Jason Cantarella, Tetsuo Deguchi and I just posted our paper "Probability Theory of Random Polygons from the Quaternionic Perspective" to the arXiv.
In this paper we give a new probability measure on polygons which is very nice to work with, both in the sense that it is easy to compute expected values of some interesting statistics with respect to this measure and in the sense that it is extraordinarily easy to sample from this measure.
In layman’s terms, we give a way of picking polygons “at random”, and we can use this to determine very explicitly, for example, the average radius of gyration of a polygon.^{1} Our method of picking random polygons is also very simple (requiring just a few lines of code) and fast (linear in the number of edges), so it is straightforward—if we want to test some hypothesis—to produce a few million polygons and see what the answer is on that sample. We can produce polygons with very large numbers of edges almost instantaneously; the animation at the top of this post shows a random 20,000-sided polygon, and here’s a random 1,000,000-sided polygon (click to see full size):
Why care about polygons? Well, polygons serve as a simple mathematical model for polymers like DNA and proteins. Suppose you have a large sample of DNA produced by some process and want to know whether the sample is random or if it’s biased in some way (which might tell you something about the process which produced the sample). To answer that question, you have to know what a random sample would look like, which means you need to know what “random” means in this context…and that’s exactly what we figured out.^{2}
You can see the definition at the above-linked Wikipedia article, but the radius of gyration of any shape is some sort of average distance of the points in the object from its center of mass. In the case of polymers, it’s a useful measure of size since it can actually be determined experimentally. ↩
Technically, we’re just proposing a possible definition of what “random” could mean in this context; there’s no claim that this is necessarily the definition that matches any sort of experimental evidence. That being said, our definition is very symmetric,^{3} which is usually a good sign. ↩
For the experts: it comes from pushing forward a homogeneous Riemannian metric (or, if you think this way instead, Haar measure). ↩