byT. A. Heppenheimer

Advanced scientific computing is helping to turn some mathematics into an experimental science.

Mathematics is different from the other sciences.

Among nonmathematicians there is a widely held-set of. opinions that follows Thomas Henry Huxley's pronouncement in 1869 that mathe­matics "is almost purely deductive. The mathematician starts with a few simple propositions . . . and the rest of his work consists of -subtle deductions from them. . . . Mathematics is that study which knows nothing - of observation, nothing of experiment, nothing of in­duction, nothing of causation/'

Mathematicians know their subject is not like that. They know that the finished demonstrations they present to the world, elegant as works of architec­ture, in reality are structures from which the scaffolding has been re­moved, for which the blueprints have been tucked out of sight. Mathemati­cians are preoccupied with such blue­prints and scaffolding.

More particularly, the lifeblood of their science is the conjecture, or-mathe­matical hypothesis. One mathematician may be exploring logical a rguments , seeking one that holds together and that will carry him further toward a proof. Another may be examining a plausible line of reasoning, seeking the counter­example that will invalidate the argu­ment and leave an error exposed. Still another may be working to construct some new form, aiming at an example that will fulfill a set of definitions whose properties then may suggest patterns leading to new conjectures. "Many mathematicians," says Michael Artin of the Massachusetts Institute of Tech­nology, "try to see a general pattern based on just a small amount of data, then try to find theoretical reasons why it might be true." In all these ways, mathematicians work as-experimenters.

Mathematics, however, remains dif­ferent: Computers, which have become so influential in-other sciences, have, been slow to achieve comparable signifi-

Volume 16 Number 4 MOSAIC 37

cance in -mathematics. There is, of course/ a vast difference between com­putation and mathematics. Says Jacob Schwartz of New York University's Cou-rant Institute, "The mathematician is al­ways interested in general principles and less in individual facts, which is why, of all the sciences, math has been the least influenced by compu te r s / ' Moreover, for a long time, computers were not able to perform serious work in the field. As Artin recalls, "I would ask the computer to do some computa­tion I knew I couldn't do by hand, and it turned out the computer couldn't do it either. That happened many t imes/ '

This situation is now changing. In­creasingly, computers are taking their place in mathematics as an experimental instrument, a laboratory tool. By using computers in adroit and sophisticated ways, mathematicians are bringing new creativity to this oldest of the sciences.

The contributions of computers to mathematics take various forms. An im­portant one is simply to bolster a mathe­matician's courage, his readiness to undertake a project or proof. With the effort under .way, the computer fre­quently serves as a tool of exploration, displaying in concrete form the struc­tures or patterns to be studied. Com­puter graphics are particularly valuable, offering detailed maps of what until re­cently may have.been terra incognita. Furthermore, present applications of ar­tificial intelligence point to a vast future increase in the computer's flexibility. This will be the level at which it will manipulate symbolic expressions and logical statements with the facility long shown in the handling of numbers.

Bolstering courage

"Conjectures are often made on the basis of little .evidence," says Michael Artin. "The question arises: How much work is one willing to put into -a con­jecture without having some assurance that it might be true? If you can't com­pute even a single case, then you would probably hesitate to spend six months on the project." In the view of Jacob Schwartz, "The biggest obstacle to a mathematical proof is the loss of mental energy when you're searching through hundreds of possibilities, a lot of which don't work. It's like finding a needle in a haystack. But if you have good reason to believe' that something is true, it focuses

your energy. The computer functions as a metal detector that points you toward the needle, or .at least confirms that the needle is there."

Two Brown University statisticians, Ulf Grenarider and Stuart Geman, are among a number of mathematicians .whose searches have been guided by this metal detector. They work with ma­trix algebra, the study of numerical ar­rays that .crop up in a wide variety of problems featuring large amounts of data or collections, of equations. Gre-nander recalls being approached by an economist from MIT who had been studying certain matrices and had con­cluded, from some 25 numerical cases, that a certain sum of elements would always be positive. The economist was eager to-find a formal mathematical proof because it would have immediate application in economics, and he asked Grenander to prove his conjecture.

Grenander tried, but he could not get the proof to go through. "We tried a few cases," he recalls. "Yes, every case we tried had this property." These trial cases were supporting the conjecture, but It was resisting proof. "What were we to do?" says Grenander. "We were a little desperate." Fortunately, he was able to program a computer in an at­tempt to show that the conjecture was wrong. He set up an algorithm that would drive that sum of elements to be as- small as possible, perhaps even to be negative. "We ran it—and got the counterexample," Grenander says. The conjecture indeed was wrong.

Stuart Geman-had a different experi­ence, In another problem in matrix theory that proved quite recalcitrant. "When you try to prove something that's difficult," he says, "when you be­come stymied over and over again, you become suspicious of whether the result is true. If I hadn't had the computer to help me, .my suspicions would have overcome me. I couldn't have spent the months of tedious work trying to find my way through the proof. The doubt would have just completely invaded me."

