In 1675 the English scientist Robert Hooke published a puzzle. Tucked into a book about telescopes was a jumble of letters, an anagram hiding what he called the true shape for an arch. He never explained it while he was alive. When his executor finally unscrambled it in 1705, it turned out to be one line of Latin:
In other words: hang a chain between two points, let it settle, then turn that curve upside down. Build it out of stone and it stands. Try it below. Drag the two anchors, tap a link to hang extra weight on it, then flip the whole thing over.
A chain can't resist bending. Each link can only pull on its neighbours, and only along the line between them. So when a chain hangs still, it has found the one shape where every bit of its weight is carried by pure tension running along the curve. Nothing is being bent.
Flip that shape over and every force flips with it. Pulling becomes pushing. Each stone now presses squarely on the next, along the curve, with no tendency to hinge open. That matters because stone is superb at being squeezed and poor at being stretched. An arch built on the inverted chain keeps every joint in compression, which is why such arches can stand with no mortar at all.
Tap a few links to hang weights, then flip. The arch bends to match: it bulges up exactly where the extra load sits. The rule holds for any load. Whatever shape a chain takes under a set of weights, the upside-down version carries the same weights in pure compression.
The shape of a freely hanging chain is called a catenary, from the Latin catena, chain. It looks a lot like a parabola, and in 1638 Galileo noted that a hanging cord is roughly parabolic, especially when it hangs shallow. Joachim Jungius showed it isn't exactly one, and in 1691 Leibniz, Huygens and Johann Bernoulli each independently worked out the real formula, the hyperbolic cosine.
The difference comes down to where the weight is. A bare chain carries its own weight spread evenly along its length, which gives a catenary. A suspension bridge cable carries a flat deck spread evenly across the ground, and that gives a parabola. Same principle, different load, different curve.
Roman arches are semicircles, not catenaries, and many have stood for two thousand years. Switch on the semicircle in the simulation to see how far it strays from the ideal curve, especially near the base.
Engineers think about this with a line of thrust: the path the compressive force actually takes through the stones. An arch is safe as long as that line stays inside the masonry. In a catenary arch it runs right down the middle. In a semicircle it wanders, so the stones must be thick enough to contain it. Classic analyses put the bare minimum for a semicircular arch at roughly a tenth of its radius. Romans went well beyond that, and piled heavy masonry over the lower sides, which pushes the line of thrust back where it belongs.
Antoni Gaudí took Hooke literally. For the church at Colònia Güell near Barcelona, commissioned in 1898, he built a 1:10 scale model entirely upside down: strings hung from the ceiling, with small lead-filled sacks standing in for the weight of walls and roofs. The strings sagged into the shape of every column and vault at once. He photographed the model and flipped the picture to see his building.
Work only started in 1908, and money ran out. The crypt, built between 1908 and 1915, is the only part ever finished. Its leaning columns look odd until you realise each one follows a line the hanging strings found for it.
The Gateway Arch is the best known inverted chain on Earth, with one twist. Its legs are much thicker at the bottom than at the top, so more of its weight sits low down. That makes it a weighted catenary, the shape you'd get from a chain whose links get heavier towards the ends. You can approximate it in the simulation by hanging weights near both anchors.
Watch the "sideways pull" as you drag the anchors apart. A deep, narrow chain barely tugs on its hooks. A shallow one pulls hard, many times its own weight if you stretch it far enough. Flip it and the same number becomes an outward shove on the ground at each foot. That is why flat, wide arches need massive abutments or tie rods, and why Gothic cathedrals grew flying buttresses along their sides. The chain does not just tell you the shape. It tells you how hard the arch will push back.