# Does smoother objects have more chance of skipping than skidding?

Recently I was viewing this video which discusses Galileo's solution to Aristotle's wheel paradox and it got me thinking about when rolling turns into skidding. Why do geometric shapes with a finite number of edges skip and when an object has its several sides going to infinity, it skids.

Is there a precise number of sides to for which skips turn into skids in rolling motion? Or is this a consequence of the limiting behavior? How?

Reference:

Around 3:00 of this video

The skipping happens because the angle between the sides of a polygon is $$<180^\circ$$ and to proceed with rotation it has to rotate the remaining angle on its vertex. For it to be a smooth rolling motion that angle should be $$180^\circ$$ on a flat surface because that's the angle made by a straight line. That only happens when you have a smooth surface like a circle or ellipse, not pointy polygons. Theoretically, any $$n$$-sided polygon no matter how large $$n$$ is will always skip. You might not be able to tell the difference between a 50 sided coin but it's still there. If the coin was way bigger then you would notice the skip.