When Sir Christopher Wren was asked to design the Sheldonian Theatre in the 1660s, he began with visions of the great amphitheatres of Ancient Rome. Oxford having rather more rain than Rome, his design of a modern amphitheatre was in need of a roof.

Wren’s first plan was to place columns throughout the theatre to support the roof. This was vetoed by University officials, who felt it might interfere with the building's ability to be used for dancing. His next thought — that timbers could span the length of the building — was abandoned on realising that the building’s 70 foot by 80 foot dimensions were longer than the longest timbers available. To complete his design, Wren would need to build the largest unsupported roof the 17th Century had ever seen.

Only one thing could save his vision — a breathtakingly inspired piece of mathematics, by John Wallis, the then Savilian Professor of Geometry in Oxford.

Wallis designed a pattern of interlocking beams, each supported at both ends — either by the sides of the building or by other beams — and each supporting, in turn, two other beams. Each central beam uses another as a pivot, balancing the force down from the beams resting on it and the force up from the beams supporting it. In this way the weight of the roof is supported entirely by the sides of the building.

Before his design could be built, Willis needed to demonstrate the mathematical soundness of his idea with some formidable computations. He calculated the size of the beams needed to hold a given weight over a given area by solving a set of 25x25 simultaneous equations with nothing but pen and paper!

His calculations provided Wren with a solution to his design dilemma: a strong stable roof held up by mathematical principles instead of columns.

Demonstration

This design seems counterintuitive: if most of the beams are both supported by and supporting two other beams, why wouldn't the whole thing just come tumbling down?

The following video uses a simple construction demonstrating the principle of balancing forces that has kept the Sheldonian's roof aloft for the last 300 years (and hopefully a few more!). Why not try it for yourself?