In solid-state physics, including photonics and wherever periodic lattice structures occur, it is essential to establish the fundamental features associated with wave propagation through the lattice: This is achieved using Bloch waves, the reciprocal lattice, and the reduction, using periodicity, to consider the irreducible Brillouin zone. A general approach, although widely accepted as not being perfectly legitimate, is to plot the dispersion relations around the edges of the Brillouin zone. We show definitively that this can be dangerous and that an important mode of practical significance is missed if this is done in too cavalier a fashion: This missing mode is illustrated for the design of endoscopes based on spring-mass (discrete) periodic structures and photonic crystals.