The following is simply a demo of what a cycloidal pendulum looks like:

Prop XXV from Horologium Oscillatorium:
"On a cycloid with a vertical axis whose vertex is below, the times of descent in which a mobile point, starting from rest at an arbitrary point of the curve, reaches the lowest point, are all equal, and have to the times of the vertical fall along the total axis of the cycloid a ratio equal to that of the semicircumference of a circle to that of the diameter" (Struik 1986 263).