Of course, a polynomial with the vertices as its set of roots does not contain any information about the original polygon's cyclic order (or more accurately, dihedral order). I'm curious how the limiting ellipses differ for the same set of initial vertices with varying order. A priori, the ellipses might just represent the correlation coefficient of the original set of points. (Specifically, via a level curve of the best-approximating 2-dimensional normal distribution of having the initial vertices as sample points.) —Dan Henry Baker wrote: ----- Here's one rationally reversible method to 'equilateralize' a triangle in the complex plane. This method is inspired by Gosper's continued fraction root extraction hack (perhaps in HAKMEM ??). Consider the cubic polynomial p(z) having the triangle vertices as roots. ... ... -----