This is a translation of an article that appeared in the April 2000 issue of Pythagoras. It is fourth in a series of six about curves and their names.
The shape of a skateboard ramp can be deduced from its name `half-pipe': half of a cylinder. Is a semicircle the best cross section? A cycloid would be more logical: the curve enables you to achieve a perfect rhythm. But beware: the cycloid also gives you the shortest oscillation time.
In 1969 Johann Bernoulli challenged the learned minds of the world to solve the following problem. You are given two points on a wall. The task is to find a curve along which an object would slide, without friction, in the shortest possible time from one point to the other. He called this curve the brachistochrone, after the Greek words brachos (short) and chronos (time). When he published his solution some time later he wrote: "Dear reader, you will be astonished to learn that this curve, the brachistochrone, is nothing else than the tautochrone of Huygens."
What was going on? A few years earlier, in 1673, Christiaan Huygens had investigated how you could make a pendulum clock run `like clockwork', that is, with a constant frequancy, independent of the amplitude. In a normal pendulum — a weight on a rope that is suspended from a fixed point — the frequency depends on the amplitude: the larger the amplitude, the larger the period. That is not very useful if you want to make a pendulum clock. Huygens discovered that if you don't guide the weight along a circle but along a cycloid then the period (and frequency) would be independent of the amplitude. That is why the cycloid is also called the tautochrone, from tautos (equal) and chronos (time).
The cycloid (wheel curve) was, in those days, already a well-known curve. You get it by having a wheel roll, without slipping, along a straight line and following the orbit of one point on that wheel. If you ride a bike for example then the orbit of the nipple will be close to a cycloid (but not quite: the nipple never touches the ground). You can parametrize the curve as follows:
You can read this off from the following picture, r is the radius of the rolling circle.
If you hang a cycloid upside-down then you get the real tautochrone curve. It has the remarkable property that no matter how high (or low) you start, it will take the same amount of time to slide, with friction, to the lowest point. In many science museums you can see that a cycloid gives a much more efficient slide than, for example, a straight line: of two balls that are released simultaneously the one on the straight line takes noticeably more time to reach the bottom.
For every end point A there is exactly one radius r such that the corresponding cycloid passes through A.
The surprising conclusion is that the optimal curve sometimes has to go lower that the end point; the speed that you gain apparently compensates for the loss when going back up again.
Huygens also discovered how you can make the weight of your pendulum move along a cycloid: take two cycloid of the same size, hang them next to each other and fix the rope to the point where they meet. The rope should be half as long as the cycloids. Because the rope is restricted in its movements by the cycloids the weight will describe a third cycloid, of the same size as the two given ones.
Johann Bernoulli was not the only one to solve his own problem. Many others, among which his brother Jakob and Isaac Newton, succeeded in describing the brachistochrone.
