The Ascent of Man (17 page)

In a western civilisation, this room would be filled with marvellous drawings of the female form,
erotic pictures. Not so here. The representation of the human body was forbidden to Mohammedans. Indeed, even the study of anatomy at all was forbidden, and that was a major handicap to Moslem science. So here we find coloured but extraordinarily simple geometric designs. The artist and the mathematician in Arab civilisation have become one. And I mean that quite literally. These patterns represent
a high point of the Arab exploration of the subtleties and symmetries of space itself: the flat, two-dimensional space of what we now call the Euclidean plane, which Pythagoras first characterised.

In the wealth of patterns, I begin with a very straightforward one. It repeats a two-leaved motif of dark horizontal leaves, and another of light vertical leaves. The obvious symmetries are translations
(that is, parallel shifts of the pattern) and either horizontal or vertical reflections. But note one more delicate point. The Arabs were fond of designs in which the dark and the light units of the pattern are identical. And so, if for a moment you ignore the colours, you can see that you could turn a dark leaf once through a right angle into the position of a neighbouring light leaf. Then,
always rotating round the same point of junction, you can turn it into the next position, and (again round the same point) into the next, and finally back on itself. And the rotation spins the whole pattern correctly; every leaf in the pattern arrives at the position of another leaf, however far from the centre of rotation they lie.

Reflection in a horizontal line is a twofold symmetry of the coloured pattern, and so is reflection in a vertical. But if we ignore the colours, we see that there is a fourfold symmetry. It is provided by the operation of rotating through a right angle, repeated four times, by which I earlier proved the theorem of Pythagoras; and thereby the uncoloured pattern becomes in its symmetry like the
Pythagorean square.

I turn to a much more subtle pattern. These windswept triangles in four colours display only one very straightforward kind of symmetry, in two directions. You could shift the pattern horizontally or you could shift it vertically into new, identical positions. Being windswept is not irrelevant. It is unusual to find a symmetrical system which does not allow reflection. However,
this one does not, because these windswept triangles are all right-handed in movement, and you cannot reflect them without making them left-handed.

Now suppose you neglect the difference between the green, the yellow, the black, and the royal blue, and think of the distinction as simply between dark triangles and light triangles. Then there is also a symmetry of rotation. Fix your attention again
on a point of junction: six triangles meet there, and they are alternately dark and light. A dark triangle can be rotated there into the position of the next dark triangle, then into the position of the next, and finally back into the original position – a threefold symmetry which rotates the whole pattern.

And indeed the possible symmetries need not stop there. If you forget about the colours
at all, then there is a lesser rotation by which you could move a dark triangle into the space of the light triangle beside it because it is identical in shape. This operation of rotation then goes on into the dark, into the light, into the dark, into the light, and finally back into the original dark triangle – a sixfold symmetry of space which rotates the whole pattern. And
the sixfold symmetry
in fact is the one we all know best, because it is a symmetry of the snow crystal.

At this point, the non-mathematician is entitled to ask, ‘So what? Is that what mathematics is about? Did Arab professors, do modern mathematicians, spend their time with that kind of elegant game?’ To which the unexpected answer is – Well, it is not a game. It brings us face to face with something which is hard to remember, and that is that we live in a special kind of space – three-dimensional,
flat – and the properties of that space are unbreakable. In asking what operations will turn a pattern into itself, we are discovering the invisible laws that govern our space. There are only certain kinds of symmetries which our space can support, not only in man-made patterns, but in the regularities which nature herself imposes on her fundamental, atomic structures.

The structures that enshrine,
as it were, the natural patterns of space are the crystals. And when you look at one untouched by human hand – say, iceland spar – there is a shock of surprise in realising that it is not self-evident why its faces should be regular. It is not self-evident why they should even be flat planes. This is how crystals come; we are used to their being regular and symmetrical; but why? They were not
made that way by man but by nature. That flat face is the way in which the atoms had to come together – and that one, and that one. The flatness, the regularity has been forced by space on matter with the same finality as space gave the Moorish patterns their symmetries that I analysed.

Take a beautiful cube of pyrites. Or to me the most exquisite crystal of all, fluorite, an octahedron. (It
is also the natural shape 0f the diamond crystal.) Their symmetries are imposed on them by the nature of the space we live in – the three dimensions, the flatness within which we live. And no assembly of atoms can break that crucial law of nature. Like the units that compose a pattern, the atoms in a crystal are stacked in all directions. So a crystal, like a pattern, must have a shape that could
extend or repeat itself in all directions indefinitely. That is why the faces of a crystal can only have certain shapes; they could not have anything but the symmetries in the patterns. For example, the only rotations that are possible go twice or four times for a full turn, or three times or six times – not more. And not five times. You cannot make an assembly of atoms to make triangles which fit
into space regularly five at a time.

Thinking about these forms of pattern, exhausting in practice the possibilities of the symmetries of space (at least in two dimensions), was the great achievement of Arab mathematics. And it has a wonderful finality, a thousand years old. The king, the naked women, the eunuchs and the blind musicians made a marvellous formal pattern in which the exploration
of what exists was perfect, but which, alas, was not looking for any change. There is nothing new in mathematics, because there is nothing new in human thought, until the ascent of man moved forward to a different dynamic.

