Thought Toys · Strategy & computation · Exhibit 103
Three frictionless roads, one bead each, gravity only. The straight road is the shortest — and it loses. The winner starts steep, dives under its own finish line, and climbs back up. Bend the third road yourself and try to beat it.
The race — same start, same finish, speed from depth alone cycloid straight yours
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A bead slides from the upper peg to the lower one with no friction and no motor. Gravity is the only engine, so the bead's speed comes from one thing: how far it has fallen. Deep road, fast bead. The blue road is the shortest possible route — and press race: it loses, every time.
The reason is that early depth is an investment. A road that plunges at the start buys speed in the first instant, and the bead keeps that speed for the whole trip. Too much plunge wastes distance, though — a straight-down-then-flat route loses too (1.13 s against the winner's 1.02 here). The perfect trade is the amber curve, a cycloid: the path a point on a rolling wheel's rim traces. Johann Bernoulli posed this race as a public challenge in 1696. Newton solved it in a night.
Don't take the curve's word for it — the cyan road is yours. Its slider sweeps a whole family of curves, from lazy-start, through the straight line at sag 1, to plunges far steeper than the cycloid's own start. Hunt for the fastest member, or press best version of my road and let the page find it by golden-section search. The family's own optimum sits inside the slider at every finish width, so that button really is finding a best, not stopping at the edge of the track. It still arrives late. That is the brachistochrone fact: not that bending helps, but that one specific bend beats every other road through the same two pegs. The best parabola gets close — a few hundredths of a second — and close is still losing.
Now the strangest part. Drag the finish wider and watch the amber road: past a width-to-depth ratio of π/2, it dives below the dashed line — deeper than the finish itself — then climbs back up to arrive. Climbing spends speed, and it wins anyway. The extra depth made the long middle of the trip so fast that the toll at the end is repaid with interest. Bernoulli's own proof saw the road as a light ray crossing layers of ever-thinner glass: bending exactly as Snell's law bends light keeps v/sin α constant, and only the cycloid manages it — on the straight road that ratio drifts 280%.
improve/verify/103-brachistochrone.js, 19 checks) verifies the closed form against an
independent quadrature to twelve digits, matches this page's bead integrator to both, defeats the straight
line, the greedy plunge, every parabola and every power-law ramp, confirms the invariant to ten digits and
its failure on the straight line, and runs two negative controls: with gravity off the straight line wins,
and under a wrong law where speed follows the clock instead of depth, the podium inverts exactly.
Also in Strategy & computation: Sorting algorithms →
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