### Abstract

Recent multi-frequency Faraday wave experiments have produced a wide variety of interesting patterns (hexagons, quasipatterns, superlattices, etc.), and shown that many of these are selected by resonant triad interactions. It turns out that general symmetry considerations (time translation, time reflection, and Hamiltonian structure) can be used to understand (and control), in part, a class of resonant three-wave interactions relevant to pattern formation in weakly damped, parametrically forced systems near onset. The most important damped, resonant modes can be determined, as well as the manner in which the corresponding triad interactions depend on the forcing parameters; the relative phases of the forcing components, for example, may be used to enhance or suppress particular nonlinear interactions. These symmetry-based predictions are compared with numerical and experimental results.