Tai chi as a martial art seems far removed from actual fighting, since the motions are slow to the point of static. However, a few years ago during a conversation with Debra Page we came up with a cool analogy to describe part of why studying tai chi is valuable to the full speed case. I never got around to writing it up, so here goes: tai chi is like calculus.

Currently, we (or at least I) don’t know how to do multigrid on general problems, so we’re stuck using conjugate gradient. The problem with conjugate gradient is that it is fundamentally about linear systems: given $Ax = b$, construct the Krylov subspace $b, Ax, A^2 x, \ldots$ and pick out the best available linear combination. It’s all in terms of linear spaces. Interesting human scale physics is mostly not about linear spaces: it’s about half-linear subspaces, or linear spaces with inequality constraints.

After extensive testing with glasses of water, I’ve determined that it’s impossible to stably balance one infinitely thin circle on top of another. This is true whether the circles are the same size or different, and does not depend on the mass distribution of the circles. This is because two circles intersect at either zero or two points (unless they are the same, and that still isn’t a stable configuration).