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Dedication

This issue of the Mathematical Intelligencer is dedicated to Allen Shields, who wrote the Years Ago c o l u m n of the

later, I still remember that talk clearly~ because it was the first mathematics seminar talk that I understood. Mathematical Intelligencer In the last d o z e n years, from 1987 to 1989. O n 16 Allen a n d I w r o t e six reSeptember 1989 Allen died of search papers together. I'll cancer at age 62. always remember the magic A l l e n w a s o n e of t h e a n d p l e a s u r e of w o r k i n g w o r l d ' s leading authorities with Allen. It was great fun on spaces of analytic functo be in the same room with tions. He directed more h i m for h o u r s s t r u g g l i n g Ph.D. theses than anyone in against a problem. We the h i s t o r y of the M a t h e worked with each other so matics D e p a r t m e n t of the smoothly that for five of our U n i v e r s i t y of M i c h i g a n , six joint papers, Allen and I where he also served as chair wrote every word together, Allen L. Shields for a few years. In 1979 the rather than the usual fashion 1927-1989 U n i v e r s i t y of M i c h i g a n of dividing up sections be( P h o t o taken May 1 9 8 9 ) a w a r d e d Allen its Distintween the authors. guished Faculty Achievement Award. Allen's Years Ago columns sparkle with insight and I first met Allen in the fall of 1973, when he gave a demonstrate his unusual knowledge of history as well talk in the functional analysis seminar at Berkeley, as mathematics. The following articles make clear the where I was a graduate student at the time. Instead of many different ways in which Allen Shields was an impressing the experts in the audience by confusing extraordinary person. them, Allen gave a talk that the graduate students Sheldon Axler could comprehend and learn from. Seventeen years

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To My Partner, To Allen Shields Smilka Zdravkovska

Smilka Zdravkovska and Allen Shields, 1982. Photo by Allen Shields. Allen took the picture by holding the camera at arm's length and aiming carefully. I probably fell in love with Allen the first time I heard him lecture (it was on r a n d o m p o w e r series, in the Math Club1). H e r e was this tall (over 6'4", 195 cm.), lean (about 160 lbs., 73 kgs.) red-haired b o y (it was only later that I f o u n d out he w a s n ' t a junior faculty m e m b e r in the Mathematics Department, b u t rather its acting chairman; oh, well, no one is perfect), as enthusiastic as w h a t w e u s e d to call in M o s c o w an olympi-

1 The University of Michigan Math Club meets once a month, and is open to students and faculty alike. After refreshments, people are invited to give three-minute talks, and then the main speaker of the evening delivers a lecture, accessibleto a general audience of mathematics students. Allen had based his lecture on a paper by Salem (his unofficial advisor; the official one was Hurewicz) and Zygmund. |

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a d n i k (kid c o m p e t i n g in the m a t h o l y m p i a d s ) , so m u c h at e a s e . . , like a fish in water (swimming was a great h o b b y of his), or a bird in the sky (one of his fantasies was being able to fly). I didn't then k n o w the plethora of his other qualities, n o r h o w utterly compatible w e were (from trivial habits such as the orientation of the toilet paper roll to important matters such as h a v i n g similar priorities a n d world outlook). These things I discovered during a few m o n t h s of nightly nightlong p h o n e calls (we w e r e both in Ann Arbor, but I h a d b o u n d a r y conditions o n outings due to a 12year-old daughter) and from a torrent of notes/letters/ clippings with which he started bombarding me. Losing a giant is screamingly painful. Sharing excerpts from a few of the h u n d r e d s of letters and cards received a r o u n d the time of his death is a little solace.

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Allen was a mathematician, Allen was a scholar. Allen was a humanist (is it all right if I add that in his case this also meant feminist?) in a most natural a n d direct way. It was easy to be good around h i m - - a s the Montenegrin prince-bishop-cum-poet Njego~ (1813-1851) said: "'U dobru je lako dobar biti. ''4 Maybe all of us w h o k n e w him loved him so much because he brought out the best in us, he m a d e us better people. He made the world a better place. Thank you, partner. Thank you, Allen Shields.

Acknowledgment. I would like to express my (and Allen's) gratitude to all our friends who helped us in so many different ways. I would also like to thank the Mathematical Intelligencer for dedicating this issue to Allen and for inviting me to contribute to it. Mathematical Reviews University of Michigan Ann Arbor, M148107 USA 4 In good [surroundings] 'tis easy to be good.