"I would not have had the courage to continue this work," Geman says, "if it were not for the fact that I could get on the computer,. do an even larger simula­tion, and absolutely persuade myself that the conjecture was true. Every time I lost faith, I did a larger experiment." His longest test ran for some two hours on a large IBM mainframe. "Then there couldn't be any doubt. I went back to work with renewed faith, and even­tually found my way through what turned out to be a fairly difficult, con­voluted proof."

Geman is quite candid about what he has gained from this psychological sup­port: "I have proved, theorems that are harder than I'm ordinarily able to prove." Without the computer, he adds, "I would not have been good enough.

* It's iar easier to prove a difficult theorem if you know it's true."

The Bieberbach conjecture

This courage-bolstering use of the computer played a dramatic role in the recent and celebrated proof of a long­standing problem, the Bieberbach con­jecture. Set forth in 1916 by Ludwig Bieberbach, it stumped the best inves­tigators for nearly 70 years. Moreover, the difficulty in finding a proof led some mathematicians to believe the con­jecture to be false.

The Bieberbach conjecture deals with certain functions of a complex variable, z, which are written as an infinite sum of powers of z having coefficients a2/ a3, a4, . . . , or, in general, ak. (Such func­tions have the form z + a2z

2 +• a3z3

+ . . . . The Bieberbach conjecture hy­pothesizes that the magnitude, or abso­lute value, of each ak is less than or equal to k.) By early 1984 this was known to be true for the first six coeffi­cients. Many mathematicians believed, however, that it would not prove true for coefficients beyond a19. Heppenheimer is a freelance science writer.

38 MOSAIC .Volume 16 Number 4

In 1977, meanwhile, Louis de Branges of Purdue University had set out to find a proof. By the end of January 1984 he had developed a method to prove that the conjecture would hold for any given ak. He was able to verify the conjecture for the first few coefficients, but when he got to a6, as he describes it, "the cal­culation was so difficult that I could barely get through it." To go forward into the unproved realms of a7, a8/ and the rest, he would need assistance.

Early in February he went to the office of Walter Gautschi, a colleague who had excellent computer software, in the hope that Gautschi would help him. Gautschi attended a seminar de Branges gave a few days later.

As Gautschi later wrote, "I was imme­diately struck by the clarity, freshness, and elegance of Louis's talk and began to appreciate how [his method] came about/' To his delight, he recalls, the method could be described in terms of a subject much on his mind at the time.

De Branges's method was based on the testing of certain polynomials for their lack of roots between zero and one. These polynomials formed a family, with a different set for each k. Gautschi saw immediately how to perform such tests and felt that he could easily go as far as k = 100. "I was clearly fired up and was determined to carry out the computations immediately, no matter what," he says. The next day, Gautschi

ran his first program, a simple routine that evaluated each polynomial for up to 400 equally spaced points. He found no sign changes for any polynomial up to k = 19. At one stroke he had advanced into terra incognita, more than a dozen coefficients beyond what had been known only a day earlier. It cost him $3.69 on Purdue's Control Data 6500.

This was only the beginning, though, for Gautschi realized he might have missed some roots. He proceeded to im­prove his program, incorporating meth­ods to test for the presence of a root. He then pushed this improved version up to k = 31 and found the Bieberbach con­jecture to be true in each case. His most expensive run cost $10.84.

Later that month, following a sugges­tion from Jacob Schwartz, Gautschi saw an especially compelling way to prove the lack of roots for any particular k. This technique, called the Sturm se­quence algorithm, would enable him to generate a formal, rigorous proof for any particular k and thereby substanti­ate de Branges's approach.

The job was then done for particular k's, but Gautschi and de Branges really wanted something more: confidence that what they could prove true for any k would continue to be true for all k's. Final validation of de Branges's method required that to be so, and no such for­mal demonstration was known to them. Gautschi says, "Just to set my mind at

ease, I wanted to make sure that this lack of roots for all k's was not, by chance, already in the literature."

The particular polynomials of interest were derived from what are called Jacobi polynomials and were well known. Nevertheless, Gautschi knew that only a handful of mathematicians might possi­bly be familiar with the sort of result he was seeking. The mathematician he knew best was Richard Askey of the University of Wisconsin. Gautschi told him by telephone about de Branges's polynomials, explaining that it ap­peared likely that they all lacked roots between zero and one and that this would prove the Bieberbach conjecture. Askey interrupted with an emphatic "I don't believe it!" After some more coax­ing from Gautschi, however, Askey agreed to look into it.

Gautschi was working late at home that same night when the phone rang. It was Askey, and his voice was tri­umphant: "It's a theorem!"

He then pointed out a result in a 1976 paper he had written jointly with George Gasper. The paper contained, as a special case, the proof of the lack of roots for all k's and for the entire family of polynomials. Gautschi then checked the reference and confirmed the result himself. The last building block was in place. The next morning, when he saw de Branges and told him the good news, de Branges replied, rather matter-of-factly, "Well, that proves Bieberbach's conjecture."

What was the computer 's signifi­cance? In Gautschi's words, It "gave

Volume 16 Number 4 MOSAIC 39

Louis confidence in his overall proof strategy; his approach indeed seemed capable of proving the complete Bieber-bach conjecture." This conjecture today is known as de Branges's theorem.

