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:
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