Jørn Utzon's competition-winning sketch showed a series of white "sails" rising over the harbour; but how to build these free curves was a problem no one knew how to solve. The real story of the Sydney Opera House is the turning of a form into a buildable geometry.

In the early years each shell was a different, arbitrary curve; casting each with its own formwork was both impossibly expensive and structurally unresolved. The solution came from Utzon: he redefined all the shells as pieces cut from the surface of a single sphere. Taken from a sphere of the same radius, every shell now had the same curvature — which meant the shells could be built from standard, repeating, precast concrete rib segments. The arbitrary form was made producible by reducing it to a single geometric rule.

The calculations for this solution were carried out by the engineering office of Ove Arup using one of the era's first computers — an early moment in which architectural structure rested on large-scale numerical computation. The surface of the shells is clad in glossy white ceramic tiles, forming a continuous light-reflecting texture. The Sydney Opera House is thus the turning point that made free-form shell architecture buildable through geometric discipline and computing power.