Take two gears: One is fixed, the other rotates around it. If the gears are the same size the rotating gear has to rotate around its center twice as fast as it rotates around the other gear. I'm looking for intuition or reasoning why this is the case.
As you turn the gears, the point where the outer gear touches the inner gear will travel around the circumference of the inner gear, making one full rotation.
But the centre of the outer gear is twice as far (because the gears are of equal size), so it will travel twice as far too. (That is, twice the circumference of the gears.)
And because there's no slippage, this means the outer gear will have to rotate twice too.