In spite of the compliment at the end, it was a crushing blow to the younger Bolyai. Gauss was saying that his discovery of non-Euclidean geometry was nothing new. Janos never published another mathematical paper in his life. Not only had Gauss lacked the courage to publish the discovery himself, he had now compounded his mistake by discouraging an aspiring young mathematician who might have made a great name for himself.
Because Gauss was too reticent, and Bolyai gave up too easily, the third discoverer of non-Euclidean geometry deserves the most credit for bringing it to the world’s attention. He was Nikolai Ivanovich Lobachevsky, a Russian mathematician who lived in Kazan, the ancient capital of the Tatars. He first published his version of non-Euclidean geometry in 1829 in a very obscure Russian journal, but unlike Bolyai he continued to write articles and books about it and finally succeeded in getting an article into Crelle’s Journal in 1837. Even so, he did not receive the kind of acclaim during his lifetime that one might expect. Today, however, Lobachevski is considered one of the first great Russian mathematicians, and in Russia his geometry is called Lobachevskian. Western mathematicians call it, more descriptively, hyperbolic geometry.
What exactly is hyperbolic, or Lobachevskian, geometry? I think that the best way to think about it is to forget all about the Parallel Postulate and about Euclid. You must especially forget about the prejudice that you have surely been brought up with, that Euclidean is the “natural” geometry of the real world. Hyperbolic geometry is no more artificial than Euclidean. Think of it as the geometry of the ocean. If whales had invented geometry, the geometry they would have invented would be hyperbolic.
Suppose, for a moment, that you are a whale. Light is not very useful in the deep ocean, because the water is dark. So you mostly communicate and experience the world through sound. The shortest distance between two points in your world would be the path taken by sound waves. To you, this would be the analogue of a straight line.
Now here’s the catch. Sound does not travel at a constant speed in the ocean. Below a certain depth, roughly 2000 feet (600 meters), it travels at a speed that is proportional to the depth below the surface. So the path that sound waves travel is not straight, but curved. A sound wave will get from whale A to whale B quicker if it goes downward, to exploit the greater sound speed at depth, and then comes back up. In fact, we can be more precise about the nature of these curves: they are arcs of circles centered at the ocean surface! Thus, to a whale, what humans call a “circle” is actually a “line” (the shortest distance between two points).
Below Demonstration of the curves along which sound travels in the ocean.
Whale Geometry is a geometry where some surprising (to us) things happen, but they would not be the least bit surprising to whales. The sum of the angles of a triangle is less than 180 degrees. Rectangles (four-sided figures with all right angles) do not exist; however, right-angled pentagons do. Most importantly, it is a geometry of negative curvature. This means that lines that start out parallel tend to move farther and farther apart.
AMAZINGLY, ANOTHER non-Euclidean geometry, besides hyperbolic geometry, had been known for centuries—only no one ever thought of it in those terms. It is the geometry of a sphere. On the surface of a sphere (such as Earth), the sum of the angles of a triangle is greater than 180 degrees. Rectangles do not exist, but right-angled triangles do. Keep in mind the curvature of the Earth! For example, a triangle can be drawn with three right angles: start at the North Pole, travel in a straight line down to the Equator, then travel due east or west a quarter of the way around the globe, and then go due north again. You will trace out a triangle with three 90-degree angles. Spherical geometry is a geometry of positive curvature. In other words, lines that start out parallel (such as meridians, near the Equator) tend to move closer and closer together, and they eventually converge at the poles.
The reason that no one ever thought of spherical geometry as an alternative to Euclidean geometry is simple: We can see a sphere as being imbedded in three-dimensional Euclidean space, so its “non-Euclideanness” is not immediately obvious. Suppose, however, that you were unable to perceive a third dimension beyond the surface of the sphere. For example, perhaps you are an ant, living on the surface of an asteroid with no oceans (so you can go anywhere you want to). You have no concept of space, no concept of underground; everything you know is the surface of your spherical world. The curvature of that world is positive and its geometry is non-Euclidean. We could call it Ant Geometry.
