I just watched this video, and I'm a bit perplexed.
The radius of Circle A is 1/3 the radius of Circle B. Circle A rolls around Circle B one trip back to its starting point. How many times will Circle A revolve in total?
The intuitive answer is 3, but the correct answer is 4. I understand the trick -- that the center of Circle A must travel a distance of $2\pi(r_B + r_A)$, not $2\pi r_B$ -- but I'm still confused on one item.
At the risk of sounding very un-mathematical, how do the (infinite set of) points on the circumference of each circle map to each other to accomplish this?
Consider Circle A rolling along a straight line the length of the circumference of Circle B. Then it will revolve 3 times. It's like the universe "knows" when to apply a different point mapping when you change the arrangement of matter.