The four-color problem

Once a computer has given a mathe­matician a dose of courage, it then becomes a working tool, or instrument. As with any apparatus in the sciences, of course, a mathematician must learn to use this instrument intelligently. "The computer will flood you with data if you're not careful/' says Michael Artin. "Its significance is that one can make experiments that couldn't be made be­fore." Says William Thurston of the In­stitute for Advanced Study in Princeton, New Jersey, "It's not so much the results of the experiments, a lot of times. It's the discipline of thinking about your prob­lem in a concrete way." Sometimes, he says, "you have too much computa­tional power. You're tempted just to

write programs but you don' t really think carefully enough about the struc­ture of the problem."

Perhaps the most celebrated instance to date of a computer contributing to mathematics featured years of work aimed at learning first how to use the computer intelligently. This was the fa­mous proof of the four-color theorem in topology, announced in 1976 by Ken­neth Appel and Wolfgang Haken of the University of Illinois. The theorem is simple: Every map of countries in a plane can be colored with only four dif­ferent colors, so that no two adjacent countries will have the same color. The proof is complex: It took nearly 100 years to achieve it from the time the the­orem was first proposed as a conjecture.

Appel and Haken proceeded by as­suming that some maps exist that re­quire five colors. They sought to show that this assumption would lead to a contradiction of statements known to be true. If, for instance, such five-

chromatic maps exist, there would have to be one with the smallest number of countries, and this would be a minimal five-chromatic map. A part of a map, such as one country surrounded by five others, is called a configuration. It is possible to prove that certain configura­tions cannot occur in any minimal five-chromatic map, for if such a configura­tion occurred it could be replaced by one with fewer countries. The sup­posedly minimal five-chromatic map, then, would be made smaller, thereby contradicting the assertion that it is min­imal. Configurations that lead to this contradiction are called reducible con­figurations.

There are also configurations that are called unavoidable. Given a set of these configurations, at least one must occur in every map. Appel and Haken thus set out to prove the four-color theorem by finding a set of configurations that was both reducible and unavoidable. With such a set, it would follow that any five-chromatic map can be replaced by one having fewer countries, which contra­dicts the assertion that there exists a smallest five-chromatic map.

Because there could then be no small­est five-chromatic map, there could be no five-chromatic map at all. The four-color theorem would be proved. Appel and Haken's approach, therefore, was to construct an unavoidable set of config­urations and simultaneously to show that each of Its members was reducible.

The reducibility demonstrations were reasonably straightforward. At the Uni­versity of Hannover in West Germany, Heinrich Heesch had developed many of the needed procedures and insights. His methods were so complex they could be implemented only with a com­puter, but at least they were known. In addition, Heesch's student Karl Durre had already applied computers to make use of these methods. The difficult part of the problem was to generate the un­avoidable set. Heesch had introduced a rudimentary form of a technique, known as discharging, that generated unavoidable sets of configurations. Haken began in 1970 by noticing that it would be possible to improve Heesch's discharging procedure by visualizing it as a transfer of charges In an electrical network. In 1972 this led Haken to join with Kenneth Appel in an attack on the four-color theorem.

The lengthiest part of this assault was their effort to define a discharging pro-

40 MOSAIC Volume 16 Number 4

cedure that would give rise to an un­avoidable set. This would be a set whose members could be checked for reducibility in a reasonable amount of computer time. Their approach was quite robust: There were many possible versions of unavoidable sets, and many potential choices could be made. As Haken says, however, getting the right version meant the difference "between something that would take a hundred years on the computer and something that would take a thousand hours/7 It was necessary to test a variety of dis­charging procedures on a computer to learn how to develop a good one.

"I remember when Ken came into my office, and he had under his arm more than a cubic foot of computer printout/ ' says Haken. 'Then we knew there was something unfortunate in our program. This meant the number of members in the unavoidable set, by this approach, would be over 100,000, or even a mil­lion. Then we looked at it and thought of reasonable modifications to make the printout smaller. The next time, Ken had it down to only four inches thick. After the twentieth change in our pro­

cedure, it was down to Vs inch, and we thought, 'Now we can really do it/ "

Their final proof featured an un­avoidable set with 1,482 members. Their final version of the discharging pro­cedure was so efficient that they were able to implement it by hand calcula­tion. The demonstrations of reducibility, however, were still so intricate that they would have been impossible without a computer. Indeed, Appel and Haken had to rely on help from a specialist in programming, John Koch. In the end, though, Appel and Haken had what they wanted. As Haken says, they had been able to show that 'Tor any map, you would be able to find its Achilles heel, and then it couldn't be the small­est counterexample."

Wrestling with eigenwaioes

Prior to its proof, the four-color the­orem was one of the most celebrated problems in mathematics. An equally celebrated, but more abstruse, problem is the Riemann hypothesis, which has resisted proof ever since it was first put forth in 1859. It lies in the domain of analytic number theory. Eric Temple

Bell described this field in Men of Mathe­matics as "a miracle/'" for Its theorems concern discrete integers but are proved by using analysis, which deals with the continuous, the nondiscrete. In recent years Dennis Hejhal of the University of Minnesota has been a leader In studying the Riemann hypothesis and has used some of the most powerful supercom­puters available.

