Abstract: Rational functions have extraordinary power for approximation of functions with branch point boundary singularities. Thanks to the exponential clustering of poles, their convergence is root-exponential, i.e., exponential in the square root of the degree. This was discovered by Newman in 1964 for approximation of |x| on [-1,1] but only recently applied to the development of "lightning solvers" for Laplace, biharmonic, and Helmholtz problems in planar regions with corners. We will present some of these fast new numerical methods and then focus on a mathematical aspect that turns up wherever you look: why do those exponentially- clustered poles space out super-exponentially near the singularity in a manner described by a certain square root formula? This is joint work with Yuji Nakatsukasa and Andre Weideman.