It's the end of a long action movie. The hero and villain are in a mad chase over a city skyline to deactivate and trigger a nuclear bomb, respectively. The villain throws himself down on a daringly curved piece of architecture with the intent to slide to the bottom of the curve and reach the trigger first. The hero (probably played by Angelina Jolie doing parkour) throws herself into the air and hits a lower part of the curve, intending to get ahead of the villain. Alas! As they slide, the villain catches up to her, and they both end the curve at the same time. The situation can now only be resolved via fistfight with sweet flips.

Angelina has become the latest victim of the tautochrone, the most science-fiction-sounding concept ever to be invented in the 1600s. Scientists wanted a special kind of curve; one that would allow an object set down along its surface to reach the bottom in the same amount of time no matter where it was placed (using gravity alone).

Given a small amount of thought, it's clear what the approximate shape of a tautochrone should be. If something let go from a height of five feet reaches the ground at the same time as something let go from a height of five inches, the curve has to be steep high up and shallower as it moves closer to the ground. The high object will gain momentum on the steep high curve to catch up to the slower-moving object on the shallow low curve. Getting the specifics was a little harder, but in the mid-1670s, a Dutch clockmaker named Christiaan Huygens got it right. He noted that the curve should be described by a point on the edge of a wheel that was rolling in a straight line.

As shown before, when wheels move along the ground, a point along the edge of them moves at radically different speeds with respect to the ground, even though it goes at a constant speed with respect to the center of wheel. When the point is at the midpoint of the front or back edge of the wheel, it's moving perpendicular to the ground. At the top it's moving parallel to the ground. In mathematics, the shape made by the motion of the point is called a cycloid — it's the second most scifi-sounding word to actually be ancient.

The tautochrone is — for the most part — an intellectual exercise. Although Huygens tried to use it to make a more reliable pendulum clock — one that would keep time even if the pendulum was started in the wrong place — friction intervened. The curve only works perfectly if its surface is entirely frictionless. Still, it's an elegant idea, and it shares an understanding with one of the most famous theories of all time. When Einstein bent time like a cheap paperclip, he did it in a theory that can be illustrated by equating motion with time. (A pendulum on a clock that was placed on a train going the speed of light would be, on the forward swing, going faster than the speed of light itself. Since that is not possible, the only way for the motion to make sense is if the objects on the train were motionless to the outside observer. The only way for motion to be possible was if time itself was frozen, from the point of view of the observer.)

In making the tautochrone, scientists and mathematicians were essentially making a shape that was an equivalent to a certain amount of time. Depending on the size of the tautochrone, it's a five second shape, a ten second shape, or a five minute shape. It's what a certain amount of time, under earth's gravity, "looks like."

[Via Wolfram Math World]

## DISCUSSION

I think the tautochrone curve is good example of how physics messes up the beauty of a great mathematical idea. If the curve were perfectly frictionless, it would work just like the math says it should but alas reality is not so kind.

Anyway, I bet if we looked we might find a few of these curves into the bowls of skate parks.