  We investigate the dynamics of cellular solidification patterns using
three-dimensional phase-field simulations. The cells can organize into stable
hexagonal patterns or exhibit unsteady evolutions. We identify the relevant
secondary instabilities of regular hexagonal arrays and find that the stability
boundaries depend significantly on the strength of crystalline anisotropy. We
also find multiplet states that can be reached by applying well-defined
perturbations to a pre-existing hexagonal array.
