Hyperspace (7 page)

Riemann was born in 1826 in Hanover, Germany, the son of a poor Lutheran pastor, the second of six children. His father, who fought in the Napoleonic Wars, struggled as a country pastor to feed and clothe his large family. As biographer E. T. Bell notes, “the frail health and early deaths of most of the Riemann children were the result of undernourishment in their youth and were not due to poor stamina. The mother also died before her children were grown.”
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At a very early age, Riemann exhibited his famous traits: fantastic calculational ability, coupled with timidity, and a life-long horror of any public speaking. Painfully shy, he was the butt of cruel jokes by other boys, causing him to retreat further into the intensely private world of mathematics.

He also was fiercely loyal to his family, straining his poor health and constitution to buy presents for his parents and especially for his beloved sisters. To please his father, Riemann set out to become a student of theology. His goal was to get a paying position as a pastor as quickly as possible to help with his family’s abysmal finances. (It is difficult to imagine a more improbable scenario than that of a tongue-tied, timid young boy imagining that he could deliver fiery, passionate sermons railing against sin and driving out the devil.)

In high school, he studied the Bible intensely, but his thoughts always drifted back to mathematics; he even tried to provide a mathematical proof of the correctness of Genesis. He also learned so quickly that he kept outstripping the knowledge of his instructors, who found it impossible to keep up with the boy. Finally, the principal of his school gave
Riemann a ponderous book to keep him occupied. The book was Adrien-Marie Legendre’s
Theory of Numbers
, a huge 859-page masterpiece, the world’s most advanced treatise on the difficult subject of number theory. Riemann devoured the book in 6 days.

When his principal asked, “How far did you read?” the young Riemann replied, “That is certainly a wonderful book. I have mastered it.” Not really believing the bravado of this youngster, the principal several months later asked obscure questions from the book, which Riemann answered perfectly.
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Beset by the daily struggle to put food on the table, Riemann’s father might have sent the boy to do menial labor. Instead, he scraped together enough funds to send his 19-year-old son to the renowned University of Göttingen, where he first met Carl Friedrich Gauss, the acclaimed “Prince of Mathematicians,” one of the greatest mathematicians of all time. Even today, if you ask any mathematician to rank the three most famous mathematicians in history, the names of Archimedes, Isaac Newton, and Carl Gauss will invariably appear.

Life for Riemann, however, was an endless series of setbacks and hardships, overcome only with the greatest difficulty and by straining his frail health. Each triumph was followed by tragedy and defeat. For example, just as his fortunes began to improve and he undertook his formal studies under Gauss, a full-scale revolution swept Germany. The working class, long suffering under inhuman living conditions, rose up against the government, with workers in scores of cities throughout Germany taking up arms. The demonstrations and uprisings in early 1848 inspired the writings of another German, Karl Marx, and deeply affected the course of revolutionary movements throughout Europe for the next 50 years.

With all of Germany swept up in turmoil, Riemann’s studies were interrupted. He was inducted into the student corps, where he had the dubious honor of spending 16 weary hours protecting someone even more terrified than he: the king, who was quivering with fear in his royal palace in Berlin, trying to hide from the wrath of the working class.

Not only in Germany, but in mathematics, too, fierce revolutionary winds were blowing. The problem that riveted Riemann’s interest was the impending collapse of yet another bastion of authority, Euclidean geometry,
which holds that space is three dimensional. Furthermore, this three-dimensional space is “flat” (in flat space, the shortest distance between two points is a straight line; this omits the possibility that space can be curved, as on a sphere).