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Allen Lowell S h i e l d s - - S o m e Reminiscences Harold S. Shapiro

I first encountered Allen Shields in 1949 during our last undergraduate year at CCNY (City College of N e w York), w h e n we were enrolled in a course together. I don't understand how I had failed to meet him before that. The academic environment at CCNY was remarkable, perhaps less so for the mathematics faculty (although there were excellent and helpful teachers like Bennington P. Gill, Emil Post, and Lee Lorch) than for the students, many of whom were talented and ambitious. Enthusiasm ran high, and more advanced students played an active role in guiding and instructing the beginners. Their help and advice were especially valuable to me, having come to "City" as an engineering student from an outlying high school without strong academic traditions. I can only guess to what extent Allen was similarly benefitted, but it seems reasonable to assume that something rubbed off on him during those years from contacts with a student body that included Robert Aumann, Martin Davis, Leon Ehrenpreis, David Finkelstein (physics), Donald Newman, Lee Rubel, and Jacob Schwartz (naming only those that come most readily to mind). In later years Allen spoke nostalgically of City College and retained ties with N e w m a n and Rubel. I have a vivid mental picture of Allen at that period (I think his appearance changed remarkably little in the course of his life). A very tall, slim, redhead, he usually wore army clothing and plaid flannel shirts. He was very tidy and always had his homework done. With his rustic appearance and Anglo-Saxon name he seemed rather exotic in that population of typically urban, predominantly Jewish students. But we only really got acquainted later w h e n we both turned up for graduate study at M.I.T. and began a friendship that was to last for the remainder of his life. I was surprised to learn that his father was a famous "red," Art Shields, who wrote a column for the Daily Worker. In the middle-class Brooklyn home

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where I grew up we occasionally got this paper and I had read it and remembered Art's columns. I hadn't k n o w n any "card-carrying C o m m u n i s t s , " b u t m y reading included I. F. Stone's newsletters and George Seldes's books, and I was generally sympathetic to American radicalism. Although Alien and I didn't discuss politics much, it was clear that we shared many of the same attitudes. Later, I was often to visit him at his parents" fiat in New York. I was surprised that he addressed his parents as Esther and Art, never Morn and Dad. He was the only child. In the Shields home there was an atmosphere of gentleness and courtesy, and I never heard a loud voice nor a harsh word. Art could maneuver almost any topic of conversation into a political mode and would then demonstrate skillfully the "contradictions of capitalism" for which he had plenty of documentation on hand, always from impeccably bourgeois sources like the New York Times and the Wall Street Journal. Allen accepted much of Art's account of social injustices in the United States, especially when the latter spoke from experience, describing miners' strikes and the poverty he had seen among working people and poor farmers. But Allen could not accept the ideologies nor the panaceas that Art and Esther, "true believers" both, offered! This did not seem to vitiate his affectionate relationships with his parents, and he wrote me in September 1976: I had a nice three days at the beach with my folks. My dad and I had the frankest discussion about communism, the Soviet Union, Class Struggle and a' that we've had in years. We both are able to talk about these things without getting so upset as formerly.., he is less dogmatic... In later life Allen would be annoyed when people from left-wing movements "collared" him, "thought they o w n e d him" and told him of his "obligation" to support this or that cause. He came to mistrust people with strong ideological fixations. It was only personal integrity that counted.

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Meanwhile, back at the Massachusetts Institute of T e c h n o l o g y . . . it was a friendly department and we were made to feel welcome. I especially remember the wonderful departmental secretary, Ruth Goodwin. However, our baptism of graduate study was disappointing. The reigning paradigm was "generality," with attendant lack of motivation, history, application, or even examples. Also, instruction was more bloodless than at "City." Typically, an M.I.T. class consisted of the lecturer writing his notes on the board. One professor, a champion of austere formalism, often needed to draw a diagram to see what was going on, but in order not to "corrupt" us he would hide it with his body, then erase it before the class could see it. This became a standing joke between Allen and me, and perhaps it influenced the evolution of his o w n lecture style into the model of clarity and vividness that was his hallmark. Somehow, a spark of enthusiasm remained after these courses, abetted by outside reading. Allen had, during his military service in Berlin, acquired a nice collection of books in the Springer Grundlehren series. He was reading H a u s d o r f f ' s Mengenlehre, and we worked problems from P61ya-Szeg6. Later, we were privileged to study with Witold Hurewicz, who was the very embodiment of the classical European mathematical tradition we admired. Hurewicz was a lovable man and a wonderful teacher, who invited a group of s t u d e n t s to dine w e e k l y with him. We also took courses with Raphael Salem, who spent half the year in Cambridge and gave sparkling lectures on Fourier analysis, which started us reading Zygmund's treatise. Then Walter Rudin came as a Moore Instructor and showed us new perspectives in analysis. Among fellow students in Cambridge, one who had the greatest importance for my development, but also w a s v e r y s i g n i f i c a n t for Allen, w a s D o n a l d J. Newman, who had come to Harvard from City College. For him it was the integers that counted, but like his mentor Edmund Landau he cast his net wide in the waters of analysis. At Harvard, where snobbery was not unknown, he was never seen to duck into an "emergency egress" when someone approached saying "Here's a problem." The born antithesis to all that could be stultifying at graduate school, "D.J." had a genius for propounding tantalizing little problems that often became the subject of extensive discussion in our circles. One of my fondest memories from the M.I.T. years is running into Allen in the library at ungodly hours of the night. We shared reverential feelings for "classical" mathematics and had acquired the habit of browsing in sources like FundamentaMathematica and Crelle's Journal. We compared notes on what we were i

