The wheels of Aristotle

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Sometimes you come across a thing where beauty matches simplicity, and Aristotle’s Wheel Paradox is just such a thing.

I came across it this afternoon after I returned to London from two too-short weeks away and a second-hand book I’d ordered, Wheels, Life and Other Mathematical Amusements, based on Martin Gardner‘s columns for the Scientific American.

I bought it because it has three chapters on Conway’s Game of Life – but the very first pages introduced me to Artistotle’s Wheel Paradox – which I’ll now explain…

Imagine, as in the diagram above a small wheel that runs in parallel (eg., because it is attached) to a larger wheel. The large wheel runs on the line A to B, the smaller runs on C to D.

At every point the large wheel touches the line AB, an equivalent point on the small wheel touches CD: so there is a fixed and unique equivalent point on the small wheel for every point on the large wheel.

So that means when the large wheel has completed a revolution then so will the small wheel.

But if there is a fixed and unique equivalent point on the small wheel to every point on the large wheel that must surely imply their circumference is the same?

This is the paradox. And it’s one that puzzled many of the mathematical greats of past ages.

The solution is down to Georg Cantor and his formulation of the continuum. For there are an uncountable infinity of points on the rim of the two wheels. (Gardner describes this as Cantor’s Aleph-One$\aleph_1$ – and indeed Cantor thought it so – this is his continuum hypothesis – but it remains unproven that the continuum is indeed $\aleph_1$ as opposed to a higher $\aleph$.)

One final note – the book I bought is stamped “University Library Hull Withdrawn” and an online search suggests that this book is indeed no longer available to students at Hull. But how can it be justified that a major university gets rid of books in this way? Their loss, my gain though, of course.