Above Quaternionic fractals. Computer-generated image derived from a Julia Set in quaternion space.
Other mathematicians had no such compunctions. Once the floodgates were opened, anything was possible. You can have algebraic structures with three operations (addition, subtraction, multiplication) called rings; or with two operations (addition and subtraction, or multiplication and division), these are known as groups; or you can even pare it down to one operation; these structures are called monoids. With such a variety of algebraic structures to choose from, the question becomes not what is possible, but what is worth studying. Does a new structure help solve pre-existing problems? Does it have a deep, challenging, inherently beautiful theory? One new algebraic structure that has consistently scored high on both criteria is the concept of a group—and that is what will be discussed next.
* * *
* It is a splendid irony that in Hamilton’s quaternions, the three dimensions of space—the i, j, and k dimensions—are imaginary while time is the real coordinate (i.e., the number a in the expression a + bi + cj + dk). The real world and imaginary world have switched places!
14
two shooting stars group theory
In the early nineteenth century, mathematics lost two of its brightest talents at a very young age, a 26-year-old Norwegian and a 20-year-old Frenchman. Niels Henrik Abel and Évariste Galois were linked by more, though, than their untimely deaths. They combined to give a definitive answer to one of the most classical questions in mathematics: Is there a universal version of Cardano’s formula for the cubic (which I discussed in Part Two, page 61)? In the process, they opened up a new branch of mathematics, which we now call group theory.
Abel was born in 1802, the son of a long line of country clergymen. His childhood was a complicated time politically for Norway, which was a sort of pawn of the Napoleonic Wars. After almost 300 years of relatively benign Danish rule, Norway briefly became an independent state in 1814, but later that same year its parliament voted to recognize the Swedish king. Abel’s father, twice elected to parliament, became a lightning rod for scandal because of his minority pro-independence views. After Abel’s father died in 1820, his alcoholic mother left with another man, and Niels and his siblings were left in poverty.
Fortunately, Abel’s teachers recognized and encouraged his talent for mathematics. By the time he finished his university studies, it was clear to them that his abilities were much beyond any position that could be found for him in Norway. The faculty persuaded the Swedish king to give Abel a two-year travel stipend to visit Europe’s leading centers of mathematics: Göttingen (in Germany) and Paris. Abel’s extended Wanderjahr, from 1825 to 1827, started very auspiciously. One of the first people he met in Germany was Leopold August Crelle, who was about to start a new journal called Journal of Pure and Applied Mathematics, often simply known as Crelle’s Journal. However, Abel failed to impress the other leading mathematicians of the day, such as Karl Friedrich Gauss in Germany and Augustin-Louis Cauchy and Adrien-Marie Legendre in France. During his visit to Paris in 1826, Abel submitted what he considered his most important paper to the Parisian Academy of Sciences. Cauchy apparently lost it in a desk drawer, and it was not printed until 1841, long after Abel’s death.
Gal(K/Q) represents the Galois group of a polynomial over the rational numbers Q. S5 represents the group of all 120 possible permutations of five objects. Whenever a polynomial has a Galois group equal to S5, the polynomial cannot be solved using the five basic operations: +, –, ×, ÷, and n-th roots.
By the time Abel got back home to Norway his other articles had started appearing, in rapid-fire succession, in Crelle’s Journal. The Parisian mathematicians were astounded, first to read a series of breakthrough papers by an unknown mathematician from the hinterlands; then to learn that this mathematician had actually been in Paris; and finally, to learn that he had tried to present them a paper and they had lost it! Legendre sent his apologies to Abel, and along with three other mathematicians petitioned the Swedish king to find some way to assist “a young Monsieur Abel, whose works show he has mental powers of the highest rank, and who nevertheless grows ill there in Christiania [Oslo] in a position of too little value for one of his rare and early-developed talent.”
Unfortunately, the rumor that Legendre and the others had heard about Abel’s ill health was true. By 1828, Abel had developed tuberculosis, and in April of 1829 he died, just two days before a letter arrived from Crelle saying that he had arranged a professorship for Abel in Berlin.
