Extreme Maths: Infinity - The Art of the Infinite

The true mysteries of mathematics lie at the limits of our thinking - infinity. Reach beyond what you think is possible and you start to explore the wonders of maths at the extremes.

When you look down the tracks you won't see a train, since these days the trains never seem to be running - but you will see infinity. Follow that point where the tracks meet - it must be no more than a mile away; but when you get there, where has it gone? Vanished again.

The Renaissance painters made sense of pictorial space by organising it around this vanishing point. That wonderful Florentine, Leon Battista Alberti, described how he translated three dimensions into two: "I make a veil, loosely woven of fine thread, dyed whatever colour you please, divided up by thicker threads into as many parallel squares as you like, and stretched on a frame. I set this up between the eye and the object to be represented, so that the visual pyramid passes through the loose weave of the veil."

One-point perspective gave rise in time to two vanishing points – but can you picture a space with an endless number of points at infinity – one for each point on the horizon? It takes outrageous imagination and an Art of Extremes to do this - and that is exactly what mathematics is. Just as heard melodies are sweet but those unheard are sweeter still, so what the eye can be trained to see is marvellous - but the mind's eye sees wonders incomparably greater.

A painting, a piece of sculpture, a work of architecture are organised above all by gracefully balanced masses - and again the infinite in mathematics lies in the hidden, central room of this art of proportion. Which rectangle has the most pleasing ratio of length to width, for example? Experiment for yourself by making a rigid frame with a sliding fourth side, and test where your friends think best to halt it. Was the ratio of width to length close to 2 x 3, 3 x 5, 5 x 8 - the standard sizes of index cards and photos? Or to some other pair of adjacent numbers in the sequence:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...?

Leonardo of Pisa, nicknamed Fibonacci, was an Italian mathematician at the start of the thirteenth century, who showed how to form this elegant sequence of numbers, which is linked to our understanding of what makes pleasing proportions. Have you discovered how to make new terms of his sequence? The 'secret' is to add the previous two. But why should that sequence have anything to do with pleasing proportions?

The great Piero della Francesca wrote a book, "On the Divine Proportion", and in his paintings framed their parts and the whole in rectangles with these ratios. Leonardo da Vinci saw that tree branches, as they spiral up the trunk, are spaced in these proportions too. Virtually all artists work on these principles, whether they realize it or not. Pine cones and nautilus shells, antlers, the crossing rows on a sunflower's head - again and again these Fibonacci ratios appear in nature. But it is only in mathematics, the art of the infinite, that these ratios are carried beyond the visible to their invisible extreme: 1/1, 2/1, 3/2, 5/3, 8/5, 13/8 and so on approach a certain number, called The Golden Mean (roughly 1.618...): which describes the ideal proportion that the finite ratios in art and nature only approximate.