Huygens' synchrony: Emergent modes of planar symmetrically coupled pendula.

We study the collective dynamics of metronomes coupled through a common platform that is free to move in two spatial dimensions, extending Huygens' classical experiment on synchronizing pendulum clocks. Allowing planar motion of the coupling medium gives rise to emerging robust synchronization patterns. Experiments reveal a geometry-induced transition between two qualitatively distinct collective states: a translational mode, in which the metronomes synchronize with shifted phases and drive planar translation of the platform, and a rotational mode, characterized by in-phase synchronization and oscillatory rotation about the platform's center of mass. The transition between these modes is controlled by the spatial arrangement of the oscillators and is naturally interpreted as a symmetry-related bifurcation. First-principles multibody simulations reproduce the observed behaviors, while application of the Harmonic Balance method enables analytical continuation and characterization of both stable and unstable synchronized solutions. These results highlight how multidimensional dynamical coupling and symmetry alone can organize collective motion, providing a minimal mechanical setting for studying emergent synchronization phenomena in nonlinear oscillator networks.