Sunday, March 6, 2011

Of Wheels and Circles and Points, Oh My!

Steve Colyer, at "Multiplication By Infinity" blog, tackles Aristotle's classic "Wheel paradox" here:

The above image-in-motion makes the paradox especially powerful (making two clearly different circumference sizes seem equal).

In a brief search, I couldn't find a really good explanation of the paradox for the layman, on typical math sites. Bryan Bunch treats it as well as any in the first chapter of his old volume "Mathematical Fallacies and Paradoxes," which is available online here:

The explanation lies in the need to follow the path of a single point on a circumference as it moves from point A to point B, following its actual curved or cycloidal path and NOT a straight line, in order to understand the relative unwinding of the two circumferences.
The paradox also touches upon the topic of infinity and cardinality (and their fuzziness in our mind!).

Some of the old paradoxes are still among the best!

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