Pattern formation in buckled membranes was studied along with the morphology of flowers formed at the tip of silicon nanowires and ripples formed in suspended graphene sheets. Nash's perturbation method was tested for a simple case where initial and final metrics embed smoothly and there is a smooth path from one surface to another and was found to work successfully. The method was tested in more realistic conditions where a smooth path was not known and the method failed. Cylindrical flower-like membranes with a metric of negative Gaussian curvature were simulated in three and four dimensions. These four dimensional flowers had 2 orders of magnitude less energy than their three dimensional counterparts. Simulations were used to show that the addition of a fourth spatial dimension did not relieve all bending energy from the cylindrical membranes. Patterns formed at the tip of silicon nanowires were studied and found to be of the Dense Branching Morphology type. The rate of branching is dependent on the curvature of the gold bubble on which they are grown. Graphene was simulated using the modified embedded atom method potential and buckles were found to form if the carbon bonds were stretched. An energy functional was found for the energy of a sheet with a metric different from that of flat space. / text