Thought Toys · Shape & space · Exhibit 123

The chain that refuses to be a parabola.

Hang a chain between two hooks and it settles into the one curve that carries its own weight without bending anywhere. Nearly everyone draws that curve as a parabola, and Galileo did too. It is not one. The parabola turns out to be the right answer to a different question: the shape of a cable holding up a bridge deck. Fix the two hooks and the lowest point, then slide the weight from the chain to the deck. The curve between those three points moves, and the gap it leaves behind is the whole story.

Two hooks, one lowest point, and a curve between them ● the chain as it hangs   ● the parabola through the same three points

biggest gap— length of curve— sideways pull— pull at a hook—

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↳ your turn — drag Where the weight lives toward the deck and watch the chain turn into the parabola

What you're seeing

The amber curve is a chain hanging from two hooks. It is computed, not drawn by hand: at every point along it the pull of the chain exactly balances the weight it carries. The dashed cyan curve is the parabola through the same three points — both hooks and the lowest point. Every sketch of a hanging chain gets those three points right. The disagreement is everything in between.

Look at where the shading is thickest, about seven-tenths of the way out from the middle. The chain sits below the parabola there. It hangs flatter at the bottom and then climbs more steeply into the hooks. At the default depth the biggest gap is five percent of the sag, which is easy to see and impossible to fix by nudging a parabola. The chain is not a parabola that has been drawn badly. It belongs to a different family of curves, the catenary, from catena, the Latin for chain.

Now drag Where the weight lives to the right. The beads fade, a deck appears underneath, and hangers carry the deck's weight up to the cable. As the weight moves out of the chain and into the deck, the amber curve slides across the shaded gap and lands exactly on the dashed parabola. The parabola was never wrong. It is the shape of a weightless cable holding up something level and heavy, which is a suspension bridge, not a chain.

The reason is where each bit of weight sits. In a chain, the weight is spread evenly along the chain, so the steep parts near the hooks pack more weight into each step across. That extra weight is what bends the curve so hard near the ends. A deck spreads its weight evenly along the ground instead, the same amount per step across everywhere, and the curve that balances an even load is a parabola.

Then drag How deep it hangs down toward taut. The gap collapses. Below about four tenths of a half-span the two curves are within one percent of each other, and by the time the chain is nearly straight they are indistinguishable. The gap shrinks with the cube of the sag. This is why the parabola is such a good lie: for the shallow chains and cables most people ever see, it is right to within the thickness of the line.

Finally, press Flip it into an arch. A hanging chain carries its weight in pure tension, with no bending anywhere. Turn it upside down and every pull becomes a push, so the same curve stands as an arch in pure compression, again with no bending. Robert Hooke wrote that down in 1675, as an anagram whose solution was only published after his death: as hangs the flexible line, so but inverted will stand the rigid arch. The Gateway Arch in St Louis is a close cousin of this curve, thickened toward its feet.

The rule, exactly. Take the lowest point as the origin, x across and y up. The sideways pull H is the same everywhere along the curve. Write w for the weight per unit of length and μ for the share of it that hangs from a level deck. Then:

H·y″ = μw + (1 − μ)w·√(1 + y′2)

The square root is the length of chain in one step across, which is why the chain's own weight grows where the curve is steep. The two ends of the slider are the two curves with names:

a bare chain (μ = 0): y = a·cosh(x/a) − a, with H = wa      a cable with a deck (μ = 1): y = wx2 / 2H

The page solves the equation directly, by choosing the pull H that lands the curve on the hooks, and never uses either closed form. The pull at a hook for a bare chain is w·(a + sag): the weight of a length of chain equal to the hook's height above the curve's hidden baseline. Both pulls on the page are given in units of w·h, the weight of one half-span's worth of load, so a sideways pull of 0.62 means the hooks are pulled inward with 62% of that weight. The model is an ideal one. The chain is perfectly flexible and does not stretch; the deck is level and rigid; the hangers and the cable in the deck case weigh nothing.

Verified in node (improve/verify/123-catenary.js, 8 checks, three of them negative controls, all before this page existed). Every "miss" below is the largest vertical distance between the relaxed chain's nodes and a curve, as a share of the span or of the sag as stated. The 5.0% above is the peak vertical gap between the two curves, as a share of the sag.

  • With all the weight in the chain, the page's solver reproduces a·(cosh(x/a) − 1) through the same three points to a few parts in a million, and finds H = wa.
  • With all the weight in the deck, the same solver reproduces wx2/2H to round-off, with H = w/2s.
  • At fixed span and sag the parabola lies above the chain at every interior point; the gap peaks at about 0.71 of the half-span and grows 7.95-fold when the sag doubles from 0.05 — the cube law, exact in the shallow limit.
  • As the load slides from chain to deck, the curve moves one way only, and gets shorter. The chain is the longer curve through the same three points. My first draft of the proof asserted the opposite, and the gate refused it.
  • The oracle. A chain of 48 masses joined by stiff springs, dropped under gravity with damping and integrated until it stops, knows nothing about cosh. Its largest miss from the catenary of its length is 0.1% of the span; its largest miss from the parabola through its own three points is 22 times bigger.
  • Negative control one. The same mass-and-spring chain made to carry a level deck came to rest on the parabola to round-off, and off the catenary. The load picks the curve, not the chain.
  • Negative control two. Shallow chains miss the catenary through their own three points by at most 0.002% of the sag, and the parabola's miss shrinks as they get shallower. A first draft compared against the catenary of the chain's nominal length and read a 0.17% stretch of the springs as a 4% miss in the sag; a shallow chain's sag is that sensitive to its length.
  • The relaxed chain's actual spring forces match the closed forms for the sideways pull and the pull at a hook to about 1%.

Galileo took the hanging chain for a parabola in 1638. The seventeen-year-old Huygens showed in 1646 that it cannot be one, and the curve itself was found in 1691 by Leibniz, Huygens and Johann Bernoulli, answering a public challenge from Jakob Bernoulli. Galileo was not careless: for the shallow chains he could measure, the cube law above makes the two curves agree to within his instruments.

Also in Shape & space: Cross one line, and its territory closes →

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  2. 123The chain that refuses to be a parabola — you are here
  3. 51Cross one line, and its territory closes
  4. 78Every world map is lying. You get to pick the lie.
  5. 85The shape that has only one side

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