This Demonstration illustrates the fact that a cycloid is a tautochrone (isochronous) path. Six beads slide without friction down and up six cycloidal wire paths, starting from different positions and with zero initial speed. It can be seen that the six beads always reach the bottom simultaneously.
The tautochrone path is the cycloid formed by a circle of radius 1 rolling on the axis from to . With as the position of a bead at time , this cycloid has the parametric equations . Using Lagrangian dynamics, with , the resulting equation of motion is .