The mathematician Henri Poincare was once asked, after a public lecture on space and time, how we know that we live in a space of three dimensions. He supposedly replied, "Because prison walls are two-dimensional."

He chose a colorful way to state a fact already well accepted by mathematicians. In a space of two dimensions, such as the surface of a sheet of paper, we can completely enclose any given area by using a one-dimensional line, such as a circle. In a space of three dimensions, we can completely enclose any given volume by surrounding it with a two-dimensional region, such as the surface of a sphere.

Poincare was the most famous mathematician of his day and a man with great geometric insight (despite being personally so nearsighted he could never as a student see what his professors were writing on the blackboard). Even so, I think he would have been astonished to learn that according to one of today's leading theories, we don't live in a space of three dimensions at all; we live in one that has either ten or eleven dimensions.

Scientists did not come to this conclusion easily or willingly. They were driven to it because the two cornerstones on which twentieth-century physics had been built, namely, general relativity and quantum theory, are inconsistent with each other. General relativity was developed to describe large-scale features of the universe, where gravity plays a controlling influence. Quantum theory describes the world of the very small, atoms and their components, where gravitational forces can be ignored. However, each theory should in principle extend to cover the range of the other. Unfortunately, when general relativity was applied at very small scales it predicted a chaotic universe, where calculations often led to infinite answers. Physics was flawed, and at its most basic level.

This was not news to scientists. They had been struggling to reconcile relativity and quantum theory from the late 1920s onward, with little success. A breakthrough of sorts was made in 1968, though it was not seen that way at the time. Gabriele Veneziano was trying to make sense of some experimental results of nuclear physics. He noticed that an element of pure mathematics known as the beta function seemed to apply. The beta function had been studied since the eighteenth century, and its properties were well known; but what on earth did it have to do with tiny interacting particles? Veneziano could not say.

A couple of years later, three other physicists made a strange and at first tentative suggestion. If the "particles" that quantum theory deals with, such as electrons and quarks, were not particles at all, but were formed from tiny vibrating strings, then the beta function would appear in the calculations, exactly as Veneziano had noted.

The reaction of most scientists was, "interesting, but wrong." The new "string theory" made other predictions that were at odds with observation. Few people pursued the idea further.

Three who did were John Schwarz, Michael Green, and Joel Scherk. However, not until 1984 were Schwarz and Green able to put together a model free of the earlier false predictions. More than that, the new model was unlike all earlier versions of quantum theory, because gravity appeared in it naturally and inevitably.

There was only one hitch. The string loops of the theory were incredibly small - a hundred billion billion times as small as an atomic nucleus. That was all right, and even desirable, since it explained why no one had ever observed such loops. What was not all right was that the little vibrating loops could not exist in ordinary space. They needed extra dimensions, in which they could be curled up. How many dimensions? Well, a total of ten or eleven, depending on the version of string theory. Later it proved desirable that the strings themselves could also be two- or three-dimensional membranes ("branes"), instead of simple lines.

The idea of extra, curled-up dimensions was not totally alien to physicists. It had actually been proposed in 1919, by the Polish mathematician Theodor Kaluza, and extended in 1926 by Oskar Klein. Their objective was a theory that united general relativity and electromagnetism. Unfortunately, results calculated from the theory were in conflict with experiment, and in that situation theory always loses.

The universe was not as described by Kaluza and Klein. However, if you are willing to swallow the extra dimensions needed by string theory, and the mathematical complications that go with them, then the situation is much more promising. The inconsistencies that marred the marriage of quantum theory and general relativity all go away. The long-sought "unified field theory" has been found.

Or has perhaps been found. One problem that plagued string theory, and continues to plague it today, is that the mathematics involved is fiendishly difficult, so much so that the right equations can hardly be stated, still less fully solved. In fact, maybe the appropriate mathematics has yet to be invented.

Although this mathematical lack may seem natural, it is not the historical experience of science. Almost always, the mathematics needed for a new physical theory has sat there, ready and waiting. The Greek geometers had long ago provided the mathematics of conic sections, when Kepler needed it to state his laws of planetary motion. The calculus, invented by Newton and Leibniz, was available when Newton himself evolved his theory of universal gravitation. The absolute differential calculus of Ricci and Levi-Civita was waiting for Einstein when he had need of it in general relativity; and matrices, developed in the middle of the nineteenth century, were on hand to be used in quantum theory during the 1920s.

String theorists and brane theorists have not been so lucky. They are obliged to create for themselves the new mathematics needed. And they are doing it, to the point where physicists are receiving the top awards of mathematics for their work.

Daniele Amati and Ed Witten stated in the 1980s: "String theory is a part of twenty-first-century physics that fell by chance into the twentieth century."

And now, here we are in the twenty-first century. It's time for twenty-first-century physics.

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