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.

**Stringy Surfaces**

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

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.

**Extra Dimensions**

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.

**Superstringy Surfaces**

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.

**More Information**

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.

{ 7 comments… read them below or add one }

Honourable Mee,

The article ‘Strings, Surfaces and Physics’ is really interesting, informative and educative. However it is unfathomable to understand Nature’s secrecy. Your calibre to make it understandable is highly appreciated and praiseworthy.

Yours sincerely,

Dr Abhimanyu Mohanta

Dear mr.mee,

i quite didn’t understand how the theory of supersymmetry can be tested at the Large Hadron Collider and the ten dimensional thing if its not yet clear then how r u so sure about it.

Dear Raeen

Thank you for your comment. The theory of supersymmetry makes some very definite predictions such as the existence of new particles. Physicists are sifting through the data produced by the LHC searching for signs of these particles. The idea of supersymmetry grew out of research into string theory, but it is possible that supersymmetry has a role in particle physics even if string theory does not apply to the real world. Even so, if supersymmetry is discovered, many people will claim that this is the first sign that string theory is correct. If they are right, this will imply that the universe has six extra dimensions, because string theory is only possible in a ten-dimensional universe. However, there are a lot of ‘ifs’ in this statement. It is perfectly possible that none of this is correct and supersymmetry has nothing to do with the real universe. But this is the whole point of the LHC – to test these ideas and see which ones accurately describe the universe.

Dear Mr. Mee,

I would be much obliged if you kindly educate me on the properties of strings and why the string theory is restricted to ten dimensions of space and time.

There is no obvious or simple answer to this question. A classical theory of strings would be viable in any number of dimensions. But we know that in reality the world is ruled by quantum mechanics and it is very difficult to marry relativity and quantum mechanics.

The theories of modern physics are built around various fundamental symmetries. (For instance, we expect empty space to be symmetrical under translations and rotations.) It is possible to incorporate the appropriate symmetries into a classical theory of strings in any number of dimensions, but if we attempt to construct a

quantumtheory of strings it turns out that these symmetries are lost and so the theory is inconsistent and useless, except in very special circumstances. The original version of string theory is now known as thebosonicstring. In the quantum version of this theory the vital symmetries of the classical theory are only retained in the quantum theory if the universe has 26 spacetime dimensions.The bosonic string has various problems and it does not include any fermions (matter particles, such as electrons or quarks), which means that it cannot be a viable theory of the real universe. But when a new symmetry known as

supersymmetryis incorporated into string theory, then fermions are automatically included in the theory and some of the other problems of the bosonic string are cured. This new theory issuperstringtheory. It is now only possible to incorporate the symmetries of this theory into the quantum version of the theory if spacetime has ten dimensions. There is no simple intuitive reason why the magic number is ten. I will offer a step towards a partial answer.The critical dimensions for the bosonic string and the superstring are the ones in which certain vibrations of the strings correspond to massless particles. This is very important, because the exchange of these massless particles is what produces forces. For instance, the electromagnetic force is the result of the exchange of the massless particles of light that we know as photons. Massless particles vibrate transversely i.e. they only oscillate in the directions perpendicular to their direction of motion. What this means is that a photon has two possible polarizations, because in three dimensional space there are only two directions perpendicular to the direction in which the photon is travelling. In 26 dimensional spacetime there is one time dimension and 25 space dimensions, so a photon would have 24 transverse modes of vibration. In 10 dimensional spacetime there are 9 space dimensions, so a photon would have 8 transverse modes of vibration. It is actually these numbers, 24 in the case of the bosonic string, and 8 in the case of the superstring that are special. In short, 8 dimensional space and 24 dimensional space have additional symmetries and it seems to be the existence of these symmetries that enables string theory to be viable in spacetimes that have these numbers of transverse dimensions.

Dear Mr. Mee,

Thank you for sharing your concept with me. I have knowledge about the eight dimensions. What can possibly be the 26th dimension?

Mathematicians regularly consider the geometry of spaces with arbitrary numbers of dimensions. This is in essence an abstract mathematical game that does not necessarily have any bearing on the structure of the real universe. The mathematics of the bosonic string model only works in an abstract spacetime of 26 dimensions. But even there it has fundamental problems that make it unsuitable as a model of the real universe.