During the last century Georg Friedrich Bernhard Riemann began the study of the Riemann zeta-function, which he called zeta(s). He then raised the question, What are the zeroes of the function, and for what values of the complex variable s does zeta(s) = zero? The Riemann hypothesis Is the predic­tion that all such zeroes either are nega­tive Integers or are complex numbers' having real parts equal to Vi. That is, If these zeroes are plotted on an ordinary graph, they all would fall on the straight line x = Vi.

The Riemann hypothesis is important in mathematics because there are signifi­cant, Important connections between these zeroes and the familiar prime numbers, and the study of these con-

Volume 16 Number 4 MOSAIC 41

nections thus can offer new insights into the primes. (The actual connection between the zeroes and primes is given by an equation known as Riemann's ex­plicit formula.) Much, therefore, de­pends on whether the Riemann hypoth­esis is true. Is there any way to prove the hypothesis?

Early in this century mathematicians George Polya and David Hilbert inde­pendently offered a suggestion: Perhaps the zeroes of zeta(s) can be regarded as eigenvalues of some symmetric oper­ator. If so, this would prove the Riemann hypothesis . Eigenvalues of symmetric operators are analogous to the natural frequencies of vibration of a drum. When a drumhead is hit, it vi­brates in characteristic patterns, each with its own frequency. Eigenvalues thus are analogous to the frequencies of fundamental tones.

In about 1950 Atle Selberg of the In­stitute for Advanced Study showed that eigenvalues could indeed be connected to quantities resembling prime num­bers. Selberg dealt with a wide class of surfaces in space known as RIemannian manifolds. It Is possible to define classes of loops, or closed curves that lie on such a manifold and that are analogous to different paths over the surface of a doughnut, or torus. Within each such class is a curve of shortest length, called a geodesic. There are procedures for picking out what amounts to a funda­mental set of such geodesies, known as prime geodesies, each with a well-defined length; there are infinitely many of them, but even the longest is of finite length. Moreover, by regarding

the manifold as a kind of drumhead, it is possible to obtain eigenvalues of finite magnitude, and here too there are infi­nitely many of them.

Selberg used the lengths of these prime geodesies to define a new type of zeta-function, one having the same structure as the Riemann zeta-function. He showed that, for his function, the Riemann hypothesis indeed is true; all of its zeroes are indeed of the appropri­ate form. In addition he proved a the­orem closely similar to Riemann's ex­plicit lOrmuiai i ic u.enveu. an ecjuation almost identical to that given by Riemann but one dealing with eigen­values and geodesic lengths, a set of mathematical objects very different from Riemann's primes and zeroes.

Moreover, Selberg showed that the geodesic lengths obey a variety of the­orems that hold for the prime numbers. In Dennis Hejhal's words, "One begins to think that there are deep connections between primes, zeroes of the Riemann zeta-function, eigenvalues derived from a manifold, and lengths of prime geo­desies on that manifold/' Furthermore, these connections can be explored with powerful computers. As Hejhal says, "The most natural thing to do is to work with a manifold that has number-theoretic significance. It is only here that we stand any chance of finding a precise connection. So, why not compute the damned eigenvalues and see if they match up with the zeroes of the zeta-function!"

"Still, people are lazy," Hejhal con­tinues. "The Selberg formula was pub­lished in 1956. At first people didn't un-

derstand it. Then, after they did under­stand it—do you think anyone bothered to compute with it? Oh no!"

Those computer kids

The needed computations were much harder than they first appeared. The first mathematician to try them was Pierre Cartier of the Institut des Hautes Etudes Scientiflques near Paris. In 1972 he made some approximate computa­tions using what is called the" modular surface, which Hejhal describes as "the most natural number-theoretic surface to which Selberg's formula applies." In about 1977 Hartmut Haas of the Univer­sity of Heidelberg made a much better set of computations using an IBM 370 computer. To Hejhal's surprise and cha­grin, one of Haas's eigenvalues was the first zero of the Riemann zeta-function, correct to six places.

Hejhal had been studying such mat­ters and had decided, like Selberg be­fore him, that there was no direct connection between these eigenvalues and the zeroes of zeta(s). As Hejhal re­calls, though, "Here was this numerical data staring me right in the face. I had a very big headache because of this. My view of reality would be turned upside down if this was correct." He soon had an even bigger headache. He did some library work and found that of the other eigenvalues in Haas's table, several matched up with zeroes of a variant of the Riemann zeta-function, called the Dirichlet L-function. To Hejhal "it was very exciting, very depressing, and very nerve-wracking, all at the same time."

Fortunately, he had a calculus student who liked computers—William Cou­ture, now a graduate student at the Uni­versity of Minnesota. "When you want to learn about computers," Hejhal says, "you don't talk to the experts. You talk to one of those computer kids who knows how to use the system." Couture got time on a Prime computer ("a very interesting name," says Hejhal), and he and Hejhal proceeded to try to destroy Haas's results.