ÉVARISTE GALOIS’ STORY is also one of unbelievably bad luck, compounded by poor judgment. Born in 1811 near Paris, he was apparently a very difficult student in high school, described by his teachers as “original” and “bizarre.” At the age of 17, he sent a paper on the solvability of polynomials to Cauchy—the same Cauchy who had lost Abel’s manuscript a couple years earlier. Galois’ paper was lost, too, though this time it was not Cauchy’s fault. “I cannot in truth conceive of such carelessness on the part of those who already have the death of Abel on their consciences,” Galois later wrote. The accusation is, of course, completely unfair. Though the loss of Abel’s paper was a scandal, the Academy was in no way to blame for Abel’s death and, as noted above, had even tried to intervene on his behalf. However, this invective does give us an insight into Galois’ character. He was a rebel against authority, and the Academy became for him a symbol of tyrannical power.
In 1829, Galois joined a revolutionary organization called the Society of the Friends of the People. The following year, rioting broke out in the streets of Paris, and King Charles X was forced to abdicate the throne. Extreme Republicans (such as Galois) wanted to abolish the monarchy altogether, but moderate Republicans led by the popular Marquis de Lafayette prevailed. They named Louis-Philippe as the “citizen king” of France, a king who would be bound by constitutional restrictions.
Galois was unable to participate in the July 1830 revolution because he was a student at the École Normale, and the school’s director literally locked the students in. However, by 1831 he had graduated, and he no longer had to sit on the political sidelines. He was arrested twice that year, once for threatening the life of King Louis-Philippe and the second time for participating (heavily armed) in a demonstration on Bastille Day. While he was in prison, he received news that the French Academy had rejected his latest paper on the theory of equations.
Galois was released from prison in April 1832, and by the end of May he was dead. The events that led to this outcome are far from clear. One historian has written a book arguing that it was a police-organized provocation, while others deny it. Galois himself wrote to his friends, in moving words, that he was forced to take part in a duel over a woman:
“I beg patriots, my friends, not to reproach me for dying otherwise than for my country. I die the victim of an infamous coquette and her two dupes. It is a miserable piece of slander that I end my life … I would like to have given my life for the public good. Forgive those who kill me for they are of good faith.”
On May 30, the day after Galois wrote this letter, he was shot in the stomach by a man whom the writer Alexandre Dumas identified as Pescheux d’Herbinville, a hero of the Republican cause. Galois’ opponent left him on the ground to die. He was found several hours later, still alive, but he died the following day.
Above Part of a manuscript written by the French mathematician Evariste Galois (1811–1832).
ALTHOUGH IT IS TEMPTING to wonder what Abel and Galois might have achieved if they had both lived, in fact they both accomplished a great deal in their short lives. Their lasting fame rests on the theorems they proved, and not on the way that they died.
Both Abel and Galois were fascinated by the problem of finding the solutions of polynomial equations. In Part Two, I recounted how Cardano “stole” the secret for solving cubic (third-degree) polynomials, and how his servant Ferrari subsequently discovered a method for solving quartics (fourth-degree). In both of thes
e formulas, the solutions, or “roots,” can be expressed using only the operations of algebra (+, – ×, ÷, and nth roots for any n). Often the nth roots, or “radicals,” are nested inside one another, a square root inside a cube root inside a fourth root; this accounts for the term “solution by radicals.” However, no one in the intervening three centuries had found a universal solution by radicals for fifth or higher degree equations, and some were starting to suspect that no solution could be found.
It is difficult indeed to prove that a task is impossible. It is not just a matter of trying and failing to solve it. You must discover some inherent inadequacy of the tools that you have been given. In fact, Abel and Galois did not prove that quintic polynomials have no solutions. Instead, they proved something more subtle: that the five operations listed above are inadequate to express the solutions, assuming they exist. Their proofs involved a very deep and novel exploration of the idea of symmetry.
