Last year I was asked to produce three animations for an exhibition at the Royal Society in London. The exhibition was called Intersections: Henry Moore and Stringed Surfaces and was staged as a partnership between the Isaac Newton Institute in Cambridge, the Henry Moore Foundation, the Science Museum in London and the Royal Society. It was curated by Barry Phipps. The exhibition explored an inspirational encounter between the stringed surfaces created by the 19th century French mathematician Théodore Olivier and the young British sculptor Henry Moore. To give the exhibition a modern flavour a connection was drawn between these stringed surfaces and the mathematics of string theory. This article is a brief account of the exhibition.
The French mathematician Gaspard Monge (1746 – 1818) invented the branch of mathematics known as descriptive geometry. He illustrated his discoveries with surfaces created by stretching strings over a curved frame. Théodore Olivier (1793 – 1855) who was a pupil of Monge developed the use of string models further and incorporated moving components, so that the frames could be twisted and the surfaces distorted. This made the models interactive and showed how one surface could be transformed into another. In his time Olivier was famous for his beautiful models which were sold to universities and engineering students throughout Europe and North America. The Science Museum in London has a collection of about 30 models. They were manufactured by Fabre de Lagrange of Paris in 1872 and are based on Olivier’s designs. Several of these models were on display in the Intersections exhibition.
Sculptural Bird Cages
Henry Moore (1898 – 1986) was probably the most famous British sculptor of the 20th century. He began to incorporate taut strings into his abstract sculptural forms in 1937. This use of string was greatly influenced by seeing Olivier’s models in the Science Museum. He recalled: “I was fascinated by the mathematical models I saw there, which had been made to illustrate the difference of the form that is halfway between a square and a circle. One model had a square at one end with 20 holes along each side… Through these holes rings were threaded and lead to a circle with the same number of holes at the other end. A plane interposed through the middle shows the form that is halfway between a square and a circle… It wasn’t the scientific study of these models but the ability to look through the strings as with a bird cage and see one form within the other which excited me.”
A couple of the Henry Moore sculptures from the exhibition are shown here.
String theory is the best candidate that we have for an ultimate theory describing all the fundamental particles and all the forces that act on them. The new feature that string theory brings to physics is that it is based on fundamental one-dimensional entities known as strings, rather than point particles. If string theory is correct, then all fundamental particles are just different modes of vibration of a single type of object – the string. For instance, when a string vibrates in one way we might see it as an electron and when it vibrates another way we might see it as a quark or a photon.
One of the curious features of string theory is that it only works if there are ten dimensions of space and time. In order to describe the physics of the real universe, string theorists have devised clever ways in which six of the nine spatial dimensions are curled up so tightly that we are not aware of them. This means that they must be much smaller than an atomic nucleus. Even though we cannot see these tiny extra dimensions, according to string theorists their shape might determine the properties of the particles and forces that we do see in particle accelerator experiments.
The shape that these extra six dimensions are expected to form is known as a Calabi-Yau manifold. The reason for this particular shape is that it leads to a special kind of symmetry known as supersymmetry that would resolve some of the outstanding issues in particle physics. Supersymmetry is a symmetry between fermions, or matter particles, such as electrons and quarks, and bosons, the particles that produce forces when they pass between other particles. These exchange particles include photons – particles of light – that produce the electromagnetic force, and gluons, the particles that produce the strong force. My article Super Symmetry! gives more details about the significance of supersymmetry.
The torus shown in the animation above is the two-dimensional equivalent of a Calabi-Yau manifold. It is the first of my series of three animations that were displayed in the Intersections exhibition. The only four-dimensional Calabi-Yau manifold is known as a K3 surface. The second animation in the series shows a projection of this four-dimensional surface.
What physicists are really interested in are six-dimensional Calabi-Yau manifolds as one of these could potentially bridge the gulf between the natural home of string theory in ten-dimensional spacetime and the four-dimensional spacetime studied by physicists in their laboratories. There are very many six-dimensional Calabi-Yau manifolds – no-one knows how many. One of the first such Calabi-Yau manifolds to be studied is known as the quintic hypersurface, and a projection of this manifold is shown in the final animation below.
As yet string theory remains an intriguing, but unproven, approach to physics that has captivated many of the world’s best mathematicians and physicists for several decades. So far there is no experimental evidence to show that it genuinely describes the way our universe is constructed. But many physicists expect that the next big discovery at the Large Hadron Collider will be supersymmetry. If they are right, then many will claim that this is the first sign that string theory really does have something to say about how the universe works.
This article is based on the talk given by Nicholas Mee at the Matematica e Cultura 2013 Conference held at the Instituto Veneto di Scienze, Lettere e Arti in the Palazzo Franchetti on the Grand Canal, Venice from 22-24 March 2013.
If you would like a first hand account of how Calabi-Yau manifolds were discovered take a look at: The Shape of Inner Space by Shing-Tung Yau and Steve Nadis.