If you want to roll a ball down a slope, what shape of slope will get the ball from point A to point B in the least time? This is the so-called brachistochrone challenge.

At first you might naively assume that a straight line would get the ball to its destination in the least time, because the shortest distance between two point is a straight line, but the ball is not moving at a constant speed.

In fact, because it needs to get rolling, the ball will travel down a *concave* ramp much faster than down a *convex* ramp. But which concave curve?

The winning curve is an inverted cycloid, the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line.

(Hat tip to Charles.)

Deriving things like the brachistochrone and catenary or fibonacci equations were some of the most motivating puzzles of my teenage years. There’s something especially beautiful about understanding the logic of curves with natural significance and which appear all around us. A million people can look at a bridge, but how many have ever give through the motions of understanding exactly why it looks precisely like that.

Inverted cycloid is also the curve where whatever place along it you put a ball they get to the end at same time.

Handle: same here.

Lucklucky: IIRC that was actually the original motivation, to use in a clock. An ordinary pendulum’s frequency depends on the amplitude, so Huygens developed his cycloid pendulum solution.

Thanks for the pointer, Candide III.