Brachistochrone — the fastest slideRelease a bead from the top corner and let gravity pull it to the lower one. Which ramp gets it there first? Not the straight line, and not the steepest plunge — the winner is a very particular curve called a cycloid, the path traced by a point on a rolling wheel.Johann Bernoulli posed this as a public challenge in 1696. His trick: treat the bead like a light ray through layers of ever-faster glass. Snell's law of refraction, applied continuously, forces the fastest path to be a cycloid — and the calculus of variations was born.