Strange Duality

This month's MCCC theme is "Go-To Polygon", which got me thinking about my favorite way of generating random polygons. As I've mentioned before, there is a correspondence between $n \times 2$ matrices with orthonormal columns (a.k.a., the Stiefel manifold $V_2(\mathbb{R}^n)$) and $n$-gons in the plane of perimeter 2. This is explained fairly abstractly in our paper and a little more concretely in this talk, but the upshot is that it's easy to generate random $n \times 2$ matrices with orthonormal columns, so it's easy to generate random $n$-gons.

Really, the particular $n \times 2$ matrix you pick doesn't matter: what's important is the plane in $\mathbb{R}^n$ that the columns span. If you pick a different (orthonormal) basis for the plane, you just rotate the corresponding polygon.

Now, in this story quadrilaterals are special. Since a quadrilateral corresponds to a 2-dimensional plane in $\mathbb{R}^4$, and since the orthogonal complement of a 2-plane in $\mathbb{R}^4$ is another 2-plane in $\mathbb{R}^4$, we see that in this formalism, every quadrilateral has a dual quadrilateral coming from orthogonal complementation.

In the case of the square, its dual quadrilateral is another square, given by reflecting in the $x$-axis. The above animation shows the one-parameter family of quadrilaterals given by the closed geodesic connecting the square to its dual. Here's the code: