An interesting query by Bob High, posed in the May-June 1994 issue of MIT Technology Review: Suppose a billiard ball with a small black dot precisely on its top is rolled around the full circumference of a circle of the same radius. Assuming no slippage or twisting, where is the dot when the ball returns to its starting point?
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As you might guess, the dot is back on top.
“Surprisingly, however, the ball makes two complete revolutions in achieving this result,” writes Puzzle Corner columnist Allan Gottlieb. “This can be seen by comparing the described situation with the cycloid generated by the black dot, if the ball were rolling on a flat surface. When the ball is halfway through its travels, the dot touches the fixed surface, which is at the top of the ball for the actual circular surface. Hence, one complete revolution has occurred when the bottom of the circle is reached, and the second revolution occurs in the remaining travel back to the top of the circle.”
The same puzzle has been rolling around in two dimensions since at least 1868.
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