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 explain the basics of differential geometry and show some examples of how it arises in applied math. Then we specialize to surface theory and explain its relation to the problem at hand. Finally we finish with our current focus: and optimization problem formulated on piecewise solutions of the Sine-Gordon equation. This talk will be less formal than the analysis seminar talk and more focused on the basics of the theory we are interested in.