The Gherkin is one of a new breed of buildings that is first entirely constructed as a mathematical model on a computer. Every aspect of these models is linked mathematically – the shape, the height, the width, the curvature, the construction method – a change to any one of these automatically ripples through the model altering the other aspects in response.

These computer models of the building, called parametric models, allowed the designers to play: to explore different shapes for aesthetic reasons as well as practical ones. Not only could the designers explore how different shapes would look but also how they would react in the physical environment.

One particularly important aspect was the aerodynamics of the building. The movement of wind around tall straight-sided buildings can result in turbulence at their base, buffeting passers-by at ground level, or even producing wind tunnel effects. The designers of the Gherkin were able to experiment with the curved shapes and test them in computer simulations of wind movement to find a shape that reduced the turbulence on the ground. Similar experimenting with the aerodynamics of the internal structure resulted in six spiralling ventilation shafts that draw air through the building. This natural ventilation reduces the heating and cooling costs of the building by 40%.

Mathematics not only helped the design of the building, but also made the construction of such a novel and complex structure possible. The way the building is modelled mathematically can be turned into a step-by-step guide to construct it. How do you construct this amazing curved building? You might be surprised that the sleek curved exterior only contains one piece of curved glass – the curved panel at the very top of the building. Instead a mathematical understanding of the curved surface allowed it to be approximated by triangles and diamonds of flat glass.

But of course architecture and construction is a delicate balancing act. The architects needed to balance the approximation of the curved surface with the cost of the construction — more panels means higher costs, and other aesthetic considerations — like trying to maximise the amount of glass and reduce the struts joining panels.

For the designers of the Gherkin, mathematics set their imagination free – it allowed them to play with new, innovative ideas, they were free to take risks and make mistakes with their computer models without physical consequence, and to find the best solutions to practical and aesthetic questions.

Demonstration

Props: Round balloons (ideally with icosahedrons printed on them) and permanent markers for all participants. One icosahedron of roughly the same size as an inflated balloon.

It's all very well for architects and designers to come up with these striking forms, but how exactly do you go about building one in real life? Curved panels of glass are expensive to manufacture and fit. Surprisingly, the curved surface of the Gherkin has been created almost entirely out of flat panels of glass — the only curved piece is the cap on the very top of the building. And simple geometry is all that is required to understand how.

One way of approximating a curved surface, such as the surface of a balloon, using flat panels is using the concept of geodesic domes and surfaces. A geodesic is just a line between two points that follows the shortest possible distance — on the earth the geodesic lines are great circles, such as the lines of longitude or the routes aircraft use for long distances. A geodesic dome is created from a lattice of geodesics that intersect to cover the curved surface with triangles.

This rotating image from Wikipedia shows you what one looks like.

To try to build the curved shape of our balloon out of flat panels, you first need to imagine an icosahedron (a polyhedron made up of 20 faces that are equilateral triangles) sitting just inside your balloon, so that the points of the icosahedron just touch the balloon's surface. We can draw lines on the balloon above the edges of the imaginary icosahedron to help us picture it.

But an icosahedron, with its relatively large flat sides, isn't going to fool anyone into thinking it's curved. You need to use smaller flat panels and a lot more of them. So divide each edge of the icosahedron in half, and join the points, dividing each of the icosahedron's faces into four smaller triangles. If you push the corners of these smaller triangles out so they just touch the balloon's surface, you now have a polyhedron with 80 triangular faces (which are no longer equilateral triangles) that gives a much more convincing approximation of the curved surface of the balloon. Again we can draw lines on the balloon above the edges of this imaginary polyhedron to help us picture it.

You can carry on in this way, dividing the edges in half and creating more triangular faces, until the surface made up of flat triangles is as close to the curved surface as you would like.

But instead we will stop here at 80 faces and create a mathematical souvenir of the Gherkin. Colour in the crowning circle of triangles and then colour in alternating stripes of triangles spiralling down from the top of your balloon. Then gently pull the top of the balloon away from the bottom, and you have yourself your own model of the Gherkin!

You can read more about geodesic domes on Wikipedia. And you can read more about the Gherkin, geodesics and engineering and architecture in Plus Magazine.