TAUTOCHRONE
Huygens’ tautochrone: beads from any height reach the bottom in the same instant — exact SHM in arc length.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
An upside-down cycloid — the curve a wheel-rim traces, flipped into a bowl. Drop a bead in and it slides to the bottom; drop another from higher up. Common sense says the higher one takes longer. Huygens found otherwise in 1659, and it built the first accurate clock.
Arc length
The trick is to measure distance ALONG the curve, not straight down. In that arc-length coordinate the height above the bottom comes out proportional to the distance squared — so the restoring pull is exactly proportional to the distance. That is the signature of a perfect spring.
SHM
So the bead runs simple harmonic motion in arc length — exactly, at every amplitude, with none of the small-angle fudge a pendulum needs. The period is 4π√(a/g), and a quarter of it, π√(a/g), carries the bead to the bottom.
Tautochrone
And that quarter-period has no amplitude in it. Release four beads from four heights at once and they cross the bottom in the same instant — the tautochrone, "equal time." The cosine curves at right are different sizes that all reach zero together.
Audit
Audited: the arc-length equation of motion is exact SHM to 1e-12, and the descent time is amplitude-independent to the bit. Hang a bob on cycloidal cheeks and you get Huygens’ isochronous clock; the pendulum module’s 15%-at-90° error is precisely what this shape erases. It is also the brachistochrone — the fastest slide — though that we state, not prove.