Aristotle's wheel paradox

Aristotle's wheel paradox is a paradox from the Greek work Mechanica traditionally attributed to Aristotle. There are two wheels, one within the other, whose rims take the shape of two circles with different diameters. The wheels roll without slipping for a full revolution. The paths traced by the bottoms of the wheels are straight lines, which are apparently the wheels' circumferences. But the two lines have the same length, so the wheels must have the same circumference, contradicting the assumption that they have different sizes: a paradox.

The fallacy is the assumption that the smaller wheel indeed traces out its circumference, without ensuring that it, too, rolls without slipping on a fixed surface. In fact, it is impossible for both wheels to perform such motion. 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. A modern approximation of such an experiment is often performed by car drivers who park too close to a curb. The car's outer tire rolls without slipping on the road surface while the inner hubcap both rolls and slips across the curb; the slipping is evidenced by a screeching noise.

Mathematically, the number of points in the inner circle is equal to the number of points in the outer circle, i.e. there is a bijection between them. This does not apply to physical wheels because they are made of discrete atoms, so the number of atoms of the larger wheel is greater if the wheels have the same density, width and thickness (only their radii are different).

Further reading

• Rota Aristotelica, The Archimedes Project, Digital Research Library
• Weisstein, Eric W., "Aristotle's Wheel Paradox" from MathWorld.