Plotting daily stock returns against themselves with one day's lag reveals a striking pattern. Evenly spaced lines radiate from the origin; the thickest lines point in the major directions of the compass. This “compass rose” pattern appears in every stock. It is caused by discreteness. However, counter-examples demonstrate that the existence of exchange-imposed tick sizes (e.g. eighths) is neither necessary nor sufficient for the compass rose. The compass rose cannot be used to make abnormal profits: it is structure without predictability. Among other consequences, the compass rose may bias estimation of ARCH models, and tests for chaos.