"Three Points are taken at random on an infinite Plane. Find the chance of their being the vertices of an obtuse-angled Triangle."

 

This is the text of Lewis Carroll's Pillow Problem #58, from 1884. Of course, the obvious probabilistic interpretation of the problem is invalid, since the uniform distribution on the plane is not a probability measure. Various authors have given answers when the points are independently chosen from a Gaussian distribution, or uniformly from some bounded, convex domain, but this seems unnatural: surely the problem Carroll is getting at is that of choosing random triangles, not random points from some tractable distribution on the plane.

 

As Stephen Portnoy observed in 1994, the problem of choosing random triangles is problematic largely due to the apparent lack of a natural transitive group action on the set of triangles. In this talk, we show how to construct a measure on the set of triangles with exactly such an action by identifying triangle space with a particular Grassmannian. We can then give a precise answer to Carroll's question and indeed answer a number of related questions about planar n-gons. This approach also generalizes to random polygons in space, which are used to model ring polymers in biophysics, and to random closed smooth curves both in the plane and in space. This is joint work with Jason Cantarella (University of Georgia), Thomas Needham (Ohio State), and Gavin Stewart (NYU).