Free Novel Read

Why Beauty is Truth Page 15


  Groups had already proved their worth by resolving an age-old conundrum, the solvability of the quintic. It soon became clear that the same circle of ideas disposed of several other age-old problems. You didn’t always need group theory as such, but you needed to think like Abel, Galois, and their successors. And even when you thought you weren’t using groups, they often lurked in the background.

  Among the unsolved problems the Greek geometers bequeathed to posterity, three had become notorious: the problems of trisecting the angle, duplicating the cube, and squaring the circle. Even today, trisection and circle-squaring attract the attention of numerous amateurs, who seem not to have grasped that when mathematicians use the word “impossible,” they mean it. Duplicating the cube seems not to have the same allure.

  These three are often referred to as the “three problems of antiquity,” but this phrase exaggerates their importance. It makes them appear to be on a par with major historical puzzles such as Fermat’s Last Theorem, which went unanswered for more than 350 years. But this puzzle was explicitly recognized as an unsolved problem, and it is possible to identify the precise point in the mathematical literature where it was first posed. All mathematicians were aware not just of the problem but the presumed answer—and who had first asked the question.

  The Greek problems are not like that. You won’t find them listed in Euclid as unsolved problems that need attention. They exist mainly by default: they are obvious extensions of positive results, but for some reason Euclid avoided them. Why? Because no one knew how to solve them. Did it occur to the Greeks that they might not have solutions? If so, no one made much fuss. It undoubtedly occurred to people like Archimedes that no straightedge-and-compass solutions existed, because he developed alternative techniques, but there is no evidence that Archimedes considered the issue of constructibility important in its own right.

  Later it became important. The lack of solutions to these problems pointed to major gaps in humanity’s understanding of geometry and algebra; they gained currency as “folklore” problems, known to the professionals through a kind of cultural osmosis. By the time they were solved, they had taken on an aura of historical and mathematical significance. Their solutions were seen as major breakthroughs—especially squaring the circle. And in all three cases, the answer was the same: “It can’t be done.” Not with the traditional tools of straightedge and compass.

  This may seem rather negative. In most walks of life, people seek to answer questions or overcome difficulties by whatever means comes to hand. If a tall building cannot be constructed from bricks and mortar, engineers use steel frames and reinforced concrete. No one gains fame by proving that bricks are not up to the job.

  Mathematics is not quite like that. The limitations of the tools are often just as important as what they can accomplish. The importance of a mathematical question often depends not on the answer as such, but on why the answer is correct. So it was for the three problems of antiquity.

  The scourge of trisectors everywhere was born in Paris in 1814, and his name was Pierre Laurent Wantzel. His father was first an army officer and later professor of applied mathematics at the École Speciale du Commerce. Pierre was precocious; Adhémard Jean Claude Barré de Saint-Venant, who knew Wantzel, wrote that the boy showed “a marvelous aptitude for mathematics, a subject about which he read with extreme interest. He soon surpassed even his master, who sent for the young Wantzel, at age nine, when he encountered a difficult surveying problem.”

  In 1828, Pierre successfully applied to enter the Collège Charlemagne. He won first prize in both French and Latin in 1831, and he came in first in both the entrance examination for the École Polytechnique and that for the science section of what is now the École Normale, which no one had ever done before. He was interested in just about everything—mathematics, music, philosophy, history—and he liked nothing better than a good, hard-fought debate.

  In 1834, he turned his mind to engineering, attending the École des Ponts et Chaussées. But soon he was confessing to his friends that he would be “but a mediocre engineer,” decided that he really wanted to teach mathematics, and took a leave of absence. The switch worked: he became a lecturer in analysis at the École Polytechnique in 1838, and by 1841 he was also a professor of applied mechanics at his old engineering school. Saint-Venant tells us that Pierre “usually worked during the evening, not going to bed until late in the night, then reading, and got but a few hours of agitated sleep, alternatively abusing coffee and opium, taking his meals, until his marriage, at odd and irregular hours.” The marriage was to his former Latin coach’s daughter.

  Wantzel studied the works of Ruffini, Abel, Galois, and Gauss, developing a strong interest in the theory of equations. In 1837 his paper “On the means of ascertaining whether a geometric problem can be solved with straightedge and compass” appeared in Liouville’s Journal de Mathématiques Pures et Appliquées. It took up the story of constructibility where Gauss had left off. He died in 1848 at the age of 33—probably as a result of overwork from an excess of teaching and administrative duties.

  On the questions of trisection and duplicating the cube, Wantzel’s impossibility proofs resemble Gauss’s epic work on regular polygons, but are much easier. I’ll start with the duplication of the cube, where the issues are very transparent. Does there exist a straightedge-and-compass construction for a line of length ?

  Gauss’s analysis of regular polygons is based on the idea that any geometric construction boils down to solving a series of quadratic equations. He pretty much takes this for granted, because it follows algebraically from properties of lines and circles. Some reasonably easy algebra implies that the “minimum polynomial” of any constructible quantity—the simplest equation that it satisfies—has degree equal to a power of two. That equation may be linear, quadratic, quartic, octic (degree 8), of degree 16, 32, 64, . . . but whatever the degree is, it is a power of two.