Left Spherical geometry and the curvature of the Earth.
Instead of one geometry of nature, we can now see there is a whole spectrum of geometries with different amounts of curvature, ranging from Ant Geometry (spherical) to Human Geometry (Euclidean) to Whale Geometry (hyperbolic). But that’s not all. These are only the geometries of constant curvature. We can also imagine geometries whose curvature varies from place to place. They can be two-dimensional, three-dimensional, or even higher. Gauss (perhaps influenced by his unpublished thoughts on hyperbolic geometry) was the first mathematician to understand the concept of varying curvature in a two-dimensional space, and his student Bernhard Riemann extended the concept to higher dimensions in 1854. Both of them thus anticipated one of the epochal discoveries of the twentieth century: Albert Einstein’s theory of general relativity, which postulates that our four-dimensional spacetime has curvature that varies from place to place. Without Lobachevski, Bolyai, Gauss, and Riemann, Einstein would never have been able to write down the equations for his theory.
Above An engraving displaying an “Allegory of Geometry,” by F. Floris, 16th century.
16
in primes we trust the prime number theorem
Gauss’s mishandling of the discovery of non-Euclidean geometry was one of the few black marks on an otherwise remarkable career. He contributed so much to so many parts of the subject, and he virtually created the modern subject of number theory, which deals with the properties of whole numbers and especially with the solution of equations in whole numbers.
Gauss was born in 1777 in Brunswick, Germany. As a child prodigy, Gauss attracted the attention of the Prince of Brunswick, who supported him through preparatory school and the University of Göttingen, where he earned his doctorate in 1799 with his first, not entirely satisfactory proof of the Fundamental Theorem of Algebra. (He later gave three more proofs.)
Gauss always had a special place in his heart for number theory, a subject he called the “queen of mathematics.” His earliest significant discoveries came in this subject. In 1796, still a student at the university, Gauss proved that a regular 17-sided polygon can be constructed by a ruler and compass—a discovery that had eluded the ancient Greeks, who first took an interest in such construction problems. Although it looks like a theorem of geometry, this theorem is closely linked to the solvability of polynomials.
The key question is whether the angle (360/17)° can be constructed with ruler and compass. If so, the 17-gon can be constructed simply by piecing together 17 isosceles triangles with this angle at their vertex. As long ago as 1637, in his book La Géometrie (which Gauss had surely studied), René Descartes had found a simple criterion for constructability of a line segment. Namely, a segment can be constructed with a ruler and compass from a given line segment of unit length if its length can be expressed using only whole numbers and the five algebraic operations +, –, ×, ÷, and √. This should look somewhat familiar—it looks a lot like the toolbox for solving polynomial equations by radicals. But it is more restricted, because only square roots are allowed—no third or higher roots. Likewise, an angle is constructible if its cosine and sine are constructible lengths.
The function π(n) [not to be confused with the number π] represents the number of primes less than n. The prime number theorem says that this total is roughly equal to the integral of a density function, 1/ln(x). Though it is only approximate, the formula gets more and more accurate (on a percentage basis) as n gets larger.
&nb
sp; Gauss’s audacity is amazing. To prove that (360/17)° is a constructible angle, he solved the polynomial equation x17 = 1. At this point, in 1796, no one knew whether degree-five equations were solvable, even using cube roots, fourth roots, and fifth roots. Gauss was proposing to solve a degree-seventeen equation, with fewer tools. And he succeeded!
Five years later, in 1801, Gauss published his first book, Disquisitiones arithmeticae. It was the first systematic book on number theory, establishing its methods and identifying its interesting questions. His theorem on 17-gons appears there, along with a generalization: an n-sided polygon is constructible if all of the odd prime factors of n are one greater than a power of 2, and furthermore are raised to only the first power. Only five such primes are known: 3 (21 + 1), 5 (22 + 1), 17 (24 + 1), 257 (28 + 1), and 65,537 (216 + 1). This result is as far from being practically useful as any theorem can be. It would probably take a lifetime to perform the construction of a 65,537-sided polygon, and when you finished it would be impossible to tell the result apart from a circle!