Left to right: A. Shields, J.-P. Kahane, A. O. Gel'fond, Y. Katznelson, H. S. Shapiro at Lake Sevan, Armenia, 1965.

reading or working on. I'll never forget that atmos p h e r e - w e were the only living souls around; it seemed truly amazing that we had access to the departed, to Banach and PoincarG to the Urysohn metrization theorem as seen through the eyes of Urysohn, and rather clearer than in our topology course. After M.I.T., Allen went to Tulane and I went to Bell Labs for two years during the Korean war. In the years that followed I became deeply involved with a nascent radical organization. Allen and I discussed this sometimes, and he listened politely but wasn't "buying" any new utopias. "The paradise that Cantor had created" was good enough for him. N o w Allen's Wanderjahren were over. He had acquired a family, a professorship at Michigan, and a mathematical profile combining operator theory and complex analysis. During 1959-61 Allen was in N e w York, I had returned to mathematical work, and we did our first joint research: on interpolation in H | Several years later I obtained, with his help, an appointment at Michigan. I left this wonderful, supportive department in 1972 for personal reasons, and thereafter saw Allen infrequently. Looking back on Allen's life one sees that "the elements were mixed in him" and he could have done well at several professions. He loved history and, like his father, had a phenomenal memory. He was a fine amateur linguist. He could have been a journalist--he loved to travel about the country (later, the world) drinking in local atmosphere and history, hitching or offering rides, always curious to talk with the people he met en route. "[I picked up a] hitch-hiker in Virginia," he wrote me, in the best Art Shields manner, after driving down to New Orleans in September 1953, "[who had] left wife and five kids in North Carolina with groceries for one month. Came to Va. looking for w o r k - - h a s to buy school books for daughter in high school, $12 a year . . . . "

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Walter Rudin at M.I.T., 1951. Photograph by Allen Shields.

Allen's twin interests in history and people (without the definite article!) continually recur as I re-read his letters from a thirty-five year period. On one of my last visits to Ann Arbor he introduced me to a new friend, who owned a farm and made excellent maple syrup. He admired rugged individuals, especially craftsmen and artisans who took pride in their workmanship. Allen was not especially handy, but his mathematical papers are "crafted" with loving care. He also had high ethical principles and went out of his way to acknowledge the priority of others, or ideas from other sources. I feel very privileged to have had Allen as a friend. The countless practical things he did on my behalf when I left Ann Arbor for the summers, all the help my wife and I later received when we settled t h e r e . . . he was always ready to lend a hand. But his friendship went far b e y o n d practical matters, in ways I cannot adequately convey without going into autobiographical detail that would be out of place here. He had difficulty in saying "no" to any request. Paradoxically, this good-natured trait sometimes led to conflict when he acquiesced in more commitments than he could handle. And Allen certainly did not like to create conflicts nor hurt anyone's feelings! This trait is typified by an amusing incident, which I cannot refrain from telling here. When we were roommates at M.I.T., I somehow assumed the role of cook (for which I had absolutely no qualifications). One evening I got the hare-brained idea to add yeast to a meatloaf recipe so it would rise and become porous. The yeast did produce a sharp odor that permeated the apartment, but nary a millimeter did the meat rise. It tasted horrible! Allen tasted his portion and made a wry face. When I asked what he t h o u g h t of it he replied, "Well, it's not the greatest." But he ate it, and in fact took a second helping. No account of Allen would be complete without a

Allen Shields (left) and Harold Shapiro at Hadrian's Wall, 1976. Photograph by Peter Duren.

glimpse of his lighter side. His playfulness is typified by the r e b u s e s and p u n s (often outrageous) that abound in his letters. There appear formulations like "Consider a b o u n d e d miserable function in the eunuch's ball . . . "; references to mathematicians like Show K, Crimson Cow, and others with names truly horrendous, although there seldom was any malice in the Shields transform. At Ann Arbor, if one of the "locals" was to hold a seminar, Allen would usually launch a paper airplane at a strategic moment as the speaker was about to begin. (Years later, in the frostier seminar climate of a certain northern European country, I sometimes miss those airplanes.) I often find myself quoting Allen's metaphors. A mathematical problem is a "jackpot" which gains in value as more of us throw our quarters into it. Or "functional analysis moves you horizontally, hard analysis vertically." (Thus, functional analysis gives you different equivalent formulations of your original problem, but it doesn't solve it for you. It shows y o u places where y o u can dig, but dig you must, somewhere, and that is the task of hard analysis. It may be in all cases you have to dig through granite, and that requires the services of an "F.O.N." --force of nature.) Allen retained his sense of humor, even after he had to drink the bitter cup that was his lot. He bore his suffering with colossal fortitude, and at the conference in his honor in August 1989, gravely ill, he greeted the delegates with a pun. In the summer of 1988 he was stricken, in his best years, and with so much unfinished, by the illness that soon was to tear him away from us. For me, the grief of this loss will always remain, but it is tempered by thankfulness that such a rare individual did exist, and I was so fortunate as to have known him.

