Creating the Gherkin's curves, London
Look along the opposite river bank. What shapes do you see? Although most buildings are based on rectangles and squares, two curved icons dominate the skyline: the spherical dome of St Paul's Cathedral and tapering curved shape of 30 St Mary Axe, aka the Gherkin.
Creating enormous, curved buildings is a far more difficult task than one based on vertical straight lines. To ensure a spherical dome will support its own weight it needs to be built with a thickness of at least 4% of its radius. The dome of St Peter's, Rome, has a thickness of about 15% of its radius. Wren used a different approach in his construction of the dome of St Paul's as we'll see a little later in this tour. But what if you want to build a much larger curved form out of the thinnest possible shell, made of just steel and glass?
The strength of triangles
The answer lies in that most uncurvy of shapes: the triangle. [How many pieces of curved glass are used in the Gherkin?] Although the Gherkin appears to have a sleek curved form it is actually made up of hundreds of flat panels, with just a single curved piece of glass right at the very top of the building. Not only do these panels give the iconic building its curved appearance, the strength of their triangular shape also allows the building to have such a light and airy structure.
[Why are triangles so strong?] Triangles are inherently strong because they form a fixed rigid shape. This can be demonstrated by building a triangle out of garden canes, securing the corners with rubber bands. The shape is fixed by the length of the sides and the triangle withstands quite substantial forces applied to it.
[Why is a square unstable?] However if you built a square in the same way, a gentle push at one corner could easily change the shape into a paralleogram. There are infinitely many four-sided shapes with equal sides, a square is just one of them, and so the shape is easily transformed from one to the other with minimal force.
A square lacks the rigid strength of a triangle. But by adding diagonal bracing, a common feature in bridges and buildings, the structure can again rely on the strength of a triangle to hold its shape.
Creating the new world
Triangles have always been a fundamental tool in architecture and construction, but mathematics has allowed them to be used to construct some of the complex and daring shapes we see in architecture today. [Can you spot any other triangles along the river or in the city?] Triangulation of surfaces, such as the Gherkin or the roofs of the British Museum courtyard and the new concourse at Kings Cross Station, has allowed spectacular curved shapes to be built with a minimal amount of material.
Triangulation of surfaces is also responsible for the virtual worlds many of us enjoy. Digital images and computer generated animation all rely on clever mathematics to construct and manipulate triangulated models of our favourite characters and their environments, and then to mathematically paint them to bring them to life.
Props: 20 garden canes and lots of rubber bands.
We've included questions (in italics in the description above) that you can ask the group to get them involved.
Divide the group into two groups and ask one to construct a tetrahedron (a pyramid made out of four equilateral triangles) and the other a cube.
When finished, ask the the first group to let go of their tetrahedron. The structure will stand unsupported due to its rigid shape and will even bear a substantial amount of force. This is because there is only one possible shape that can be constructed with four triangular faces with equal length sides.
The carbon atoms in diamond are arranged in a continuous lattice of tetrahedra. This is why diamonds are so hard.
When the second group let go of their cube, the structure will collapse. The shape isn't rigid as the corners are flexible. The shape can easily be skewed into any one of the infinitely many shapes that can be built with six quadrilateral faces with equal length sides. (These squashed cubes are called parallelepipeds.)