This example gives an inkling of the central role of prime numbers in number theory. These are the numbers that combine (by multiplication) to form all others, and in this sense they are as fundamental as the elements in chemistry. They are important both as a tool for solving other problems, and as a subject of study in their own right. One of their enduring mysteries is to understand how they are distributed.
Above A diagram showing a method for identifying prime numbers, described by the ancient Greek mathematician Eratosthenes (276 BC–194 BC).
Here is the paradox. Primes behave very much as if they were randomly distributed on the number line. The distribution is not completely uniform; large numbers are less likely to be prime than small numbers, because there are more possible prime divisors. Gauss conjectured on the basis of empirical evidence that the “density” of prime numbers decreases in proportion to the natural logarithm of n, written ln(n). This means that a ten-digit number is half as likely to be prime as a five-digit number, and five times less likely to be prime than a two-digit number. We can think of 1/ln(n) as the “probability” that n is prime. And yet this statement is absolutely paradoxical, because there is no probability involved! Either a number is prime or it is not.
Nevertheless, Gauss’s conjecture, called the Prime Number Theorem after it was finally proved in 1898, provides remarkably accurate estimates of the distribution of primes. For example, the density formula says that the number of primes less than 1 million should be about 78,628. In reality, the number of primes is 78,498—an error of less than 0.2 percent. If we go up to 1 billion, the predicted number is 50,849,235. The exact number is 50,847,534—so the estimate is off by less than 0.004 percent! I hope that you are as amazed by this fact as I am. Think of how difficult it is to determine whether even one large number is prime. Even with current computer technology, no one can tell whether a randomly chosen 200-digit number is prime. And yet, using the Prime Number Theorem, we can find a very accurate (though not perfectly accurate) count of all the primes less than that number!
THE STORY OF the Prime Number Theorem is a little reminiscent of Fermat’s Last Theorem. At some unknown date, Gauss wrote the cryptic comment, “Prime numbers less than a ≈ a/ln a,” which is roughly a statement of the Prime Number Theorem. There is no indication of a proof, and he probably based his assertion on numerical evidence. Around 1850, the Russian mathematician Pafnuty Chebyshev proved that the error in the above approximation, for large enough numbers n, is never greater than 11 percent. Of course, the examples above intimate that the error is in fact considerably smaller. Chebyshev’s work was a big step in the right direction, in part because he used the zeta function as a tool for counting the number of primes.
In 1859, Bernhard Riemann took another amazing step forward, which explains why the zeta function (discussed in Chapter 12) is now named after him rather than Chebyshev. He discovered an exact formula for the number of primes less than n. However, there is a catch. To compute the number exactly, you need to know the infinitely many places in the plane where the Riemann zeta function takes the value zero. (These places are called the “zeros” of the zeta function, as shown below). If you know approximately where the zeros are, Riemann’s formula tells you approximately how many primes there are.
In 1898, Jacques Hadamard and Charles de la Vallée Poussin, working separately, both proved that all the zeros lie in an infinite strip, to the right of the line x = 0 and to the left of the line x = 1. Even this rough information on the location of the zeros was good enough to prove the Prime Number Theorem, one of the landmark theorems of the nineteenth century. Fortunately for Hadamard and de la Vallée Poussin, Gauss was no longer alive to say, “I knew that 100 years ago!”
The story of the theorem is not quite over, though. The more accurately you can pin down the location of Riemann’s zeros, the more you know about prime numbers. Hadamard and de la Vallée Poussin showed the zeros lie on an infinitely long and straight “street” in the plane (the shaded strip in the figure below). Riemann had conjectured, but could not prove, a much more precise statement: All of the zeros lie exactly in the middle of this street! If this statement, called the “Riemann Hypothesis,” is true, it would provide the finest control over the distribution of primes.
