Rolling a Coin around a Coin

The red circle rolls on the outside of the black circle of radius 1. (The blue point shows how it rolls.) An invisible point at distance from the center of the black circle is attached to the red circle, tracing out a path as the red circle rolls; such curves are called epitrochoids.

When the circles have the same radius, you can see that the blue point is oriented in the same direction after half a revolution. Therefore a coin rolling completely around another coin of the same size goes through two complete revolutions.