Contour integrals and period formulae of the tautochrone

In 1673, C. Huygens determined that a pendulum whose string is restricted to wind around a cycloid curve has a peculiar property: its period is ``tautochronic'', that is, independent of its amplitude. Though initially of interest for precision time, we discuss this curious result from the perspective of symmetries and topology of the period integral. We demonstrate this phenomenon in measurements by J. George using a 3d printed cycloidic guide.