Let’s start with the original quintic polynomial:
Assuming that it has five roots, r1, r2, r3, r4, and r5, then each of the coefficients of the original polynomial is a symmetric function of the roots. For example:
and so on. Looking at these formulas, you may notice that each of the roots participates equally. More precisely, Galois observed that if you permute the roots in any way (e.g., by replacing r1 with r2 and r2 with r1), the expressions do not change. (The terms will be listed in a different order, but the sums will still be the same.). There are 120 different ways to permute five numbers and thus a typical quintic polynomial has 120 symmetries. However, it is worth noting that some polynomials have fewer symmetries. (The reason is technical, but some permutations may be forbidden because of extra algebraic relations between some of the roots—for example, one root might be the square of another.)
Abel realized, and Galois clarified, that if a polynomial is solvable by radicals, it creates a hierarchy of intermediate polynomials and a hierarchy of “number fields” corresponding to the roots of those polynomials. This is the reason for the nesting of radicals within radicals in Cardano’s and Ferrari’s formulas; each time you peel off a radical (like peeling the layers of an onion) you move to a lower number field. The symmetries of the original polynomial have to respect this hierarchical structure.
Now comes the difficult, but clinching point of Galois’ argument. The full group (a term coined by Galois) of 120 permutations of the roots does not allow a tower of subgroups of the requisite type. It’s as if you were trying to build a wedding cake 120 feet high; you can’t do it. As it turns out, the maximum height (the maximum number of allowable permutations for a quintic polynomial to be solvable by radicals) is 20.
GALOIS’ SOLUTION ACTUALLY provided a clear-cut criterion to determine which polynomials can and which ones cannot be solved by radicals. If you have a polynomial whose “wedding cake” (or Galois group) has 20 elements or less, you can solve it. Galois thought that his criterion was hopelessly impractical—but nowadays, thanks to the computer, the calculation of the Galois group can be automated. Thus, for example, the Galois group for the polynomial x5 – x + 2 contains all 120 permutations, and therefore the solutions to the equation x5 – x + 2 = 0 cannot be written in terms of the five algebraic operations.
The equation x5 – x + 2 = 0 does have solutions. They just can’t be expressed with the limited palette of +, –, ×, ÷, and radicals. In 1858 Charles Hermite proved that the solutions to any quintic can be written down by using a new kind of function, called elliptic functions, which Abel had discovered.
This is a normal human response to a problem: If you can’t overcome the difficulty with the tools you have, invent new tools. However, mathematicians are not completely like ordinary people. Because their problems are often several steps removed from practical application, they often care more about how a problem is solved than whether it is solved. Abel’s and Galois’ proof that quintic equations cannot be solved by radicals has completely eclipsed Hermite’s discovery of how they can be solved by elliptic functions.
But there is another reason for the enduring fame of Galois’ proof. His concept of a group has now become the main tool that mathematicians use to express the ancient idea of symmetry. I find it very curious that the first explicit use of symmetry groups came in such a difficult context. It is as if no one had ever invented the wheel until the Wright brothers incidentally came up with it as a way to get airplanes off the ground. We would say, “Wow! Someone should have come up with that earlier!”
Opposite Capturing symmetry: A panel of Isnik earthenware tiles from the baths of Eyup Eusaki, Istanbul, ca. 1550–1600.
The idea of a symmetry group should be one of the most basic things in mathematics. In fact, the ability to perceive a symmetric object may even precede our ability to count. Perhaps its very obviousness made it difficult for mathematicians to discover. They could not formalize the meaning of symmetry until they encountered it in a context (the solution of polynomials) where its meaning was so far from obvious.
Fittingly, Galois’ legacy was every bit as revolutionary as the political causes that he fought for. The tool he invented, group theory, has more than fulfilled the visions of its creator. Chemists now use group theory to describe the symmetries of a crystal. Physicists use it to describe the symmetries of subatomic particles. In 1961, when Murray Gell-Mann proposed his Nobel Prize-winning theory of quarks, the most important mathematical ingredient was an eight-dimensional group called SU(3), which determines how many subatomic particles have spin 1/2 (like the neutron and proton). He whimsically called his theory “The Eightfold Way.” But it is no joke to say that when theoretical physicists want to write down a new field theory, they start by writing down its group of symmetries.
