What shape does a chain hanging under its own weight take — the parabola Galileo guessed?
▶ Launch the interactive simulationHang a flexible chain (a Verlet rope of equal links between two pinned ends, settled with position-based distance constraints) and, as the span slowly breathes, compare its shape to the catenary y = a·cosh(x/a) and to the least-squares best-fit parabola, by the RMS residual of each.
the CATENARY, not the parabola: the chain matches y = a·cosh(x/a) to a tiny residual (≈0.006) while even the BEST-FIT parabola is tens of times worse at deep sag — and only catches up when the chain is nearly flat, where cosh ≈ parabola. The cosh wins because the chain has uniform mass per unit LENGTH (a parabola is the cable of a suspension bridge, uniform mass per unit horizontal span); it is also the least-potential-energy shape, which the rope finds on its own (Bernoulli/Huygens/Leibniz, 1691)