In fact, after the Bible, Euclid’s
Elements was
probably the most influential book of all time. For 2 millennia, the keenest minds of Western civilization have marveled at its elegance and the beauty of its geometry. Thousands of the finest cathedrals in Europe were erected according to its principles. In retrospect, perhaps it was too successful. Over the centuries, it became something of a religion; anyone who dared to propose curved space or higher dimensions was relegated to the ranks of crackpots or heretics. For untold generations, schoolchildren have wrestled with the theorems of Euclid’s geometry: that the circumference of a circle is π times the diameter, and that the angles within a triangle add up to 180 degrees. However, try as they might, the finest mathematical minds for several centuries could not prove these deceptively simple propositions. In fact, the mathematicians of Europe began to realize that even Euclid’s
Elements
, which had been revered for 2,300 years, was incomplete. Euclid’s geometry was still viable if one stayed within the confines of flat surfaces, but if one strayed into the world of curved surfaces, it was actually incorrect.

To Riemann, Euclid’s geometry was particularly sterile when compared with the rich diversity of the world. Nowhere in the natural world do we see the flat, idealized geometric figures of Euclid. Mountain ranges, ocean waves, clouds, and whirlpools are not perfect circles, triangles, and squares, but are curved objects that bend and twist in infinite diversity.

The time was ripe for a revolution, but who would lead it and what would replace the old geometry?

Riemann rebelled against the apparent mathematical precision of Greek geometry, whose foundation, he discovered, ultimately was based on the shifting sand of common sense and intuition, not the firm ground of logic.

It is obvious, said Euclid, that a point has no dimension at all. A line has one dimension: length. A plane has two dimensions: length and breadth. A solid has three dimensions: length, breadth, and height. And there it stops. Nothing has four dimensions. These sentiments were echoed
by the philosopher Aristotle, who apparently was the first person to state categorically that the fourth spatial dimension is impossible. In
On Heaven
, he wrote, “The line has magnitude in one way, the plane in two ways, and the solid in three ways, and beyond these there is no other magnitude because the three are all.” Furthermore, in A.D. 150, the astronomer Ptolemy from Alexandria went beyond Aristotle and offered, in his book
On Distance
, the first ingenious “proof that the fourth dimension is impossible.

First, he said, draw three mutually perpendicular lines. For example, the corner of a cube consists of three mutually perpendicular lines. Then, he argued, try to draw a fourth line that is perpendicular to the other three lines. No matter how one tries, he reasoned, four mutually perpendicular lines are impossible to draw. Ptolemy claimed that a fourth perpendicular line is “entirely without measure and without definition.” Thus the fourth dimension is impossible.

What Ptolemy actually proved was that it is impossible to visualize the fourth dimension with our three-dimensional brains. (In fact, today we know that many objects in mathematics cannot be visualized but can be shown to exist.) Ptolemy may go down in history as the man who opposed two great ideas in science: the sun-centered solar system and the fourth dimension.

Over the centuries, in fact, some mathematicians went out of their way to denounce the fourth dimension. In 1685, the mathematician John Wallis polemicized against the concept, calling it a “Monster in Nature, less possible than a Chimera or Centaure…. Length, Breadth, and Thickness, take up the whole of Space. Nor can Fansie imagine how there should be a Fourth Local Dimension beyond these Three.”
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For several thousand years, mathematicians would repeat this simple but fatal mistake, that the fourth dimension cannot exist because we cannot picture it in our minds.

The decisive break with Euclidean geometry came when Gauss asked his student Riemann to prepare an oral presentation on the “foundation of geometry.” Gauss was keenly interested in seeing if his student could develop an alternative to Euclidean geometry. (Decades before, Gauss had privately expressed deep and extensive reservations about Euclidean geometry. He even spoke to his colleagues of hypothetical “bookworms” that might live entirely on a two-dimensional surface. He spoke of generalizing
this to the geometry of higher-dimensional space. However, being a deeply conservative man, he never published any of his work on higher dimensions because of the outrage it would create among the narrow-minded, conservative old guard. He derisively called them “Boeotians” after a mentally retarded Greek tribe.
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)

Riemann, however, was terrified. This timid man, terrified of public speaking, was being asked by his mentor to prepare a lecture before the entire faculty on the most difficult mathematical problem of the century.

