Photonic crystals are a powerful tool for the manipulation of optical dispersion and density of states. By utilizing the tools of lattice simulation, we can engineer photonic systems that have a variety of interesting band structures. By using many sided regular polygons along with other lattice types, band structures with large band gaps and multiple flat bands can be created. As well, the unique control provided by these media makes a playground for strong-coupling quantum electrodynamics where a single, highly nonlinear emitter hybridizes with the band structure of the qubit.