I think of these as a Spirograph, but today got thinking about these epicycloids and hypocycloids because of Copernicus' Theorem http://www.cut-the-knot.org/pythagoras/Copernicus.shtml and the Coin Rolling Paradox http://curiosamathematica.tumblr.com/post/112681620692/coin-rotation-paradox. What patterns do you see? Can you predict the patterns based on the circle sizes? How many times do you have to roll around to get back to the starting position?
More GeoGebra at bit.ly/mh-ggb