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Your Universe of Digital Possibilities
“As hangs the flexible line, so but inverted will stand the rigid arch” — Robert Hooke’s anagram of 1675. A chain finds the shape that carries its weight in pure tension; turn that shape upside down and it carries the same weight in pure compression, which is all that stone can do. Hang a net and flip it into a vault, test Galileo’s guess about the hanging chain, and thin a stone arch until it can no longer stand.
A chain of weight w per unit length, pulled horizontally by H, hangs in a hyperbolic cosine. Its slope grows with the weight of chain below each point — a parabola would need the weight spread along the ground instead.
A cable carries its load w(x) per horizontal length in pure tension, its horizontal pull H the same everywhere. A uniform deck gives a parabola; the cable’s own weight, spread along its length, gives the catenary. Inverted, the same line carries the same load in pure compression: the arch.
Stone is strong in compression, has no strength in tension and does not slide. Then if any line of thrust in equilibrium with the loads lies inside the masonry, the arch stands; it fails only when no line fits and hinges open where the line touches the edge.
The seventh Heron machine computes with gravity. A chain has no stiffness and so cannot carry a bending moment: whatever shape it hangs in is, by necessity, the line of pure tension for its load. Invert it and it is the line of pure compression — the shape a masonry arch must contain. The net relaxes to its balance, the chain is solved link by link, the arch is searched for any line that fits; the auditor checks each against the catenary, the parabola and the thinnest arch on record. Soap films do the same with tension in two dimensions in The Soap; the fastest path under gravity, the cycloid, is The Bead’s; and the principle that nature chooses the stationary path is The Action’s.