  In this short, conceptual paper we observe that essentially the same
mathematics applies in three contexts with disparate literatures: (1) sigmoidal
and RBF approximation of smooth functions, (2) rational approximation of
analytic functions near singularities, and (3) $hp$ mesh refinement for
solution of PDEs. The relationship of (1) and (2) is as simple as the change of
variables $s = \log(x)$, and our informal mnemonic for this relationship is
``sigmoid = log(ratapprox).''
