Aristotle's wheel paradox
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Aristotle's wheel paradox is a paradox from the Greek work Mechanica. There are two wheels, one within the other, in other words two circles with different diameter. By projecting outwards from a point within both circles, a bijection between the two circles is established. Thus for each point in the small circle, there is exactly one point in the big circle and vice versa. The paradox arises when a line on which the circles travel without slipping is also visualized as a set of points. When a wheel travels, each of its points corresponds to a point on the line and since the wheels have the same number of points they should travel the same distance per revolution. Thus they would have the same circumference, which is a contradiction to our assumption that they have different sizes.
The fallacy is the assumption that, if the one-to-one correspondence is correct and the two curves have the same cardinality then they must also have the same length. Physically, if two joined concentric wheels with different radii were rolled along parallel lines then at least one would slip; if a system of cogs was used to prevent slippage then the wheels would jam.
[edit] See also
[edit] External links
- MathWorld's explanation including a picture of the problem
- Rota Aristotelica, The Archimedes Project, Digital Research Library
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