What's Up , Moonface?

The moon, viewed from the earth, always presents the same face; but this face is perceived as tilted at different angles to different observers at different times. Given a few (oversimplified) astronomical facts, it becomes a purely mathematical problem to describe this moontilt. In this article we show how to set up and solve this problem, using vectors in three-space and onedimensional calculus. The solution reveals some fascinating qualitative features-such as the fact that at times the moon does the twistM- that are subject to simple observational verification. My goal is to show that interesting mathematical problems can be found in everyday life.