Waves are all around us, in ocean waters and wind-swept sands, in light and in sound, in pressure waves, wi-fi, cellular networks, and radio signals, and in patterns of mathematics and biofields of living things. We begin with some facts about familiar waves, and we consider the wave equation that explains how they travel. The simple wave equation leads to intriguing mathematical patterns tied to a new concept of what it means for paths to cross at right angles. We discover a new world of waves where right angles are not right.