"I intended to show that the first number on his list was not an eigen­value," says Hejhal. "I intended to show that Haas had made a mistake. I set up an algorithm. If a certain set of coefficients lined up In a certain way, there'd be no way on earth this number could be an eigenvalue. And when I started experimenting—Couture helped me with the programming—there was

42 MOSAIC Volume 16 Number 4

no way [Haas's first number] could be an eigenvalue. The numbers did not match up; they would have violated cer­tain theorems known to be true/7

Hejhal then tried other numbers in Haas's table, and those that were zeroes of the L-function variant fell apart. They also failed his tests. As he recalls, though, 'The numbers in Haas's table that did not have number-theoretic sig­nificance—those I couldn't destroy." So, how did those incorrect numbers come out to six decimal places? What were they? What were the true eigenvalues?

Hejhal decided to rerun what he guessed was Haas's original procedure. He quickly modified his program to in­corporate the usual , standard tech­niques used in similar problems. "Out came Haas's numbers, exactly," he says, "zeroes of zeta(s) and all." He then could try to find what Haas had done incorrectly.

"I don't know how long I was work­ing on this," he says. "It took time. It was summer. It was hot, I was working eight hours a day, and I was disgusted. But my subconscious mind must have been working, because one day I was just fed up. I just didn't understand it. I sat down with a sheet of paper, started jotting down what could be happen­ing—and in fifteen minutes I figured out exactly what was going, on." With the insight he developed, he could go back and understand what his computa­tions were telling him.

He realized that the "miraculous" numbers had been computed correctly, but they were not true eigenvalues. True eigenvalues are supposed to. cause a mathematical function to have the value of zero everywhere on the modular sur­face. For Haas's numbers, though, the function had values that were zero ev­erywhere except at one special point, where it had the value of infinity.

Haas's computer solution necessarily had had to treat the modular surface as being made up of a large but finite col­lection of points; there was no way its infinitely many points could fit in a computer memory. There was room be­tween these points, therefore, for the one special point at infinity to consist­ently fall into the cracks. Haas's pro­gram had failed to make allowance for this possibility.

Hejhal later went on to work with Brad Berg of Cray Research Corpora­tion, the company that built the Cray-I, then, the world's fastest and most

powerful computer. Hejhal succeeded in showing what he had expected at the outset: that the true eigenvalues have no direct connection with the zeroes of the Riemann zeta-function. Nevertheless, for all this work, he had just begun. "We were just testing the most naive case," he says. "There are lots of other operators, lots of other manifolds, and. lots of other possibilities that one needs to understand."

Most recently Hejhal has been collab­orating with Enrico Bombieri of the In­stitute for Advanced Study in the first detailed study of the behavior of the zeroes of what are called Epstein zeta-functlons, a type of .function closely re­lated to zeta(s). A Cray computer is again playing a crucial role in the work. "In a certain sense," Hejhal says, "we're just continuing what Riemann started with pencil and paper. We just have a better pencil."

"Let it be said, I am fed up with the miracles of modern technology," Hejhal says.. "I believe that mathematics was better In the nineteenth century and that we should all go back there. But since we can't, we should use the best modern tools available. Therefore, I'm not interested in small computers, but in big ones." Hejhal thus is one of the few pure mathematicians In the country currently receiving National Science Foundation research support for use of a Cray supercomputer.

-Mathematicians' graphics

In Introductory texts on calculus, col­lege students are presented with abun­dant curves and diagrams to illustrate the elementary functions. In advanced texts and reference works, such as Whit-

taker and Watson's classic Modern Analy­sis, s tudents may make their way through hundreds of pages that are dense with mathematics but contain only a sporadic handful of illustrations. The reason is not that mathematicians outgrow their need for diagrams; rather it is that the needed renderings are hard to prepare.

As William Thurston says, "When mathematicians give talks, they draw pictures. The pictures may be crude, but you can learn a lot from them. But math­ematicians tend to be embarrassed to publish such pictures, and when they write papers, they tend to leave most of the illustrations out. Illustrations just tend to be an incredible pain to gener­ate, because there's no good group of technical people accustomed to doing the kinds of artwork that mathemati­cians want."

Today's high-level computer graphics are changing this situation. Moreover, they'are bringing new ease to computer experiments, giving vivid and immedi­ate displays of results. "Where numer­ical experiments are easy, one gets very sharp ideas of what is true," says John Milnor of the Institute for Advanced Study. "You can learn more by a few minutes of looking at what happens on a video screen than by hours of formal computation."

The study of higher dimensions Is one area to which such graphics have brought new insight. Thomas Banchoff, now at Brown University, has been working in this area for a number of years. He recalls his days at the Univer­sity of Notre Dame, In about 1959. "When.I studied complex variables," he says, "I asked my professor, 'When are we going to graph these functions?' I felt that accurately drawn diagrams were essential. He replied, 'Don't be silly, young man. To graph these things would take four dimensions.' But I pri­vately decided to try to visualize these things, because If I could, I could gain Insights that other people didn't have."