Mathematics Institute Royal Institute of Technology S-100 44 Stockholm, Sweden i

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In Remembrance of Allen Shields Peter Duren

Late one afternoon in the early 1950s, my father was walking through the corridors of Gibson Hall at Tulane University, where the Mathematics Department was housed. As he passed an empty classroom, his eye fell upon a body sprawled out on a large desk in front of the room. The head and torso were covered with newspaper, leaving only a few arms and legs exposed. My father was alarmed, but he was Chairman of the Department, so he felt it was his duty to investigate. When he drew closer, he was relieved to see the newspaper gently rising and falling in regular rhythm. Hesitantly lifting a corner of the paper, he discovered the new Instructor he had just hired by the name of Allen Shields. Over the years I e n j o y e d retelling that story in Allen's presence, especially among a group of unsuspecting friends, and I think Allen enjoyed hearing it told. At the punchline I always seemed to promote him to an Assistant Professor, and he would shout, "Instructor!" Then he would explain that he liked to sleep in that room because the desk was longer and more comfortable than the one in his office. I first met Allen Shields in the summer of 1953. My father introduced us at the Mathematics Meetings in K i n g s t o n , O n t a r i o - - p r o b a b l y after o n e of Paul Halmos's lectures on axiomatic set theory. I was just 18 years old; Allen was 26. I vividly remember the tall, erect young man with flaming red hair w h o spent a few minutes in friendly conversation with me. Later on, after I joined the Michigan Department in 1962, Allen and I got to be good friends. We wrote 5 papers together. George Piranian tells the story of h o w Allen happened to wind up in Ann Arbor. In 1955, on the first day of the American Mathematical Society (AMS) S u m m e r M e e t i n g in A n n Arbor, G e o r g e a s k e d

Chairman T. H. Hildebrandt for a leave of absence for the Winter Term of 1956. Immediately, Hildebrandt declared that he could not grant the request unless George found a replacement. On his way from Hildebrandt's office to one of the lecture sessions, George ran into Allen Shields, w h o m a year earlier he had met at the Summer Meeting in Laramie. Allen was cooperative, and George dashed back to report that he had found a substitute and that after fifteen more minutes the substitute would present a ten-minute paper. In those days, many AMS sessions ran badly behind schedule. Hildebrandt suffered through a long presentation in which a promising lad tried to tell everything he knew. Then came Allen's turn. Young as he was, Allen had already mastered the art of beginning his blackboard work in the upper left-hand corner and ending neatly at the lower right, with one minute to spare. Hildebrandt was so impressed that on the spot he offered Allen a one-term appointment. Later, the department persuaded both Shields and Hildebrandt to extend the arrangement. Allen received a Ph.D. from M.I.T. in 1952 under the formal supervision of Witold Hurewicz, but his actual thesis advisor was Rapha61 Salem, who was then dividing this time between Paris and M.I.T. In the fall of 1952, Allen went to New Orleans to accept one of Tulane's newly created Research Instructorships, with a reduced teaching load funded by the Office of Naval Research. Other Research Instructors there with Allen w e r e Leon Brown, Paul Conrad, Don Edmondson, Paul Mostert, and David Newburgh. At Tulane, Allen learned a lot of functional analysis from Bill Pettis and wrote some papers with Paul Mostert on topological semigroups. Only after his arrival in Ann Arbor did he begin to do the kind of mathematics for which he is best known. A two-year stay

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(1959-61) at N.Y.U. had a decisive influence on the direction of Allen's future research. It was there that Allen and his old friend Harold Shapiro wrote their two well-known papers [5,6] on interpolation in lip spaces and on the zero-sets of analytic functions with finite Dirichlet integral. A few years later, w h e n Harold came to Ann Arbor, the three of us spent an exciting year tracking down non-Smirnov domains, those elusive objects whose existence was then known only through a complicated geometric construction of Keldysh and Lavrentiev. We were delighted to find [3] that the problem reduces to the construction of a singular nondecreasing function in the Z y g m u n d class of smooth functions. After that first collaboration, Allen and I spent a lot of time together: the memories come flooding back. I recall the little things. Lunch over a pad of calculations. Allen eating an apple, everything but the stem. The wad of annotated index cards in his shirt pocket. Then there were the transom signals. We would often close our office doors when we wanted to concentrate, and we didn't always respond to knocks. In such a situation I once heard a knock not on my door but on the transom above it! I knew instinctively that it had to be Allen, and of course I opened the door (not the transom). After that, transom knocks became our regular calling cards. (Recalling the Tulane days, Paul Mostert says that "transoms were Allen's favorite obj e c t s - w i n d o w s to the little people. He would peer over the transoms at me during m y lectures with strange grimaces designed to drive me into hysterics before my students!") Allen liked to play with words. His letters were spiced with all sorts of intentional and outrageous distortions. A Banach space was sometimes called a Bone Ache space. He referred to Chubby Chef's inequality. A proposed course in measure theory with modest prerequisites became a course on the LaVague integral. There was another example that I'll always remember. In the summer of 1967, Allen and I were writing up w h a t we called "the m o n s t e r , " a long paper [2] on the dual space of HP with p < 1. I thought the manuscript was finished when I left town to spend a few weeks with my wife and children on the coast of Maine. But Allen kept shipping me additional sections of manuscript that he proposed to give to the typist. One page of his familiar scrawl contained a sentence that read, " W i t h o u t loss of genitalia, we may assume . . . . " Later he told me he was just checking to see if I was reading my mail. The monster came to life in a curious way. In 1966, while writing my book on Hp spaces, I was surprised to discover that the bounded linear functionals on HP had never been described for p < 1; there was only a

