The brachistochrone (Greek: brakhistos "shortest" + khronos "time") is the curve along which a bead, sliding under gravity without friction, travels between two points in the least time. Surprisingly, it is not the straight line — it is an arc of a cycloid.

The 1696 challenge

Johann Bernoulli posed the problem in 1696 as a challenge to "the sharpest mathematicians in the world." Solutions came from Newton, Leibniz, l'Hôpital, Jakob Bernoulli and Johann himself — the birth of the calculus of variations.

Fermat's principle analogy

Johann's elegant trick reframed the falling bead as a ray of light. By energy conservation the speed is v = √(2g·h), so the bead moves faster lower down — like light through ever-faster media. Fermat's principle of least time then forces Snell's law, sinθ / v = const, at every point. The curve obeying this is exactly the cycloid.

The cycloid solution

A cycloid is traced by a point on a rolling circle of radius r:

x = r(θ − sinθ)
y = r(1 − cosθ)

Run upside-down from the start point, the cycloid threads through both endpoints and beats every other path — line, arc and parabola alike.

The tautochrone property

Huygens (1659) discovered the same cycloid is the tautochrone: a bead released from any point on the arc reaches the bottom in the same time, independent of where it starts. Switch to Tautochrone mode to watch beads released at different heights arrive together.

• Line — shortest path, but slowest.
• Arc — a circular dip, faster than the line.
• Parabola — close, but still not optimal.
• Cycloid — the brachistochrone, always wins.