Banchoff went on to work with Charles M. Strauss, now an indepen­dent consultant in Providence, Rhode Island. They proceeded to study the ge­ometry of a torus In four-dimensional space, and they made a film showing what would be observed if the torus could be seen moving in this hyper-space. "Once we made this- film, it was clear we were seeing things that we nev­er would have seen any other way,"

Volume 16 Number 4 MOSAIC 43

Banchoff says. 'The-first image of that torus looked like an ordinary inner tube. But then-we rotated in- four-space, and the torus appeared to "be doing things that ordinary inner tubes don't do. It got very bulbous on one side. Then it turned inside out. No one had ever seen these things rotated from one into the other. When Strauss and I saw it on our first film, it was one.of the most exciting experiences I've had in my life."

Now, fifteen years after this pioneer­ing film, Banchoff, in collaboration with colleague Hiiseyln .Kocak and others, has returned to' the study of the torus— this time as the union of orbits of a dy­namical system graphed in four-space.

Graphics . in higher dimensions may involve pitfalls. At Stanford University, for example, statistician Persi Diaconis has been studying arrays of points in as many as 20 dimensions. His 'graphics programs project, the arrays down to the 2 dimensions of a video screen, where, in his words, 'The shadow of the bell-shaped curve comes over you/ ' That is,

• with so .much compression of dimen­sions, a statistical result known as the central-limit'theorem comes into play and introduces spurious bell-shaped distributions into the statistics.

There also is the danger of finding ap­parent order where there really is only randomness. If there are 100 points scat­tered randomly in a 50-dimensional space, for example, half can be colored red and half black, again at random. It is almost always possible to pass a plane through the ensemble and separate the red from the black, perfectly. "I might well think, 'Aha, I've found structure in the data, '" Diaconis says. "But it's just that you have a lot more room in the higher dimensions."

New vistas :Even in two dimensions, computer

graphics have-opened new vistas. A dramatic example is the study- of Julia sets, named for French mathematician Gaston Julia. These sets can be under­stood by starting with an ordinary cal­culator - that has SIN, COS, and EXP buttons representing the sine/ cosine, and exponential functions. The keying in of a number followed by the repeated pressing of one of these buttons, for. as

-many times .as the presser wishes, re­sults in the Iterate of the particular func­tion, which Is the result of applying the function numerous times in succession. Does the iteration drive the result off to

Banchoff Reducing four dimensions to two,

infinity, or does the result remain close to a well-defined value? It is easy to.see, for example, that with repeated pressing of the square-root button, the iterate soon gets close to the figure one. However, if the presser repeatedly en­ters a two-step -sequence consisting of pressing' the squaring button and then subtracting two, the behavior for vari­ous starting values is rather more diffi­cult to explain.

In.dealing with the functions of a complex variable, the behavior-is even more complicated. Here is a topic that stands on the cutting edge of present research in mathematics, for there Is no straightforward way to tell how the iter­ates behave.

The usual method used Is to program a computer, have It perform the itera­tions, and then display the results graphically When this is done, It is-seen that some values of the complex variable produce iterates that escape; other val­ues give iterates that remain bounded. The' set of the values that do not escape, that remain bounded under iteration of a function, Is called the Julia set of that function.

Gaston Julia and his colleague Pierre Fatou developed a theory for these sets between 1915 and 1930, but because of the inability to create examples, the sub­ject essentially died. No one had any idea what these objects looked like.

During the 1960s Benoit Mandelbrot at IBM's Thomas J. Watson Research Cen­ter and P. T. Bierberg In. Finland re­opened this, topic by using computers to

-calculate examples. During 1980 John

Hubbard of'Cornell University worked in France with Adrian Doua-dy of the Ecole Normale Superieure.- They stud-led Julia sets of a simple function, f (z) = z2 + c, in which c is a complex con­stant. This function is so simple in ordi­nary algebra that It Is studied in high school. Iteration, however, increases Its difficulty dramatically. In the domain of . complex numbers, the Julia sets were known to have highly intricate and de­tailed shapes, - featuring fractal geome­try. (See "Fractal Symmetry" by M.ort La Brecque, Mosaic Volume 16 Number 1.)

Douady .was following this up, and Hubbard recalls what happened:

"One night I was sitting with an Ap­ple II computer. I generated on a plotter a certain collection of graphics. There are precisely 15 such Julia sets in the collection, and I drew them all. Then I really looked at them. I drew [certain patterns] by hand, I connected them up, and I gazed at what Yd seen. I wanted to understand what made these. Julia sets different. So I connected up all these things, and I saw something that was new. I saw fifteen different trees."

There is a topic In mathematics known as graph theory, which deals with collections * of points and of the lines drawn between them. Trees are a specific class of such objects. Hubbard had seen graph-theoretical trees. "\ tried to extract the structure I could see In there," he says. "I then thought out what the ones for [another collection of Julia sets] ought to be. I drew them all— there were 31 of them—and went to the computer and found them. I verified my conjecture experimentally."

In graph theory there are standard procedures whereby a tree may be de-

• rived for any' fraction. Hubbard showed that by a very complicated procedure, it Is possible to derive a complex constant, C, from that, fraction. In the Julia set of the function f (z) = z2 + c, that tree acts as a backbone, or framework, repeated over and over to form a fractal pattern, which outlines that Julia set.

