Interesting article. I wasn't even properly familiar with triangulation of (convex) polygons. I knew it was possible because 3d games triangulate things all the time. What I didn't realise is that there is a fixed number of triangles in a triangulation of a polygon. I had assumed the word "minimal" had been missing at first, but looking at the pictures I realised it would be impossible to add an extra one (unless you introduce a vertex by intersecting a line with the middle of an existing line)
Interesting scaling as diffusion in a uniform space goes like x^2 ~ t, or t~x^1/2. So I’d expect the time to equilibrate to go like a length to the 1/2. Maybe that works out if the length is some power of n in this case? Or maybe this space is just very different from a uniform one.
The nth coefficient of the ith Mandelbrot polynomial lemniscate [3] is equal to the number of possible binary trees with n nodes of height i or less. This is because z -> z^2 + c reflects coefficients recursively produced by repeated applications of the binomial theorem.
When n becomes <= i, the nth coefficient stabilizes, converging to the nth Catalan number, which is the number of possible binary trees with n nodes. The nth Catalan number is also the number of triangulations of an n+2-gon.
What is the significance of all of this? There is currently no closed form approximation for these numbers. As simple as it might sound, we have no closed form for the number of binary trees with i nodes of height <= n. If we did, we could approximate the Mandelbrot and it would unlock new frontiers of understanding chaos theory.
