Fermat's Last Theorem (27 page)

The simple square is relatively symmetric, possessing both rotational and reflectional symmetries, but it does not possess any translational symmetry. This means that if the square were to be shifted in any direction, an observer would spot the movement immediately because its position relative to the axes would have changed. However, if the whole of the space were tiled with squares, as shown in
Figure 16
, this infinite collection of squares would then have translational symmetry. If the infinite tiled surface were to be shifted up or down by one or more tile spaces, then the translated tiling would appear to be identical to the original one.

Figure 16. An infinite surface tiled with squares exhibits rotational and reflectional symmetry, and in addition has translational symmetry.

The symmetry of tiled surfaces is a relatively straightforward idea, but as with many seemingly simple concepts there are many subtleties hidden within it. For example, in the 1970s the British physicist and recreational mathematician Roger Penrose began dabbling with different tiles on the same surface. Eventually he identified two particularly interesting shapes, called the kite and the dart, which are shown in
Figure 17
. On their own, there is only one way these shapes can be used to tile a surface without gaps or overlaps, but together they could be used to create a rich set of tiling patterns. The kites and darts can be fitted together in an infinite number of ways, and although each pattern is apparently similar, in detail they all vary. One pattern made from kites and darts is shown in
Figure 17
.

Figure 17. By using two different tiles, the kite and the dart, Roger Penrose was able to cover a surface. However, Penrose tiling does not possess translational symmetry.

Another remarkable feature of Penrose tilings (the patterns generated by tiles such as the kite and dart) is that they can exhibit a very restricted level of symmetry. At first sight it would appear
that the tiling shown in
Figure 17
would have translational symmetry, and yet any attempt to shift the pattern across so that it effectively remains unchanged ends in failure. Penrose tilings are deceptively unsymmetrical, and this is why they fascinate mathematicians and have become the starting point for a whole new area of mathematics.

Curiously Penrose tiling has also had repercussions in material science. Crystallographers always believed that crystals had to be built on the principles behind square tiling, possessing a high level of translational symmetry. In theory building crystals relied on a highly regular and repetitive structure. However, in 1984 scientists discovered a metallic crystal made of aluminium and manganese which was built along Penrose principles. The mosaic of aluminium and manganese behaved like the kites and darts, generating a crystal which was almost regular, but not quite. A French company has recently developed a Penrose crystal into a coating for frying-pans.

While the fascinating thing about Penrose's tiled surfaces is their restricted symmetry, the interesting property of modular forms is that they exhibit infinite symmetry. The modular forms studied by Taniyama and Shimura can be shifted, switched, swapped, reflected and rotated in an infinite number of ways and still they remain unchanged, making them the most symmetrical of mathematical objects. When the French polymath Henri Poincaré studied modular forms in the nineteenth century, he had great difficulty coming to terms with their immense symmetry. After working on a particular type of modular form, he described to his colleagues how every day for two weeks he would wake up and try and find an error in his calculations. On the fifteenth day he realised and accepted that modular forms were indeed symmetrical in the extreme.

Unfortunately, drawing, or even imagining, a modular form is impossible. In the case of the square tiling we have an object which lives in two dimensions, its space being defined by the
x
-axis and
the
y
-axis. A modular form is also defined by two axes, but the axes are both complex, i.e. each axis has a real and an imaginary part and effectively becomes two axes. Therefore the first complex axis must be represented by two axes,
x
_r
-axis (real) and
x
_i
-axis (imaginary), and the second complex axis is represented by two axes,
y
_r
-axis (real) and
y
_i
-axis (imaginary). To be precise, modular forms live in the upper half-plane of this complex space, but what is most important to appreciate is that this is a four-dimensional space (
x
_r
,
x
_i
,
y
_r
,
y
_i
).

This four-dimensional space is called
hyperbolic space.
The hyperbolic universe is tricky to comprehend for humans, who are constrained to living in a conventional three-dimensional world, but four-dimensional space is a mathematically valid concept, and it is this extra dimension which gives the modular forms such an immensely high level of symmetry. The artist Mauritz Escher was fascinated by mathematical ideas and attempted to convey the concept of hyperbolic space in some of his etchings and paintings. Escher's
Circle Limit IV
embeds the hyperbolic world into the two-dimensional page. In true hyperbolic space the devils and angels would be the same size, and the repetition is indicative of the high level of symmetry. Although some of this symmetry can be seen on the two-dimensional page, there is an increasing distortion towards the edge of the picture.

