Aristotle's wheel paradox
Aristotle's wheel paradox is a paradox or problem appearing in the Greek work "Mechanica" traditionally attributed to Aristotle. A wheel can be depicted in two dimensions using two circles. The larger circle is tangent to a horizontal surface (e.g. a road) that it can roll on. The smaller circle has the same center and is rigidly affixed to the larger one. The smaller circle could depict the bead of a tire, a rim the tire is mounted on, an axle, etc. Assume the larger circle rolls without slipping (or skidding) for a full revolution. The distances moved by both circles are the same length, as depicted by the blue and red dashed lines and the distance between the two black vertical lines. The distance for the larger circle equals its circumference, but the distance for the smaller circle is longer than its circumference: a paradox or problem.

The paradox is not limited to a wheel. Other things depicted in two dimensions show the same behavior. A roll of tape does. A typical round bottle or jar rolled on its side does; the smaller circle depicting the mouth or neck of the bottle or jar.

There are a few things that would be depicted with the brown horizontal line in the image tangent to the smaller circle rather than the larger one. Examples are a typical train wheel, which has a flange, or a barbell straddling a bench. Drabkin called these Case II and the kind in the image Case I. A similar but not identical analysis applies.
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paradox
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Greek
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Mechanica
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Aristotle
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circle
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circumference
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Aristotle's wheel paradox
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