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Allen Shields at Tulane.

partial description due to S. S. Walters. I worked on the problem, found the Lipschitz space description (within ~) for p # 1/2, 1/3. . . . , and sent in an abstract [1] for publication. Next I told the result to Allen via a seminar talk. Within a few days, he amazed me by using it to produce an ingenious example of a "fat" subspace of HP, a proper subspace that only the zero functional can annihilate. This meant that although there are enough functionals to distinguish points of HP, there aren't enough to distinguish points from subspaces. Here was a natural space where the HahnBanach theorem fails! We got pretty excited, and together we worked out some more results. Meanwhile, I learned from Frank Forelli, and later from Walter Rudin when my abstract appeared, that a student of Rudin at Rochester named Bernhard Romberg h a d written a thesis in 1960 with a similar (slightly stronger) representation of the dual space. The thesis had never been published, but there was an abstract [4]. I wrote to Rochester for a copy of the thesis, but a copy arrived instead from Romberg himself. By that time he was doing operations research and computer work, helping to design ships for Arthur D. Little, Inc. in Cambridge, Massachusetts. He was out of touch with pure mathematics, but he expressed interest in our work. Allen and I looked through the thesis and found some nice results and new ideas that went beyond what we had anticipated. We proposed to incorporate some of Romberg's ideas into a triple paper, and he readily agreed. (Around that time I met Bernie Romberg at an AMS Meeting, but I think he and Allen

Witold Hurewicz at M.I.T., 1951.

Raphael Salem at M.I.T., 1951.

never met.) The paper was soon "finished," but Allen was blazing with inspiration. He kept turning out one addendum after another as he sought to abstract the structure of HP to the general setting of topological vector spaces. That's h o w the paper came to be such a monster. Doing mathematics with Allen was fun. He was creative and powerful, yet childlike in his enthusiasm. His lectures were models of clarity and inspiration. He was a wonderfully gregarious mathematician w h o found excitement in mathematical discussions with colleagues and students. Throughout his career, he always had a strong tendency to collaborate. The record shows that of his 80 research papers, all but 15 were written jointly with other mathematicians. He had 44 different coauthors. Some of his frequent collaborators not already identified were (in chronological order) Leon Brown, Lee Rubel, Carl Pearcy, and Sheldon Axler. Yes, he did write a paper with Paul Erd6s. All of Allen's co-workers knew what it was like to discuss mathematics with him. It seemed that every time I went into his office with a half-baked idea or a small discovery, Allen would respond with a steady stream of pertinent mathematical facts, fully detailed and totally accurate. Sometimes he would hold forth at the blackboard for half an hour. He had a wealth of information filed away in his head, all neatly arranged and cross-referenced, and he knew where to find what he wanted. With the stream of information came new insights and conjectures, ideas for proofs, and lots of enthusiasm. He was an inspirational figure, and a

marvelous mathematician. Working with him was pure joy. His phenomenal memory for detail extended well beyond the realm of mathematics. He read avidly and broadly, and he was a walking encyclopedia of information about historical events, literature, and current politics. (When our children were reading the Oz books, they found that Allen remembered all of the plots!) A typical scenario at a dinner party w o u l d begin with a self-proclaimed expert holding forth on some topic of historical interest. Allen would suffer in silence, like a musician with perfect pitch, as facts were distorted. Only upon appeal, after a dispute arose, would Allen tell what he knew. Invariably it would turn out that he knew far more than the expert, and he knew the names, the dates, the exact sequence of events. But as he set the record straight, he gave no offense. Sometimes a reference book would be hauled in to settle a continuing dispute. Allen's information always proved correct. Paradoxically, his daily life was anything but organized. His mail lay unanswered, often unopened for months or even years. It simply got buried beneath a heap of papers in his office. His files were chaotic. As an editor of the Proceedings of the American Mathematical Society, he would sometimes bypass his well-organized secretary by sending out a manuscript to a referee with a handwritten covering letter, making no record of where he sent it. (I should know; I followed him as editor.) Allen lived a simple life, without grandeur, without extravagance or pretension. He talked to everyone in

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the same straightforward manner, with never a hint of either condescension or deference. He liked to wear blue jeans and sandals. He once owned a necktie, and in earlier years he even wore it sometimes. But later he abandoned all unnecessary frills. (He did wear a tie when he and Smilka were married.) I remember that in the summer of 1976, he traveled to a big conference in England, carrying everything he n e e d e d for 3 weeks in an old battered typewriter case. That was his only luggage.

