  In recent years, nanomechanics has evolved into a mature field, with
wide-ranging impact from sensing applications to fundamental physics, and it
has now reached a stage which enables the fabrication and study of ever more
elaborate devices. This has led to the emergence of arrays of coupled
nanomechanical resonators as a promising field of research, serving as model
systems to study collective dynamical phenomena such as synchronization or
topological transport. From a general point of view, the arrays investigated so
far represent scalar fields on a lattice. Moving to a scenario where these
could be extended to vector fields would unlock a whole host of conceptually
interesting additional phenomena, including the physics of polarization
patterns in wave fields and their associated topology. Here we introduce a new
platform, a two-dimensional array of coupled nanomechanical pillar resonators,
whose orthogonal vibration directions encode a mechanical polarization degree
of freedom. We demonstrate direct optical imaging of the collective dynamics,
enabling us to analyze the emerging polarization patterns and follow their
evolution with drive frequency.
