The world around us is filled with curly, complex, and crenellated forms: the leaves of lettuce, the petals of flowers, and even the feet of sea slugs! Why do such surfaces arise and how can we model them mathematically? We take a variational approach and model observed surfaces as minimizers of an elastic energy functional. This leads to subtle and surprising mathematical results on the regularity of the minimizers. This talk establishes the relevant background to study such surfaces and then focuses on the sub-problem of constructing approximate minimizers. The latter turns out to be related to the Sine-Gordon equation and an optimization problem formulated on piecewise solutions to the Sine-Gordon equation. We explore this optimization procedure and the dynamics relevant to understanding it.