See also: https://0x0.st/ilHN.png I had a similar thought about distributions, but disregarded it after one calculation. The "score" that I mentioned previously is actually the ratio of principal moments of inertia (relatively close to what you describe). Ordering seems to matter most of all. The eigenvector idea is enticing because order is built in to a state vector. Maybe that approach will yield some insight, but I remain in doubt. --Brad On Fri, Sep 18, 2020 at 6:46 PM Dan Asimov <dasimov@earthlink.net> wrote:

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. ... ... -----

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