  On the other hand, satisfies the cubic equation x3 – 2 = 0, and this is its minimum polynomial. The degree is 3, which is not a power of 2. Therefore the assumption that the cube can be duplicated using straightedge and compass leads, by impeccable logic, to the conclusion that 3 is a power of 2. This is obviously not true. By reductio ad absurdum, therefore, no such construction can exist.

  Trisection of the angle is impossible for a similar reason, but the proof is slightly more involved.

  First, some angles can be trisected exactly. A good example is 180°, which trisects to 60°, an angle that we can construct by making a regular hexagon. So the impossibility proof begins by picking some other angle and proving that this choice cannot be trisected. The simplest angle to pick is 60° itself. One-third of that is 20°, and we will show that 20° cannot be constructed using straightedge and compass.

  This is a sobering thought. Look at a protractor, an instrument for measuring angles. Confidently marked on it are angles of 10°, 20°, and so on. But those angles are not exact—for a start, the inked lines have thickness. We can make an angle of 20° that’s good enough for an architectural or engineering drawing. But we can’t construct a perfect 20° angle using Euclidean methods—that’s what we plan to prove.

  The key to this puzzle is trigonometry, the quantitative study of angles. Suppose that we start with a hexagon inscribed in a circle of radius 1. Then we can find a 60° angle, and if we could trisect it, we could construct the bold line in the figure (next page).

  Suppose this line has length x. Trigonometry informs us that x satisfies the equation 8x3 – 6x – 1 = 0. As in the problem of duplicating the cube, this is cubic, and again it is the minimum polynomial of x. But if x is constructible then the degree of its minimum polynomial must be a power of 2. Same contradiction, same conclusion: the proposed construction is impossible.

  Trisecting an angle of 60° is equivalent to constructing the length marked x.

  The way I have presented these proofs conceals a deeper structure, and from a more abstract perspective Wantzel’s
solutions of these two problems of antiquity both boil down to symmetry arguments: the Galois groups of the equations that correspond to the geometry have the wrong structure for straightedge-and-compass constructions. Wantzel was well aware of Galois groups, and in 1845 he developed a new proof that some algebraic equations cannot be solved by radicals. The proof followed Ruffini and Abel closely, but simplified and clarified the ideas. In the introduction Wantzel states,

  Although [Abel’s] proof is finally correct, it is presented in a form too complicated and so vague that it is not generally accepted. Many years previous, Ruffini . . . had treated the same question in a manner much vaguer still . . . In meditating on the researches of these two mathematicians . . . we have arrived at a form of proof which appears so strict as to remove all doubt on this important part of the theory of equations.

  The sole remaining problem of antiquity was squaring the circle, a task that amounts to constructing a line of length exactly equal to π. Proving this construction impossible turned out to be much more difficult. Why? Because instead of π having a minimum polynomial of the wrong degree, it turned out to have no minimum polynomial at all. There is no polynomial equation with rational coefficients having a root equal to π. You can come as close as you like, but you can never get exactly π.

  The mathematicians of the nineteenth century realized that the distinction between rational and irrational numbers could profitably be refined. There were different kinds of irrational. Relatively “tame” irrationals like could not be represented as exact fractions, that is, as rational numbers, but they could be represented in terms of rational numbers. They satisfied equations whose coefficients were rational numbers—in this case, x2 – 2 = 0. Such numbers were said to be “algebraic.”

  But mathematicians realized that in principle there might exist irrational numbers that were not algebraic and whose link to the rationals was far more indirect than that for the algebraic numbers. They transcended the rational realm altogether.

  The first question was, do such “transcendental” numbers actually exist? The Greeks supposed that all numbers might be rational until Hippasus disillusioned them, and Pythagoras allegedly was so incensed that he drowned the messenger. (More likely, Hippasus was just expelled from the Pythagorean cult.) The mathematicians of the nineteenth century were aware that any belief that all numbers are algebraic was equally likely to lead to tragedy, but for many years they lacked a Hippasus. All they had to do was to prove that some specific real number—π was a plausible candidate—is not algebraic. But it’s difficult enough to prove that some number, π, for example, is irrational, and for that all you have to show is the nonexistence of any pair of integers such that one divided by the other gives you π. To prove that a number is not algebraic, you have to replace these hypothetical integers by all possible equations, of any degree whatsoever, and then derive a contradiction. It gets messy.

  The first significant progress was made by the German mathematician and astronomer Johann Lambert in 1768. In a paper on transcendental numbers, he proved that π is irrational, and his method paved the way to everything that followed. It made essential use of ideas from calculus, notably the concept of an “integral.” (The integral of any given function is a function whose rate of change yields the original function.) Starting from the assumption that π is equal to some exact fraction, Lambert proposed to calculate a fairly complicated integral that he had invented for just this purpose, which involved not just polynomials but trigonometric functions. There are two distinct ways to calculate this integral. One of them gives the answer zero. The other proves that the answer is not zero.

