  In graphene growth, island symmetry can become lower than the intrinsic
symmetries of both graphene and the substrate. First-principles calculations
and Monte Carlo modeling explain the shapes observed in our experiments and
earlier studies for various metal surface symmetries. For equilibrium shape,
edge energy variations $\delta E$ manifest in distorted hexagons with different
ground-state edge structures. In growth or nucleation, energy variation enters
exponentially as $\sim e^{\delta E / k_{B} T}$, strongly amplifying the
symmetry breaking, up to completely changing the shapes to triangular,
ribbon-like, or rhombic.
