Rod through a curve

Tasks

Before trying it out, did you believe that the rod would fit through the curve?

Rotate the rod to see it fit through the curve.

As it moves, the rod is sweeping out a shape. Can you picture the shape it is making?

Maths

It may be surprising that a straight rod can fit through a curved hole. As the rod moves through the hole, it sweeps out a shape known as a hyperboloid.

A hyperboloid is an example of a curved surface made entirely of straight lines. These kinds of surfaces are known as ruled surfaces.

Other examples of ruled surfaces include cylinders, cones and pringle-shaped saddle surfaces.
History

Hyperboloids have been studied since ancient times. Greek mathematician, Archimedes first described how to calculate the volume of a hyperboloid.

In 1669, English mathematician and architect, Christopher Wren, proved that the hyperboloid was a doubly ruled surface which means there are two straight lines that pass through each point on the surface. Wren did not use the shape in his architecture, instead he was interested in making lenses by using a straight tool on a rotating cylinder of glass.

In the 1880s, Russian engineer, Vladimir Shukhov, was the first to use beams to construct hyperboloid shaped structures such as water towers and lighthouses. And architects today continue to use the hyperboloid shape to make towers.
People


Christopher Wren: 1632 – 1723
Christopher Wren was an English scientist and architect, most famous for building St Paul’s cathedral in London.
Apart from architecture, Wren studied animals, weather, magnetism, planets and mathematics. In maths, Wren proved that the hyperboloid was a doubly ruled surface.


Vladimir Shukhov 1853 – 1939
Russian engineer and architect, Vladimir Shukhov, was the first to create hyperboloid structure made from steel beams. These structures were strong and lightweight. He made about 200 hyperboloid towers, all over the world, the most famous being the 160 metre high Shukhov tower in Moscow.
Applications

In architecture, hyperboloids are perfect for making tall towers using steel beams. The beams give the structure strength while maintaining an attractive look. Examples include cooling towers and rocket engines.

Corporation street bridge in Manchester is a hyperboloid structure on its side, connecting buildings across the street.
Maths at Home

Make your own hyperboloid town using skewers and elastic bands.