There is an intuitive way to see why this is true. Picture a circle rolling around the outside of another. Easy to picture, right? Even if it’s not clear how m

There is an intuitive way to see why this is true. Picture a circle rolling around the outside of another. Easy to picture, right? Even if it’s not clear how many times it rotates, you can still visualize it. Now imagine the outer circle has to spin around the inside of the first circle. Stack two quarters, and try to spin the top quarter around the inside of the bottom quarter. You can’t do it. There’s no room. If they’re identical size, the top quarter can’t spin "around" the inner edge of the bottom quarter. Now put a penny on top of a quarter. Spin that around. Not along the outside, but along the inside of the quarter. You’ll see the penny is tracing a very tight circle. There’s not much room. Now put the penny next to the quarter and spin it around the outer edge of the quarter. It goes easily. Plenty of room. If you do the naive calculation, you’d get the same answer for both cases. But clearly there’s a difference. The difference is more apparent if you imagine the coins had little teeth, like gears. If you go around the outside, it’ll go just fine. But if you hollow out the quarter and put teeth along the inside, and try to spin the penny around in that, you’ll find that it doesn’t make anywhere close to a full rotation each time you go around.