The Fastest Way Down

1. The race

Three beads start at rest at point A. Each slides down its own track, with no friction, to point B, which is 2 m lower and π m (about 3.14 m) to the right. One track is straight, one is part of a circle and one is a cycloid. Which bead do you think gets to B first? Make a guess, then press Play.

Why the steep start wins. The cycloid drops almost straight down at first, so its bead gains speed at once and keeps that speed for the rest of the trip. The time saved by going fast early is more than the time lost on the longer path.

The cycloid is the curve traced by a point on the rim of a rolling wheel. Here the wheel has radius R = 1 m and rolls along the underside of the level line through A:

x = R(θ − sin θ),   y = −R(1 − cos θ),   0 ≤ θ ≤ π

Bernoulli's challenge. In June 1696 Johann Bernoulli challenged the mathematicians of Europe in the journal Acta Eruditorum: find the curve along which a body slides from one point to a lower point in the least time. He named it the brachistochrone, from the Greek for “shortest time.” Correct solutions came from Johann himself, his older brother Jakob Bernoulli, Gottfried Leibniz, the Marquis de l'Hôpital and Isaac Newton. Newton's solution was printed without his name. Every one of them found the same answer: the cycloid.

How this race stays honest. Nothing here is keyframed or timed by hand. With no friction, the only rule is conservation of energy: after a bead has dropped a height h, its speed is v = √(2gh), with g = 9.81 m/s². The page cuts each track into 4,000 tiny pieces, divides each piece's length by the bead's speed there, and adds up the times. The beads are then drawn where those running totals say they are at each moment on the clock. For the straight line this gives t = L√(2 ÷ (g h)) ≈ 1.189 s, and for the cycloid t = π√(R ÷ g) ≈ 1.003 s, just as the formulas say.

2. Same time from any height

Here is the same cycloid, drawn as a full bowl. Four beads are let go at the same instant, each from a different height. The highest bead has the farthest to go. Which one reaches the bottom first?

They all arrive together. A bead that starts higher has farther to go, but it starts on a steeper part of the curve and moves faster the whole way. On a cycloid these balance exactly, so every bead takes π√(R ÷ g) ≈ 1.003 s, from any starting point. That is why the cycloid is also called the tautochrone, “same time.”

Huygens and the clock. Christiaan Huygens proved this property of the cycloid and published it in 1673 in his book on pendulum clocks, Horologium Oscillatorium. A pendulum bob that swings along a cycloid keeps the same time whether its swings are large or small.

3. Quick check