Christianity began to surge back in northern Spain about
AD
1000 from footholds like the village of Santillana in a coastal strip which the Moors never conquered. It is a religion
of the earth there, expressed in the simple images of the village – the ox, the ass, the Lamb of God. The animal images would be unthinkable in Moslem worship. And not only the animal form is allowed; the Son of God is a child, His mother is a woman and is the object of personal worship. When the Virgin is carried in procession, we are in a different universe of vision: not
of abstract patterns,
but of abounding and irrepressible life.

When Christianity came to win back Spain, the excitement of the struggle was on the frontier. Here Moors and Christians, and Jews too, mingled and made an extraordinary culture of different faiths. In 1085 the centre of this mixed culture was fixed for a time in the city of Toledo. Toledo was the intellectual port of entry into Christian Europe of all
the classics that the Arabs had brought together from Greece, from the Middle East, from Asia.

We think of Italy as the birthplace of the Renaissance. But the conception was in Spain in the twelfth century, and it is symbolised and expressed by the famous school of translators at Toledo, where the ancient texts were turned from Greek (which Europe had forgotten) through Arabic and Hebrew into
Latin. In Toledo, amid other intellectual advances, an early set of astronomical tables was drawn up, as an encyclopedia of star positions. It is characteristic of the city and the time that the tables are Christian, but the numerals are Arabic, and are by now recognisably modern.

The most famous of the translators and the most brilliant was Gerard of Cremona, who had come from Italy specifically
to find a copy of Ptolemy’s book of astronomy, the
Almagest
, and who stayed on in Toledo to translate Archimedes, Hippocrates, Galen, Euclid – the classics of Greek science.

And yet, to me personally, the most remarkable and, in the long run, the most influential man who was translated was not a Greek. That is because I am interested in the perception of objects in space. And that was a subject
about which the Greeks were totally wrong. It was understood for the first time about the year
AD
1000 by an eccentric mathematician whom we call Alhazen, who was the one really original scientific mind that Arab culture produced. The Greeks had thought that light goes from the eyes to the object. Alhazen first recognised that we see an object because each point of it directs and reflects a ray
into the eye. The Greek view could not explain how an object, my hand say, seems to change size when it moves. In Alhazen’s account it is clear that the cone of rays that comes from the outline and shape of my hand grows narrower as I move my hand away from you. As I move it towards you, the cone of rays that enters your eye becomes larger and subtends a larger angle. And that, and only that, accounts
for the difference in size. It is so simple a notion that it is astonishing that scientists paid almost no attention to it (Roger Bacon is an exception) for six hundred years. But artists attended to it long before that, and in a practical way. The concept of the cone of rays from object to the eye becomes the foundation of perspective. And perspective is the new idea which now revivifies mathematics.

The excitement of perspective passed into art in north Italy, in Florence and Venice, in the fifteenth century. A manuscript of Alhazen’s
Optics
in translation in the Vatican Library in Rome is annotated by Lorenzo Ghiberti, who made the famous bronze perspectives for the doors of the Baptistry in Florence. He was not the first pioneer of perspective – that may have been Filippo Brunelleschi –
and there were enough of them to form an identifiable school of the Perspectivi. It was a school of thought, for its aim was not simply to make the figures lifelike, but to create the sense of their movement in space.

The movement is evident as soon as we contrast a work by the Perspectivi with an earlier one. Carpaccio’s painting of St Ursula leaving a vaguely Venetian port was painted in 1495.
The obvious effect is to give to visual space a third dimension, just as the ear about this time hears another depth and dimension in the new harmonies in European music. But the ultimate effect is not so much depth as movement. Like the new music, the picture
and its inhabitants are mobile. Above all, we feel that the painter’s eye is on the move.

Contrast a fresco of Florence painted a hundred
years earlier, about
AD
1350. It is a view of the city from outside the walls, and the painter looks naively over the walls and the tops of the houses as if they were arranged in tiers. But this is not a matter of skill; it is a matter of intention. There is no attempt at perspective because the painter thought of himself as recording things, not as they look, but as they are: a God’s eye view,
a map of eternal truth.

The perspective painter has a different intention. He deliberately makes us step away from any absolute and abstract view. Not so much a place as a moment is fixed for us, and a fleeting moment: a point of view in time more than in space. All this was achieved by exact and mathematical means. The apparatus has been recorded with care by the German artist, Albrecht Dürer,
who travelled to Italy in 1506 to learn ‘the secret art of perspective’. Dürer of course has himself fixed a moment in time; and if we re-create his scene, we see the artist choosing the dramatic moment. He could have stopped early in his walk round the model. Or he could have moved, and frozen the vision at a later moment. But he chose to open his eye, like a camera shutter, understandably at
the strong moment, when he sees the model full face. Perspective is not one point of view; for the painter, it is an active and continuous operation.