We examine periodically structured media. When periods of the structure are on the same length scale as the wavelength then destructive interference can occur. This gives rise to frequency intervals where no waves can propagate inside the material. These are the well known photonic band gap crystals. These crystals offer great promise in new technology and also occur naturally and can be seen in the structural coloration of butterfly wings. In this lecture we highlight auxiliary spectral problems directly related to the physical structure of these materials. For 2 and 3-D crystals the spectra is connected to the union of structural resonances that correspond to parallelepipeds of every size and aspect ratio containing integral number of lattice periods. Control of band gaps requires characterization of resonances associated with the total collection of these parallelepipeds of every size and shape. These resonances correspond to the eigenvalues of the quasi-periodic Neumann-Poincare operator. We illustrate how this spectra can be used as a new tool in the design of photonic band gap crystals.