For such Julia sets, - Hubbard says, "I - can tell you what each one Is. More than that, I can construct for each one a par­ticular tree that will completely describe it. Each one has an individuality all its own, which Is .completely characterized by a particular tree, or alternately by a - particular rational number. This may be the first theorem to show up for which computer experimentation was.abso­lutely so fundamental."

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Volume 16 Number 4 MOSAIC 45

Sets. No one knew what, a Mandelbrot set (above) looked like.. Julia.sets {left}'are produced by applying a function repeatedly.-

The rational numbers or-fractions, however, constitute only an infi­nitesimal portion of all numbers. As Hubbard admits, the Julia sets he has been able to study are exceptional cases. "The real numbers have a behavior that is largely still not understood," he says. "A real number requires an infinite amount of information to specify it. [For example, TT = 3.14159 . .. . requires infi­nitely many digits.] So the correspond­ing math will-necessarily be infinite. Presumably a similar theory goes through, but no one has been able to prove anyth ing ." Even for the well-studied function f (z) = z2 -f c, only an infinitesimal part of the whole problem is understood. Nevertheless, this infi­nitesimal part has already proved to be of commercial significance. Hubbard's Julia sets gave rise to graphic patterns of intricately detailed color and form. Al­most at once, Hubbard was deluged with requests for .copies of his computer art, illustrating the Julia sets. "He got more mail than the rest of the depart­ment put together," declared Anil Nerode, his chairman. Thus, Hubbard has formed a company, Artmatrix, to market his graphics.

Artificial intelligence-Anil Nerode of Cornell University ex­

pects techniques from the field of ar­tificial intelligence to become important in mathematical research. The emphasis will be on the manipulation of symbols, not on computation. In a variety of areas involving topology, logic, and algebra, problems arise calling for intricate forms that are to be expressed in. symbols. "Often the needed manipulations are prohibitive," says Nerode. "You can-do only the simplest cases by hand." Nerode anticipates that the computer languages Prolog and Lisp, which have been developed in artificial-intelligence laboratories, will enable computers to handle symbol manipulations and thereby greatly extend the range of problems that can be effectively studied.

"You make a conjecture and it isn't quite right," Nerode says, "but it's very difficult for you to write down- an exam­ple by hand. Every one of us looks for small counterexamples; the computer will allow us'to look at very large ones without much more effort."

"That's what appeals to me," says Nerode, "to see why we're a little off, because we often are. For instance, sup­pose you make a conjecture about finite

Hubbard, Trees connected the Julia sets.

Galois groups. They do take a long time to compute. If you have a particular sort of counterexample in mind and you try to do it by hand, you're just going to give up. But if you try to do it with a Lisp-type program, you have a good chance of writing one out / '

The rise of symbol manipulation stands to recapitulate the rise, 30 years ago, of languages such as Fortran that allowced wording scientists to use corn-puters for the large-scale manipulation of numbers. For Fortran to become an

everyday tool, computers had to have enough memory, at least several tens of kilobytes, to support this new lan­guage. Moreover, there was a need for compilers to translate programs into machine-coded instructions in an effi­cient fashion. The computers them­selves were built with numerical coproc­essors, in which the rules for addition and multiplication were built into the very hardware of the central processing unit. This vastly speeded up those ele­mentary operations.

The symbol-manipulating language Lisp was invented by John McCarthy of Stanford in about 1962, but only now is it beginning to fulfill its potential. It is far more memory-intensive . than For­tran; even rudimentary computations demand four megabytes of main mem­ory in a single-user computer. Lisp thus takes up almost the entire memory of even- very recent mainframes, such as the IBM-3081 and the DEC-20. Says Anil Nerode, "Managers of time-shared computers hate having Lisp users be­cause they drive off the other users."

Both Japan and the United States are now building computers with enough memory to support Lisp and Prolog effectively. The IBM AT personal com­puter, using a version of the UNIX oper­ating system, supports up to 16 mega­bytes of main memory and accommo-dates these l a n g u a g e s , It features an

advanced microprocessor, along with compilers for Lisp and Prolog, which greatly speed up the execution of pro­grams. For Lisp, the Symbolics Machine Compiler also is in use. In. addition, for even greater efficiency, the basic logical operations needed for symbol manip­ulation are being hard-wired into the circuitry of advanced microprocessors.

"I have a conjecture about pure math­ematicians, " says Nerode. "About a quarter of them have run into this: They've been working very hard, but they can't see what to do next, because when they try to compute out the sym­bol manipulations it's too difficult." Nerode says 9 of 40 mathematicians in his department at Cornell have encoun­tered such roadblocks. Several of them have subsequently found a hard-won counterexample to some conjecture and are currently hoping to use Lisp to con­struct additional counterexamples. By making comparisons, they hope to re­fine and sharpen their conjectures.

"It took a couple of years for desktop computers to spread through the de-

46 MOSAIC Volume 16 Number 4

partment, with Pascal and Fortran/' de­clares Nerode. "Now, with the IBM AT, programming in Lisp and Prolog is spreading in a similar way. I encouraged people to do symbolic processing in these languages and got a couple of peo­ple to learn them. They became enthusi­astic, and soon spread the word."

