The shape a hanging chain refuses to be

2026-06-03

Hang a chain loosely between two nails and it settles into a smooth, drooping curve. Everyone's first guess is a parabola - the same U you get from y = x². Galileo thought so too. For a wide, shallow hang you can stare at it all day and never catch the difference.

So let us put the parabola right on top of the chain - same two pins, same lowest point - and then close the gap between the nails.

Wide apart, the dashed parabola hides perfectly under the chain. Bring the pins together and it peels away: the real chain hangs fuller, bellying out past the parabola down the sides before plunging to the same low point. The two curves share their endpoints and their bottom and still refuse to be the same shape.

What the chain is actually doing

A parabola is what you get when the weight is spread evenly along the horizontal - so much load per metre across. But a chain's weight is spread evenly along itself, and the steep parts near the ends pack a lot of chain into a little horizontal distance. More chain hanging there means more weight pulling down there, which is why the curve bellies out where a parabola would not. Solve what that does to the shape and out comes the hyperbolic cosine:

y = a·cosh(x / a)

the catenary, from the Latin catena, a chain. It was Huygens, Leibniz and Johann Bernoulli who pinned it down in 1691, a year after Jakob Bernoulli threw the problem open - and Huygens, aged seventeen, who had first proved it was not a parabola, against Galileo.

When it really is a parabola

The parabola guess is not foolish - it is just the wrong structure. A suspension bridge does hang in a parabola, because there the heavy part is the flat roadway below, loaded evenly across the horizontal, and the cable carries that. Weight per horizontal metre gives a parabola; weight per metre of cable gives a catenary. Change what carries the load and you change the curve.

Flip a catenary upside down and every hanging tension becomes a pushing compression, perfectly in line with the stone - which is why a free-standing arch wants to be an inverted catenary. The Gateway Arch in St. Louis is one on purpose. Gravity drew the curve; the architects just turned it over.

Drag the pins until they nearly touch. The gap readout climbs and the verdict changes. The chain was never going to be a parabola - you just had to crowd it until it admitted so.