We consider the spectrum of the Schrödinger operator on an octagonal lattice using the Floquet-Bloch transform of the Laplacian. We will first consider the spectrum of the Laplacian in detail and prove various properties thereof, including spectral-band limits and locations of singularities. In addition, we will prove that Schrödinger operators with 1-1 periodic potentials can open at most two gaps in the spectrum precisely at energies $pm1$, and that a third gap can open at 0 for 2-2 periodic potentials. We describe in detail the structure of these operators for higher periods, and motivate our expectations of their spectra. / Master of Science / In quantum physics, we would like the capability to model environments, such as magnetic fields, that interact with electrons or other quantum entities. The fields of graph theory and functional analysis within mathematics provide tools which relate well-understood mathematical concepts to these physical interactions. In this work, we use these tools to describe these environments using previously employed techniques in new ways.