The Bridge of Sighs, Oxford
Hertford Bridge, more commonly known as the Bridge of Sighs, was finished in 1914. The bridge links together the Old and New quads of Hertford College, and is one of the most popular tourist sites in the city.
Although the bridge seems relatively old nowadays, the maths that hides in its design was first noticed by Archimedes in the 3rd century BC. Archimedes, famous for running through the streets naked shouting "Eureka, Eureka" (I’ve got it, I’ve got it), would often write letters to his friend Dositheus with new problems he was working on and insights that he had.
In one letter he focused entirely on curves, and finished his letter with the proof that if you have a triangle with a given base and height, and a smooth curve with the same base and height, the area of the curve is 4/3 the area of the triangle!
The annotated picture above illustrates how this maths relates to the Bridge of Sighs. The top of the bridge is designed to be a triangle, but the bottom is a curve known as a parabola, and the triangle and the parabola have the same base and height. Therefore, by a piece of 2000 year old mathematics, the area of the curve at the bottom of the bridge is 4/3 the area of the triangle that forms the top. This property is known as the Quadrature of the Parabola.
The way that Archimedes proved this result used the concept of infinity, a new and intriguing concept that he’d come across when looking at the number pi. This was just another thing that put Archimedes well ahead of his time.
The parabolic shape of the bridge also illustrates an important mathematical idea used by stonemasons, architects and engineers. In the seventeenth century, the mathematician Robert Hooke realised that the perfect shape for an arch, in order that it support its own weight, was a catenary curve. A catenary is the shape made by a hanging rope or chain held at its two ends. When an arch has this shape the thrust caused by its own weight is directed exactly along the shape of the curve. But if an arch needs to support a load spread horizontally across it, the arch needs to be a slightly different shape — a parabola — just like the Bridge of Sighs.
The Quadrature of the Parabola can be easily seen by using an elastic band. Stand with a good front-on view of the Bridge of Sighs; hold up the elastic band and from your viewpoint make the shape of a triangle so that when you look through the elastic band you can see the triangle that forms the top of the bridge.
Then move the elastic band down holding it in the same shape and line it up with the bottom of the arch. You should immediately see that the triangle fits in well but there is area left over. In fact this extra area should be a third the area of your triangle.