The modular forms which live in hyperbolic space come in various shapes and sizes, but each one is built from the same basic ingredients. What differentiates each modular form is the amount of each ingredient it contains. The ingredients of a modular form are labelled from one to infinity (
M
₁
,
M
₂
,
M
₃
,
M
₄
, …) and so a particular modular form might contain one lot of ingredient one (
M
₁
= 1), three lots of ingredient two (
M
₂
= 3), two lots of ingredient three (
M
₃
= 2), etc. This information describing how a modular
form is constructed can be summarised in a so-called modular series, or
M
-series, a list of the ingredients and the quantity of each one required:

Just as the
E
-series is the DNA for elliptic equations, the
M
-series is the DNA for modular forms. The amount of each ingredient listed in the
M
-series is critical. Depending how you change the amount of, say, the first ingredient you might generate a completely different, but equally symmetrical, modular form, or you might destroy the symmetry altogether and generate a new object which is not a modular form. If the quantity of each ingredient is arbitrarily chosen, then the result will probably be an object with little or no symmetry.

Modular forms stand very much on their own within mathematics. In particular, they would seem to be completely unrelated to the subject that Wiles would study at Cambridge, elliptic equations. The modular form is an enormously complicated beast, studied largely because of its symmetry and only discovered in the nineteenth century. The elliptic equation dates back to the ancient Greeks and has nothing to do with symmetry. Modular forms and elliptic equations live in completely different regions of the mathematical cosmos, and nobody would ever have believed that there was the remotest link between the two subjects. However, Taniyama and Shimura were to shock the mathematical community by suggesting that elliptic equations and modular forms were effectively one and the same thing. According to these two maverick mathematicians, they could unify the modular and elliptic worlds.

In September 1955 an international symposium was held in Tokyo. It was a unique opportunity for the many young Japanese researchers to show off to the rest of the world what they had learned. They handed round a collection of thirty-six problems related to their work, accompanied by a humble introduction –
Some unsolved problems in mathematics: no mature preparation has been made, so there may be some trivial or already solved ones among these. The participants are requested to give comments on any of these problems.

Four of the questions were from Taniyama, and these hinted at a curious relationship between modular forms and elliptic equations. These innocent questions would ultimately lead to a revolution in number theory. Taniyama had looked at the first few terms in the
M
-series of a particular modular form. He recognised the pattern and realised that it was identical to the list of numbers in the
E
-series of a well-known elliptic equation. He calculated a few more terms in each series, and still the
M
-series of the modular form and
E
-series of the elliptic equation matched perfectly.

This was an astonishing discovery because, for no apparent reason, this modular form could be related to an elliptic equation through their respective
M
-series and
E
-series – these series were identical. The mathematical DNA which made up these two entities was exactly the same. This was a doubly profound discovery. First, it suggested that deep down there was a fundamental relationship between the modular form and the elliptic equation, objects which come from opposite ends of mathematics. Second, it meant that mathematicians, who already knew the
M
-series for the modular form, would not have to calculate the
E
-series for the corresponding elliptic equation because it would be the same as the
M
-series.

Relationships between apparently different subjects are as creatively important in mathematics as they are in any discipline. The relationship hints at some underlying truth which enriches both subjects. For instance, originally scientists had studied electricity and magnetism as two completely separate phenomena. Then, in the nineteenth century, theorists and experimentalists realised that electricity and magnetism were intimately related. This resulted in a deeper understanding of both of them. Electric currents generate magnetic fields, and magnets can induce electricity in wires passing close to them. This led to the invention of dynamos and electric
motors, and ultimately the discovery that light itself is the result of magnetic and electric fields oscillating in harmony.

Taniyama examined a few other modular forms and in each case the
M
-series seemed to correspond perfectly with the
E
-series of an elliptic equation. He began to wonder if it could be that every single modular form could be matched with an elliptic equation. Perhaps every modular form has the same DNA as an elliptic equation: perhaps each modular form is an elliptic equation in disguise? The questions he handed out at the symposium were related to this hypothesis.

The idea that every elliptic equation was related to a modular form was so extraordinary that those who glanced at Taniyama's questions treated them as nothing more than a curious observation. Sure enough Taniyama had demonstrated that a few elliptic equations could be related to particular modular forms, but they claimed that this was nothing more than a coincidence. According to the sceptics Taniyama's claim of a more general and universal relationship seemed to be largely unsubstantiated. The hypothesis was based on intuition rather than on any real evidence.