A proposed course in measure theory with modest prerequisites became a course on the La Vague integral. Allen was good at languages, including English, and he liked to use them. He was fluent in Spanish, German, Russian, and French. After spending some time with Smilka and her daughter Bojana (who came to the United States from Skopje, Yugoslavia), he learned to communicate in Macedonian and even Serbo-Croatian. He was probably best at German and Russian. His German was perfected when he enlisted in the Army in 1944 (at age 17) and spent some time in Berlin after the war was over. But it was always the Russian language, and the Soviet Union, which held a particular fascination. Allen's father, Art Shields, was a well-known Communist who for many years worked in Moscow as a correspondent for the Daily Worker. Allen himself was keenly interested in all things Russian, and he had an expert knowledge of Russian history and culture. It was a passion that he and Smilka shared. When at last they made an extended trip together to Russia, it seemed a cruel twist of fate that it had to end in a medical nightmare. What set Allen apart from other people was the absolute integrity in everything he did a n d said. He didn't lie, cheat, or steal; but it went much further. In discussing any issue, he was always careful to state the raw facts as he knew them, unembellished, unexpurgated, unslanted. It wasn't his nature to plead for one side or the other, to color the facts, or to make convenient omissions. He gave the straight story, and he wanted the straight story from everyone else. He had the patience to suspend judgment until he felt fully informed. His integrity extended to his mathematics, and to everyday life. It also guided his actions as an administrator. When as Department Chairman he had to request funds from the Dean, he felt compelled to present all sides of the case, pro and con. Needless to say, this is highly unusual. The Dean must have been III

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surprised at first, but one hopes he came to respect Allen's credibility. More should be said about Allen as Chairman of the Department. Certainly, his strength did not lie in the efficient shuffling of papers, in the rapid transaction of business, or in the timely response to deadlines. Without Ethel Rathbun behind him, Allen's Chairmanship might have ended in disaster. But there is a n o t h e r side of the Chairmanship, a more h u m a n side, where Allen excelled. He was a good listener, and he really cared about people. I know of two cases (I'm sure there were others) d u r i n g his time as Chairman where he went far beyond the call of duty to h e l p m e m b e r s of t h e f a c u l t y w i t h p e r s o n a l problems. In both instances, he knew that his role as supervisor put him in a special position to help. To Allen's credit, he was willing to get involved, and his efforts really made a difference. Both cases required time and took Allen away from his more formal duties as Chairman. But they were important because they had a direct bearing on people's lives. Allen was much more than a creative and productive mathematician. He was a warm human being, a loyal son, a loving father, a d e v o t e d h u s b a n d , a staunch friend. Above all he lived a life of kindness and concern for his fellow-creatures. He never let selfish desires or ambitions cause him to hurt another person in any way. Even as death approached, he found himself consoling those who grieved for him, making little jokes to cheer them up. He showed us how to live, and he showed us how to die. There was a giant among us.

References 1. P. Duren, Linear functionals on Hv spaces with p < 1 (abstract), Notices Amer. Math. Soc. 14 (1967), 265. 2. P. L. Duren, B. W. Romberg, and A. L. Shields, Linear functionals on Hv spaces with 0 < p < 1, J. Reine Angew. Math. 238 (1969), 32-60. 3. P. L. Duren, H. S. Shapiro, and A. L. Shields, Singular measures and domains not of Smirnov type, Duke Math. J. 33 (1966), 247-254. 4. B. W. Romberg, The dual space of Hp (abstract), Notices Amer. Math. Soc. 9 (1962), 210. 5. H. S. Shapiro and A. L. Shields, On some interpolation problems for analytic functions, Amer. J. Math. 83 (1961), 513-532. 6. H. S. Shapiro and A. L. Shields, On the zeros of analytic functions with finite Dirichlet integral and some related function spaces, Math. Z. 80 (1962), 217-229. Department of Mathematics University of Michigan Ann Arbor, MI 48109 USA

