Aristotle's Wheel Paradox | STEM Montessori Toy

There is a dilemma represented in this wooden puzzle that will have you wondering how it can be possible. You won't be the first. Aristotle took a whack at this in the Aristotelian Mechanica, as well as in the Mechanica of Hero of Alexandria and Galileo gave it a try as well. See what you can make of it...

Not everyone agrees that Aristotle invented this little paradox, but everyone agrees that it would be just like him to come up with something like this. The paradox involves two different-sized wheels, one inside another. The two rotate in sync and you can see as you turn the wheels to travel the length of the opening in the puzzle that the etched line on the wheels turns one time, thus the circumference of each wheel has rotated over an equal length.

Both wheels use exactly one circumference to trace the same amount of distance - the length of the space in the puzzle. But, clearly one circumference is smaller than the other. Either that means that the wheels have the same circumference, which they don't, or that different circumferences "unroll" to the same length, which they can't.

That can't possibly be right.... The smaller radius can't possibly be equal to the larger one, so what's going on?

To further illustrate the paradox, I have included a separate copy of the inner wheel. We know that when attached to the bigger wheel that it turns one rotation to cover the length of the opening starting at the hash mark. But now take the small wheel and position it at the same starting point and rotate one revolution.... it does not travel the same distance! Such a dilemma.... Can you explain it??

This wooden model made in our Hudson Florida shop nicely presents this paradox in a physical format that will likely leave you or friends or students struggling for an explanation. Great fun for thinkers!

Measures about 11" x 3.5"