  We present measurements on parametrically driven surface waves (Faraday
waves) performed in the vicinity of a bi-critical point in parameter space,
where modes with harmonic and subharmonic time dependence interact. The primary
patterns are squares in the subharmonic and hexagons in the harmonic regime. If
the primary instability is harmonic we observe a hysteretic secondary
transition from hexagons to squares without a perceptible variation of the
fundamental wavelength. The transition is understood in terms of a set of
coupled Landau equations and related to other canonical examples of phase
transitions in nonlinear dissipative systems. Moreover, the
subharmonic-harmonic mode competition gives rise to a variety of new
superlattice states. These structures are interpreted as mediator modes
involved in the transition between patterns of fourfold and sixfold rotational
symmetry.
