We describe challenges and opportunities that arise if one attempts to visualize number theory. The goal is to create graphics which convey a theorem or argument, which tell a story, which are clear and honest, and which enable exploration. In the context of statistics, this is the mainstay of data visualization. In the number-theoretic context, problems of visualization are less familiar but tractable. From our pursuit of graphical excellence, we derive a few broad principles for the visualization of pure and experimental mathematics.