Tautochrone: Cycloid Isochronism

What you are seeing: frictionless beads released simultaneously from different heights along the same cycloid bowl, then left to swing. They keep perfect step forever: every bead crosses the bottom at the same instant and reaches its own turning point at the same instant, swing after swing, no matter the amplitude. Huygens proved in 1673 that the equation of motion in arc-length $s$ from the bottom is $\ddot{s} = -(g / 4R) s$ , an exact simple harmonic oscillator independent of amplitude.

The second bowl is a plain circular arc, the shape of an ordinary pendulum, with the same beads dropped from the same heights. It matches the cycloid for tiny swings but not for large ones: the higher beads swing more slowly, so beads that started together drift out of step over successive swings and the bottom crossings smear apart. The diagnostic makes the difference exact, the cycloid's descent time is flat against release height while the circle's rises. That flatness is what makes the cycloid the tautochrone, the curve of equal time, and it is why Huygens built pendulum clocks with cycloidal cheeks so their period would not drift with swing amplitude.

Figure 1. Tautochrone. Method: closed-form

s(t) = s_0 \cos(\omega t)

with

\omega = \sqrt{g / 4R}

, mapped through the cycloid parametrization.

speed1

beads5

WHAT TO TRY

Watch the top bowl: the beads start at wildly different heights yet swing in perfect step, all crossing the bottom together in one synchronized flash, swing after swing.
Now watch the circular bowl below it with the same drops: the higher beads swing more slowly and drift out of step, so after a few swings their bottom crossings are scattered. Same heights, different curve, different answer.
Add more beads and read the lower plot: the cycloid's descent time is a flat line against release height, while the circle's climbs. The flatness is the whole point.