  If π were not a fraction, then neither method would apply, so no problems would arise. But if π were a fraction, zero would have to be different from itself. No way.

  The details of Lambert’s proof are technical, but how it works is very informative. To get started, he had to relate π to something simpler, and trigonometry came to his rescue. The next problem was to fix things up so that something special would happen if π were rational. This was where the polynomial bit came in, along with the clever idea of forming an integral. After that, the proof was a matter of comparing two distinct methods for computing the integral, and showing that they gave different answers. That bit was messy and technical, but routine for experts.

  Lambert’s proof was a major step forward, but plenty of irrational numbers can be constructed, the most obvious being , the diagonal of a unit square. So proving π irrational did not prove that it was unconstructible. It meant there was no longer any point in trying to find an exact fraction for π, but that was a different issue altogether.

  At this point, mathematicians faced an unusual dilemma. They had made a distinction between algebraic numbers and transcendental ones, and they believed it would be important. But they still did not know whether any transcendental numbers existed. In practical terms, the supposed distinction might be meaningless.

  It took until 1844 to prove the existence of transcendentals. The breakthrough was made by Liouville, who had previously salvaged Galois’s work from the academic rubbish heap. Now Liouville managed to invent a transcendental number. It looked like this:

  0.110001000000000000000001000. . .,

  where longer and longer sequences of 0’s are separated by isolated 1’s. The important point is that the lengths of the blocks of zeros have to increase very rapidly.

  Numbers of this kind are “almost” rational. There exist unusually good rational approximations—basically thanks to those blocks of zeros. The long block above, for instance, with 17 consecutive zeros, implies that what comes before it—0.110001—is a much better approximation to Liouville’s number than you might expect of a random decimal fraction. And 0.110001, like any finite decimal, is rational: it is equal to Instead of being accurate to six decimal places, it is accurate to 23 decimal places. The next nonzero digit is a 1 in the 24th place.

  Liouville had realized that algebraic numbers, other than rational ones, are always rather badly approximated by rationals. Not only are such numbers irrational; to get a good rational approximation you have to use very big numbers in any fraction that gets close. So Liouville deliberately defined his number to have extraordinarily good rational approximations, much too good for it to be algebraic. Therefore it had to be transcendental.

  The only criticism we can direct against this clever idea is that Liouville’s number is very artificial. It has no evident connection with anything else in mathematics. It is plucked from thin air for the sole reason that it can be very well approximated by rationals. No one would care about it at all save for that one remarkable feature: it is provably transcendental. So now mathematicians knew that transcendentals did exist.

  Whether interesting transcendentals existed was another matter, but at least the theory of transcendental numbers had some content. Now the task was to provide interesting content. Above all, is π transcendental? If it were, that would knock the old squaring-the-circle problem on the head. All constructible numbers are algebraic, so no transcendental is constructible. If π is transcendental, it is impossible to square the circle.

  The number π is justly famous because of its connections with circles and spheres. Still, mathematics contains other remarkable numbers, and the most important—probably even more important than π—is known as e. Its numerical value is approximately 2.71828, and like π it is irrational. This number arose in 1618, in the early days of logarithms; it determines the correct interest rate if compound interest is applied over ever-shorter intervals. It was called b in a letter Leibniz wrote to Huygens in 1690. The symbol e was introduced by Euler in 1727, and it appeared in print in his Mechanics of 1736.

  By using complex numbers, Euler discovered a remarkable relation between e and π, often considered the most beautiful formula in mathematics. He proved that eiπ = –1. (This formula does have an intuitive explanation, but it involves differential equations.) After Liouville’s discovery, the next step to the
proof that π is transcendental took a further 29 years, and it applied to the number e. In 1873 the French mathematician Charles Hermite proved that e is transcendental. Hermite’s career has remarkable parallels with that of Galois—he went to Louis-le-Grand, was taught by Richard, tried to prove that the quintic is unsolvable, and wanted to study at the École Polytechnique. But unlike Galois he got in—by the skin of his teeth.

  One of Hermite’s students, the famous mathematician Henri Poincaré, observed that Hermite’s mind worked in strange ways: “To call Hermite a logician! Nothing can appear to me more contrary to the truth. Methods always seemed to be born in his mind in some mysterious way.” This originality served Hermite well in his proof that e is transcendental. The proof was an elaborate generalization of Lambert’s proof that π is irrational. It also employed calculus; it evaluated an integral in two ways; and if e were algebraic, those two answers would be different: one equal to zero, one nonzero. The difficult step was to find the right integral to compute.

  The actual proof occupies about two printed pages. But what a wonderful two pages! You could search for a lifetime and not discover the right choice of integral.

  The number e is at least a “natural” object of mathematical study. It crops up all over mathematics and is absolutely vital to complex analysis and the theory of differential equations. Although Hermite had not cracked the problem of π, he had at least improved on Liouville’s rather artificial example. Now mathematicians knew that the everyday operations of mathematics could throw up entirely reasonable numbers that turned out to be transcendental. Soon a successor would use Hermite’s ideas to prove that one of those numbers was π.