci) that equivalently define *E* happen to be — no, don’t just *happen to be*, they *have to be*! — the answer to “what are the zeroes of *p*′?”

**Footnote.** It turns out that the Steiner ellipse *E* of Δ is also the John ellipse of Δ, meaning that it is the ellipse of maximal area that is contained within Δ. Higher-dimensional convex bodies also have associated John ellipsoids of maximal volume, but I don’t know if those ellipsoids have such nice *algebraic* relationships with the vertices of the containing bodies.

I learned a wonderful application of this result to fluid dynamics today in a fascinating talk at the Newton Institute by Morten Brøns. Place three identical point vortices in the plane, and make a triangle with them as vertices. Then the foci of the Steiner inellipse of that triangle are precisely the stagnation points of the resulting fluid flow. 😀

See H. Araf and M. Brøns, “On stagnation points and streamline topology in vortex flows”, J. Fluid Mech. 370:1-27, 1998. http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=14363

Very nice. I wonder what happens if you have point vortices of differing signs or magnitudes…