Aristotle's Wheel Paradox

This Demonstration shows Aristotle's wheel paradox. Although the concentric circles have differing diameters, the parallel straight lines produced by rolling are of equal length, suggesting that the circles have equal circumferences. This is, however, an illusion, since the smaller circle is being dragged as it rolls, causing it to slide in order to compensate for the larger distance traced by the larger circle.
This sliding behavior is more easily understood by observing the analogous figures using polygons. With each polygon, it can be seen how the inner polygon, when rotating with the outer polygon, must be lifted in order to compensate for the extra distance covered by the outer polygon. For the cases of a triangle through a hexagon, each inner polygon "hops" an equal horizontal distance but a decreasing vertical distance (the height of the red bumps decreases while the length remains constant). Therefore, as the number of sides goes to infinity, producing a circle, the vertical distance traveled (height of the bump) goes to zero, and the inner shape slides rather than hops.


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This was a project for Advanced Topics in Mathematics II, 2017–2018, Torrey Pines High School, San Diego, CA.
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