To do proofs

Nerode, along with his colleague Richard Platek, also sees great promise in the use of artificial intelligence as an aid in proving theorems. "When you try to do proofs/7 says Nerode, "you very often have to divide the proofs into many cases. It's tremendously helpful if you can reduce the number of cases as much as possible/ ' This can be done by using the unification algorithm, dis­covered by Jacques Herbrand in 1929 and implemented as a computer al­gorithm by John Robinson of Syracuse University. "It means," says Nerode, "that there is this thing that comes up all the time in symbolic computation. These are things that we do badly and the machine does well."

There is already a field known as au­tomated theorem-proving. Its main use thus far has been In verifying computer algorithms. For example, one might be presented with a computer program that is supposed to carry out some pro­cedure and is written in microcode, ma­chine language. People are very poor at reading microcode, but computers are very good at it and can generate formal proofs that the algorithm is correct.

Richard Platek describes the current state of the art: "Some theorems have been proved, based on conjectures found in minor journals of math—repu­table, but minor. But outside the field of automated theorem-proving, they'd be of very limited Interest. It just shows the field has matured sufficiently to get minor theorems. When computers find a proof, it's very impressive; when they don't, they run all night."

Nevertheless, Platek and Nerode an­ticipate that these theorem provers will advance to the point of being genuinely useful to mathematicians. Nerode looks to the advent of a mathematician's help­er. A researcher will put in a desired result, along with a collection of appro­priate theorems and results, and a sketch of the proof. The computer then will fill In the logical gaps and verify that the proof Is correct. Platek has a similar vision. "You tend to write out a

proof on paper, and If it doesn't work you stare at it," he says. "What I want is that you write It out on this machine, and if it's incomplete, the machine will tell you why it's incomplete." When is something like this likely to happen? "In ten years," says Platek.

Beyond the mathematician's helper, there already are hints of computer pro­grams that will formulate and prove conjectures, In the fashion of a mathe­matician. At Stanford University, Doug­las Lenat has developed a program to formulate concepts and theorems in mathematics. He set the program to considering the concepts of divisors of a number. Working on its own, the pro­gram discovered the existence of prime numbers and focused its attention on them. In about an hour of running time, the program reproduced several well-known conjectures about prime num­bers and, furthermore, showed that every integer can be factored into a unique set of primes.

The true significance of this lies in the future, but Lisp and Prolog are already in existence. As Nerode describes them, "they are literally outside the Intellec­tual sphere of 99.9 percent of all mathe­maticians. Yet they are made for symbol manipulation; they are enormously powerful. For Instance, in proving the four-color theorem, Ken Appel did the symbol manipulation by transferring in­formation in and out of registers."

Language considerations

"If you look at the manipulations," says Nerode, "writing all that stuff out in Lisp is trivial. The work on problems of that sort is vastly compressed by using the modern languages. When you use these logical techniques, the search for the right configuration to a large ex­tent comes out of this unification al­gorithm, which the machine does effi­ciently and we don ' t . The preferred language is Prolog. It's so simple; a math­ematician can learn it in an hour. Nev­ertheless, current versions of Prolog leave much to be desired for the pure mathematician. Their Unification Al-gorithims, as presently Implemented, can prove false theorems. Today's Pro­logs have been written to support expert systems and similar applications. To achieve high speed, they thus leave out a rarely used feature that could give very long run times—but which is nonetheless essential for a correct Unifi­cation Algorithm.

"We are not a market that's large enough for commercial programmers to justify developing versions of Prolog suited to our needs / ' says Nerode. "Ei­ther we'll have to get it as a gift, or we'll have to do it ourselves." Thus, to use Prolog, mathematicians may have to check by hand at every step in a proof to see that the Unification Algorithm has been applied correctly. In a proper ver­sion of Prolog, the machine would do this automatically. The Japanese Prolog machines Implement the basic Prolog in­structions in microcode.

Patterns

"Theorems help keep mathematicians honest," says William Thurston. "We want theorems; but if you can actually see the mathematical patterns, the the­orems become obvious." The signifi­cance of computers is that they greatly increase the ease with which a mathe­matician can see the patterns.

The patterns may show that a certain conjecture is almost certainly right, that even extensive computation has been unable to give a counterexample. This can then give a mathematician needed assurance, so he does not lose heart in the face of difficulties. The patterns may also show a counterexample, however, in which case the mathematician has been spared the prospect of putt ing forth effort in a fruitless quest.

With a quest under way, the computer becomes an exploratory tool, to study the patterns themselves, to examine means for generating other patterns, to test and refine conjectures, methods, approaches. Rarely if ever will a mathe­matician write a program, allow it to chug away, and be prepared to accept the result as it stands. Rather, the math­ematician will use the computer to do experiments, particularly new ones, that could not have been done otherwise. The mathematician and computer, then, are partners in a dialogue in which the goal is insight.

In time, there may be cybernetic aids to theorem proving and even to the de­velopment of conjectures, of the in­sights themselves. Meanwhile, there already are mainframes, supercom­puters, high-level graphics, and pro­grams for symbol manipulation. Their significance can only grow. As Anil Nerode says, "Here you are as a work­ing mathematician, and you never dreamed so much power could be avail­able so cheaply." •

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