Over the next several months, Riemann began painfully developing the theory of higher dimensions, straining his health to the point of a nervous breakdown. His stamina further deteriorated because of his dismal financial situation. He was forced to take low-paying tutoring jobs to provide for his family. Furthermore, he was becoming sidetracked trying to explain problems of physics. In particular, he was helping another professor, Wilhelm Weber, conduct experiments in a fascinating new field of research, electricity.

Electricity, of course, had been known to the ancients in the form of lightning and sparks. But in the early nineteenth century, this phenomenon became the central focus of physics research. In particular, the discovery that passing a current of wire across a compass needle can make the needle spin riveted the attention of the physics community. Conversely, moving a bar magnet across a wire can induce an electric current in the wire. (This is called Faraday’s Law, and today all electric generators and transformers—and hence much of the foundation of modern technology—are based on this principle.)

To Riemann, this phenomenon indicated that electricity and magnetism are somehow manifestations of the same force. Riemann was excited by the new discoveries and was convinced that he could give a mathematical explanation that would unify electricity and magnetism. He immersed himself in Weber’s laboratory, convinced that the new mathematics would yield a comprehensive understanding of these forces.

Now, burdened with having to prepare a major public lecture on the “foundation of geometry,” to support his family, and to conduct scientific experiments, his health finally collapsed and he suffered a nervous breakdown in 1854. Later, he wrote to his father, “I became so absorbed in my investigation of the unity of all physical laws that when the subject of the trial lecture was given me, I could not tear myself away from my research. Then, partly as a result of brooding on it, partly from staying indoors too much in this vile weather, I fell ill.”
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This letter is significant, for it clearly shows that, even during months of illness,
Riemann firmly believed that he would discover the “unity of all physical laws” and that mathematics would eventually pave the way for this unification.

Force = Geometry

Eventually, despite his frequent illnesses, Riemann developed a startling new picture of the meaning of a “force.” Ever since Newton, scientists had considered a force to be an instantaneous interaction between two distant bodies. Physicists called it action-at-a-distance, which meant that a body could influence the motions of distant bodies instantaneously. Newtonian mechanics undoubtedly could describe the motions of the planets. However, over the centuries, critics argued that action-at-a-distance was unnatural, because it meant that one body could change the direction of another without even touching it.

Riemann developed a radically new physical picture. Like Gauss’s “bookworms,” Riemann imagined a race of two-dimensional creatures living on a sheet of paper. But the decisive break he made was to put these bookworms on a
crumpled
sheet of paper.
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What would these bookworms think about their world? Riemann realized that they would conclude that their world was still perfectly flat. Because their bodies would also be crumpled, these bookworms would never notice that their world was distorted. However, Riemann argued that if these bookworms tried to move across the crumpled sheet of paper, they would feel a mysterious, unseen “force” that prevented them from moving in a straight line. They would be pushed left and right every time their bodies moved over a wrinkle on the sheet.

Thus Riemann made the first momentous break with Newton in 200 years, banishing the action-at-a-distance principle. To Riemann,
“force” was a consequence of geometry
.

Riemann then replaced the two-dimensional sheet with our three-dimensional world crumpled in the fourth dimension. It would not be obvious to us that our universe was warped. However, we would immediately realize that something was amiss when we tried to walk in a straight line. We would walk like a drunkard, as though an unseen force were tugging at us, pushing us left and right.

Riemann concluded that electricity, magnetism, and gravity are caused by the crumpling of our three-dimensional universe in the unseen fourth dimension. Thus a “force” has no independent life of its own; it is only the apparent effect caused by the distortion of geometry.
By introducing the fourth spatial dimension, Riemann accidentally stumbled on what would become one of the dominant themes in modern theoretical physics, that the laws of nature appear simple when expressed in higher-dimensional space. He then set about developing a mathematical language in which this idea could be expressed.