Can you color the plane with three colors so that every equilateral triangle with sides of length 1 has one vertex of each color? In the talk I'll answer this question and describe several generalizations. Some of these have implications for the foundations of quantum mechanics (the Kochen-Specker theorem). I'll explain these using only linear algebra and some elementary number theory, without assuming any knowledge of quantum mechanics. Recent comments about these observations invoke Euclid's Postulates. I'll conclude with a brief discussion of this connection. Refreshments will be served.