TAUTOCHRONE

A curve on which an object falling under gravity will reach the bottom in the same amount of time, no matter from where it starts.

This may seem like the most abstruse of mathematical ideas, but it makes accurate pendulum clocks possible. It goes back to the seventeenth century Dutch scientist Christiaan Huygens, who knew that the pendulum is not quite the perfect keeper of time that one would like. It works well enough if its movements are kept small, but as the size of the pendulum’s swing varies, so does the time it takes — only slightly, but it’s a serious problem if you’re trying to make your clock precise.

Huygens discovered that there is one curved shape, and only one, which is perfect in this respect: the cycloid, the curve traced out when a point on the edge of a wheel rolls along a road. If you position a cycloidal curve like an inverted arch, and release a marble from any point on it, it will always take exactly the same time to reach the bottom, no matter where on the curve you start from. So the cycloid is said to be a tautochrone. Huygens used this discovery to construct curved jaws from the point of support of the pendulum; these forced its string to follow the right curve no matter how large or small the swing.

The word comes from the Greek tauto, “the same” (which we have inherited in words like tautology) and chronos, “time” (as in chronometer); so the word means a curve of equal time.

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