Aristotle's Wheel Paradox

[fdesilva]

Weisstein, Eric W. "Aristotle's Wheel Paradox." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Aristotles ... radox.html

A paradox mentioned in the Greek work Mechanica, dubiously attributed to Aristotle. Consider the above diagram depicting a wheel consisting of two concentric circles of different diameters (a wheel within a wheel). There is a 1:1 correspondence of points on the large circle with points on the small circle, so the wheel should travel the same distance regardless of whether it is rolled from left to right on the top straight line or on the bottom one. this seems to imply that the two circumferences of different sized circles are equal, which is impossible.
The fallacy lies in the assumption that a one-to-one correspondence of points means that two curves must have the same length. In fact, the cardinalities of points in a line segment of any length (or even an infinite line, a plane, a three-dimensional space, or an infinite dimensional Euclidean space) are all the same: (the cardinality of the continuum), so the points of any of these can be put in a one-to-one correspondence with those of any other.

I would explain the above paradox as follows. Consider the inner circle fixed and forming an axle, around which the outer circle is rotating. Then this inner axle will be seen as slipping along the rail. As such only a single point would be in contact with all the points on the rail. Now if this axle were to also rotate very slowly, then more than one point on it will be mapped to the rail. However each point on the axle will get mapped to several points on the rail. It is best seen as slipping and turning at the same time.  
Thus if the bigger wheel is driving then each point on it circumference will map to a point on the rail. While in the smaller circle a given point will map to more than a single point of its rail as it is also slipping.
If on the other hand the smaller wheel is driving, the distance along the rail travelled by one turn would be reduced to that of the length of the smaller circle. In this case while the points on the smaller circle will map to the reduced rail distance, many points of the lower rail will map to single point on the large circle.

[elliebean]

Why complicate it? The points on the rail correspond equally to the center point of both circles. It's simple and not at all paradoxical. What more is there to know about it?

[fdesilva]

Why complicate it? The points on the rail correspond equally to the center point of both circles. It's simple and not at all paradoxical. What more is there to know about it?

Yes, thats a great way to look at it. The reason for the complication is the way it is demonstrated, the depiction is more a illussion than a paradox.