Now that Fermat’s Last Theorem has been proved, the Riemann Hypothesis is at the top of number theorists’ “most wanted list.” In 2000, the Clay Mathematics Foundation named it one of seven “millennium problems,” and offered a reward of a million dollars for its solution. Because it is so technical, there are few elementary examples of problems that the Riemann Hypothesis would solve. However, here is one example.
Right The zeros of the zeta function.
Back when I was in second grade, I noticed that the decimal expansions of certain fractions take a long time to repeat, while others do not. For example, 1/3 = 0.3333… is a quick repeater, as is 1/37 = 0.027027… On the other hand, 1/7 = 0.1428571428… is a slow repeater, cycling through six digits before it starts over. An even slower repeater is 1/19 = 0.0526 …, which goes a full 18 digits before starting over with …0526…
In fact, the decimal expansion of any number 1/n will eventually settle into a repeating cycle of no more than (n – 1) digits. The numbers that take the full (n – 1) digits are always prime, like 7 and 19. However, not all primes are slow repeaters. For example, 1/37 starts repeating long before the 36th digit—it takes only three digits! There is no known formula to determine which prime numbers are fast repeaters or slow repeaters. However, if the Riemann Hypothesis is true, then about 37.4 percent of all primes are slow repeaters. This is typical of the amazingly precise information about primes that number theorists can squeeze out of the Riemann Hypothesis. (This result was proved by Christopher Hooley in 1967.)
Above Carl Frederich Gauss (1777–1855).
How good is the evidence for the Riemann Hypothesis? To date, ten trillion zeros of the zeta function have been found, and they all lie exactly in the middle of the “critical strip,” just as Riemann predicted. Any reasonable scientist, in any other subject, would have declared the problem solved long ago. However, in such matters mathematicians are not reasonable.
17
the idea of spectra fourier series
So far, the stories I have told in this chapter have not portrayed French mathematics in a very positive light. First the French Academy of Sciences lost Abel’s memoir on elliptic functions, and then it spurned Galois’ revolutionary discovery of group theory. Some of the most unexpected breakthroughs in the first half of the 1800s came from mathematicians elsewhere—Hamilton in Ireland, Abel in Norway, Bolyai in Hungary, Lobachevsky in Russia, and of course Gauss in Germany. Nevertheless, the center of mathematics at this time was undoubtedly in Paris. Any student of mathematics today will inevitably encounter a whole suite of French names from this period, including Lagrange, Laplace, Legendre, Cauchy, Liouville, Poisson, Fourier.
It is re
markable that French mathematics remained so strong throughout a period of huge political upheaval. It weathered the French Revolution, the Terror, then Napoleon’s rise, his exile and return, the restoration of the monarchy, the abdication of King Charles and ascension of King Louis-Philippe, and finally the Second Republic and Second Empire. All of these swings of political fortune had repercussions for individual mathematicians, whose fortunes rose and fell with their leaders of choice. Nevertheless, French mathematical culture as a whole prospered. Perhaps one reason was France’s increased social mobility, which gave access to education and professional opportunities to anyone with talent and a little bit of luck.
For any function f(x), “f-hat” is its Fourier series. It decomposes f into a spectrum of sine and cosine waves of different frequencies. The second formula tells how to reconstruct the original function f from its spectrum. In a sense, it says that “f-hat-hat” equals f again.
A PERFECT EXAMPLE would be Joseph Fourier, a tailor’s son who was born in 1768 and orphaned at age nine. Brought up in a convent and educated in a military school, he supported the Revolution and managed not to be executed during the Terror, although he was arrested twice. He rose through France’s top educational institutions, the École Normale where he was a student, and the École Polytechnique where he became a junior professor.
The Universe in Zero Words Page 11