We consider the problem of Gaussian beam scattering by finite arrays of pinned points, or platonic clusters, in a thin elastic plate governed by the biharmonic plate equation. Integral representations for Gaussian incident beams are constructed and numerically evaluated to demonstrate the different behaviours exhibited by these finite arrays. We show that it is possible to extend the scattering theory from infinite arrays of pinned points to these finite crystals, which exhibit the predicted behaviour well. Analytical expressions for the photonic superprism parameters $p$, $q$ and $r$, which are measures for dispersion inside the crystal, are also derived for the pinned plate problem here. We demonstrate the existence of negative refraction, beam splitting, Rayleigh anomalies, internal reflection, and near-trapping on the first band surface, giving examples for each of these behaviours.