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Roulette: A Comfortable Ride on an n-gon Bicycle

A special case of a roulette curve involves an improvised bicycle with two -gonal wheels rolling on a catenary-stitched ground. The wheels' centroids follow a straight line to ensure smooth cycling. However, rolling on triangular wheels is physically impossible. To see why, zoom in and move the time slider, possibly holding down the Alt key for fine adjustments.

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Wishing to ride comfortably on a bicycle with -gonal wheels is a true dilemma: a lemma with two solutions. Miki Muster, a Slovenian cartoonist, suggested the first solution in his comic To the Moon, which featured three buddies venturing to the Moon and meeting its inhabitants, who drive around noisily and ignorantly on square-wheel bikes. One of the heroes shows them the beauty of a circular wheel.
The second solution was suggested by a mathematician, Stan Wagon (see this on his homepage), who built a square-wheel bike and catenary ground to earn himself an entry in Ripley's Believe It or Not.

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