On Being Allen's Student Joel H. Shapiro

The Ann Arbor I remember of the 1960s was a wonderful place to learn mathematics. In addition to Allen Shields, the faculty included such distinguished analysts as Paul Halmos, Ron Douglas, Peter Duren, George Piranian, and Carl Pearcy. Advanced graduate students, such as Lew Coburn, Steve Parrott, Peter Rosenthal, and Joe Horowitz, all of w h o m were to make their mark later on, encouraged us rookies and passed on to us their gung-ho attitude toward mathematics. I always thought of Allen as a member of this latter group. His boyish enthusiasm and gentle, gregarious manner seemed more appropriate to our bull sessions in the Michigan Union cafeteria than to the intimidating confines of a Professor's Office. The easy informality of the Union carried over naturally to Allen's office on the third floor of Angell Hall. I spent many hours in that room, in front of his big blackboard, where he'd show me mathematical gems, and I'd tell him about my attempts to get started on a thesis. When I lectured to him he would fold his lanky frame into one of those little straight-backed institutional chairs you offer your students during office hours, and in order not to trip me, he'd scrunch his long legs up in front of him, so that his knees seemed to end up under his chin. To get comfortable, he'd tilt the chair b a c k . . , further and further. Inevitably, just when I thought I was about to say something important, he'd get on the wrong side of the equilibrium point and start to topple over backward, only to flail out one of those improbably long arms, grab the edge of the table, or part of the filing cabinet, and save the day. In the classroom, Allen's lectures were things of beauty. We all appreciated the tremendous effort he put into his courses, and we worked extra hard to learn the material, both for our own benefit, and to show him that we cared. One semester he gave twice the usual number of problems, of which we were supp o s e d to h a n d in our best 50%. Well, e v e r y o n e handed in all the problems, and Allen was swamped trying to grade them. W h e n he assigned the next

problem set, he told us that although he'd still grade every submission (he never wanted to discourage hard work), he wouldn't choose the best 50% himself. Each p e r s o n w o u l d g e t an a v e r a g e b a s e d on all t h e problems he or she submitted. He thought that would do the trick, since many of these problems were quite difficult, so there was a good chance of being penalized for handing in more than the minimum number. But it didn't work: everyone still handed in all the problems. Allen produced more Ph.D. students than anyone else in the history of the Michigan Mathematics Department. But I want you to know that beneath that gentle exterior lurked a very tough advisor! If y o u were his student, then under his guidance, you settled on your o w n topic, within which you found your o w n problem. And there was no guarantee that you could crack that problem. Allen kept to the sidelines b u t always made himself available to guide us to the literature, offer opinions, suggest options, lighten the mood with gentle wisecracks, and generally shout encouragement. Any student w h o got a degree with him really earned it, and in the process learned how to find the dreaded "post-thesis" research problem, and the ones after that, too. Having described something of what it was like to study with Allen, I'd like to talk a bit about the paper of his that I liked best. Being sort of abstractly minded in those days, I particularly liked his paper [3] with Lee Rubel about the strict topology on the space of bounded analytic functions, but my favorite was his subsequent article [1] with Peter Duren and B. W. Romberg on the dual space of lip with 0 < p < 1. Having just finished Allen's course on topological vector spaces, and Peter's course on HP theory, I was well prepared to appreciate this long work, affectionately d u b b e d by them "The Monster." I loved The Monster's fascinating blend of classical function theory, abstract functional analysis, and just plain chutzpah. Who would have thought that a space that was not even locally convex could contain such

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Allen with some of his students at a conference at the University of Arkansas in 1989. From left to right: Dragan Vukotic, Alex Stanoyevitch, Alfredo Octavio, Allen Shields, Joel Shapiro, Kit Chan, George Cochran. Photo by Ben Lotto.

interesting structures! Being lucky enough to be on the scene while Allen and Peter did this research and having just acquired the background to understand it, I felt like a junior member of an expedition to another planet. The Monster used classical techniques in the tradition of Hardy and Littlewood to characterize the dual space of tip (0 < p < 1), and, as I'll elaborate later, in the process created a whole new area of functional analysis. Allen frequently c o m m e n t e d that he felt Hardy and Littlewood could have got these results themselves, if only they'd had the functional analysis available to provide a framework for asking the questions. This was one of the principles that he tried incessantly to get into m y head (as an antidote, I think, to my early inclination to try to use the Closed Graph Theorem to solve every problem): "hard" analysis is what's interesting, and "soft" analysis serves both as a language for suggesting meaningful "hard" problems and as a device for transforming such problems (e.g., by duality) into something you can handle. But, with notable exceptions, which Allen certainly appreciated, soft analysis rarely solves the problems for you. 1 Allen was drawn to these wayward cousins of the more respectable Hardy spaces by an abstract version of Beurling's invariant subspace theorem that he and Rubel proved in [3]. This Beurling theorem of Rubel and Shields w o r k e d in general topological vector spaces of holomorphic functions on the disc, subject to some hypotheses. One of these hypotheses looked 1 Appropriately enough, Allen's former student Russell Lyons, a premier harmonic analyst, lectured on a spectacular such exception at the A n n Arbor conference in Allen's honor last August. Lyons described recent work of Kechris and Louveau, who used functional analysis to shorten considerably the proofs of some very deep results on sets of uniqueness for Fourier Series.