15
the geometry of whales and ants non-euclidean geometry
At the same time that a revolution was going on in algebra, similar events were taking place in geometry. Two millennia earlier, Euclid had written down a short set of axioms from which, supposedly, all of geometry could be derived. These axioms were intended to be self-evident truths that did not require any proof.
For centuries Euclid’s Geometry was considered the ne plus ultra of deductive reasoning. The eighteenth-century philosopher Immanuel Kant built up a theory of knowledge, in which he cited Euclid’s geometry as an example of “synthetic a priori” truth—in other words, infallible knowledge about the universe that is derived from pure reason rather than observation.
However, one axiom had always appeared a little bit clumsier than the others. The axiom in question is the “Parallel Postulate,” which Euclid does not use until late in his first book: “If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than two right angles.” This assumption is used, for example, to prove that the sum of the angles of a triangle equals 180 degrees.
Many mathematicians felt the Parallel Postulate was true but far from self-evident, and thus a flaw in Euclid’s otherwise sterling system of axioms. They took up the challenge of proving it from the other axioms that Euclid had provided. This mathematical grail quest lured the famous and obscure alike. Legendre (whom we have met already) believed that he had proved it. So, at one time or another, did less-famous mathematicians like John Wallis, John Playfair, Girolamo Saccheri, Johann Lambert, and Wolfgang Bolyai. In all cases, they made hidden assumptions that, under the harsh light of scrutiny by other mathematicians, were no better motivated than Euclid’s postulate.
dx and dy represent the sides of an “infinitesimal” triangle, and ds represents their hypotenuse.
In the first half of the nineteenth century, three men separately and independently dared to think the unthinkable. Perhaps a valid geometry might exist in which the Parallel Postulate was actually false. This would be a non-Euclidean geometry—that is, a geometry in which one of the axioms laid down by Euclid, m
ore than two millennia earlier, is expressly violated.
This idea was just as heretical as Hamilton’s idea of an algebra with no commutative law. However, denying the Parallel Postulate took perhaps even more courage, because it had the great weight of Euclid, Kant, and two thousand years of tradition behind it.
The first of the three revolutionaries was Karl Friedrich Gauss, the most famous mathematician of his era. Gauss, a friend of Bolyai from their student years, dabbled at proving the Parallel Postulate in the early 1800s. But gradually, around 1820, he seems to have become convinced that an alternative, non-Euclidean geometry could be constructed. However, he never published this idea, and only alluded to it somewhat vaguely in letters. The best evidence of his reasons comes from a letter he wrote in 1829 to his friend, Friedrich Bessel, in which he says that he feared the “howl from the Boeotians” (a pejorative term for stupid people) that would ensue if he published his work.
THE SECOND DISCOVERER of non-Euclidean geometry was Janos Bolyai, the son of Gauss’s old school chum. Wolfgang, who became a mathematics teacher in Hungary, tried to warn his son against trying to prove the Parallel Postulate: “For God’s sake, I beseech you, give it up. Fear it no less than sensual passions because it, too, may take all your time, and deprive you of your health, peace of mind, and happiness in life.” But his son ignored the advice, and he eventually wrote a 24-page treatise on what he called the “absolute science of space,” which his father generously published as an appendix to one of his textbooks in 1832.
The elder Bolyai naturally sent a copy to his old friend Gauss, who responded in unexpected fashion: “To praise [this work] would amount to praising myself. For the entire content of the work, the approach which your son has taken, and the results to which he is led, coincide almost exactly with my own meditations … It was my plan to put it all down on paper eventually, so that at least it would not perish with me. So I am greatly surprised to be spared this effort, and am overjoyed that it happens to be the son of my old friend who outstrips me in such a remarkable way.”
The Universe in Zero Words Page 10