The dome of St Paul’s cathedral is full of mathematics. In fact three important mathematical curves are key to Christopher Wren’s design.
Christopher Wren studied mathematics at Wadham College Oxford, graduating in 1651, but his interests ranged wide from astronomy to architecture. Newton regarded him as one of the leading mathematicians of the day and it was mathematics that is central to Wren’s design of the dome of St Paul’s.
The dome of St Paul’s cathedral is actually made up of three domes. The dome that you see from the outside is spherical in shape. A sphere has a beauty and perfection about it which is particularly appealing when seen from a distance but the shape also tapped into the idea of the church representing the shape of the cosmos.
But the dome you see within the cathedral is not actually the inside of the external dome. It is in fact a second dome whose shape is made up from a new curve discovered by contemporaries of Wren. Called the catenary curve, it is the shape that a chain makes when you hold it at both ends. Galileo had thought this shape was a parabola but this was disproved in 1669 by the German mathematician Joachim Jungius.The equation for the curve, y = cosh (x/a) for a constant value a, was calculated by several of the great mathematicians of the age: Leibniz, Huygens and Johann Bernoulli.
The catenary curves y = cosh (x/a) for different values of a
Wren particularly liked the shape of the catenary because when you look up it creates a forced perspective making it feel higher that it actually is. The use of mathematics to create optical illusion is a big theme of the architecture of the Baroque period. But it is also an important shape structurally as Wren’s collaborator Hooke explained to the Royal Society. Turned upside down this shape makes an arch that supports its own weight with pure compression and no bending. The Spanish architect Antoni Gaudi used the shape often in his architecture, for example in the Sagrada Familia in Barcelona.
In addition to the two domes you can see there is in fact a third dome which is hidden from view sitting between the outer and inner dome. Spherical domes have a problem: they are inherently weak at their highest point. So Wren created a third dome sitting between the inner and outer dome whose shape is more conic, made from rotating the positive half of the cubic equation y=x3. Hooke and Wren thought that this was the perfect shape for a dome just as the catenary is the perfect shape for an arch. This dome supports both the top of the external spherical dome and also the lantern that sits on top of the dome.