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s o m e t h i n g like the separation condition that the Hahn-Banach theorem guarantees for locally convex spaces. Since Beurling's theorem was known to hold for HP, even if p < 1, Allen was curious to see if this "Hahn-Banach separation property" held as well. This question required better information about the dual space of HP (0 < p < 1) than was currently available. To this point not much had been worked out in the literature beyond the fact that these spaces were not locally convex but still had enough continuous linear functionals to separate points. Initially, a reasonable function space description of their duals seemed unattainable. This was the "pre-BMO" era of Hardy space theory, so no such description was available even for the dual of H i . Moreover, the lack of local convexity suggested sinister complications. Nevertheless, Duren, Romberg, and Shields succeeded in identifying the dual space of lip (0 < p < 1) as a nice space of analytic functions, and consequently succeeded in answering Allen's original question in the negative: these non-locally convex Hardy spaces harbored proper closed subspaces not separable from their complements by continuous linear functionals. In fact, some of these subspaces were even weakly dense: any continuous linear functional unfortunate enough to vanish on one of them had to vanish on the whole space! Allen and Peter then discovered some previous work of N. T. Peck, who had found failures of the Hahn-Banach theorem in the sequence space ~P (0 < p < 1). Noting that the same p h e n o m e n o n seemed to be coming from two different sources, they boldly asked if the Hahn-Banach theorem had to fail in every non-locally convex F-space (a.k.a. complete, metrizable, topological vector space). 2 I worked on this problem for a couple of years, and finally got a Ph.D. by proving that, under an additional hypothesis, the answer was "yes." A few years later Nigel Kalton gave a brilliant argument that removed the offending hypothesis, and answered the question completely: If an F-space is not locally convex, then there is a closed subspace whose complement can't be separated from it by the dual space. Or, sacrificing precision for pizazz: The Hahn-Banach theorem characterizes local convexity for F-spaces. I think this beautiful result of Kalton's ought to be mentioned in every beginning textbook on functional analysis. Kalton went on to find out a lot more about nonlocally convex F-spaces and, along with Tenney Peck and Jim Roberts, transformed the subject from a collection of scattered results and curious counterex2 I've somewhat oversimplified the way things actually happened. The Duren-Romberg-Shields collaboration is an interesting story in itself, as you will see from Peter Duren's account in this issue.

amples into a coherent theory that could meaningfully interact with other areas like Banach space theory and harmonic analysis. You can find a reasonably up-todate description of what has been done in this field, along with the references I've omitted, in Kalton, Peck, and Roberts's charmingly titled F-Space Sampler [21. So, although The Monster was not the first paper about a non-locally convex topological vector space, it was the first detailed treatment of an interesting such space, and from it arose a whole new area of functional analysis. The Monster influenced important work in at least two other directions. First, there are the weakly dense subspaces mentioned above. The existence of such crazy objects is remarkable in itself, but even more amazing is the fact that Duren, Romberg, and Shields didn't just cook up ad hoc counterexamples, they discovered them hiding naturally among the most interesting subspaces any Hardy space can possess, the invariant subspaces of Beurling's theorem! So the non-locally-convex Hardy spaces suggested a n e w question that their more upscale Banach counterparts couldn't see: h o w do you tell the Beurling subspaces that are weakly closed from the ones that are not? About ten years later Jim Roberts and Boris Korenblum independently, and using different methods, answered this question. You can find this result, with Roberts's proof and a reference to Korenblum's, in [2]. Both proofs have implications that extend far beyond the immediate problem, and the result itself gives an important new stratification of the so-called "inner functions" that figure in the statement of Beurling's theorem. A functional analyst of my acquaintance is fond of observing that work on non-locally-convex spaces sometimes makes it possible to get at Banach spaces like L1 "from below."3 The Monster already contains a nice example of this principle: it tells us what the dual space of/-/1 has to be! The dual space of PIP, at least for 1/2 < p < 1, was identified in The Monster as the space of analytic functions on the unit disc that extend continuously to the closure and obey a Lipschitz condition with exponent 1/p - I on the boundary (and therefore on the whole closed disc). These functions act as linear functionals on HP by integration on the boundary (actually by integration against their complex conjugates, lest the Cauchy integral theorem make everything trivial). At least they act this way on polynomials, and the bounds obtained in The Monster allow the re-

sulting functionals to be extended by continuity to the whole space. If you let p tend to 1 in this description of the dual space, you get "Lip zero," which one might broadly interpret as the space of bounded analytic functions on the unit disc. Unfortunately, this is not the right answer: although every b o u n d e d analytic function induces, by boundary integration, a continuous linear functional on H ~, analyticity creates some boundary cancellation that makes it possible for some unbounded functions to work as well. What the authors of The Monster did not know was that a few years earlier, Campanato and Meyers independently had found a "mean oscillation" characterization of functions in Lip (lip - 1). If you let p tend to 1 in the Campanato-Meyers characterization, you get the space BMO, the functions of bounded mean oscillation. N o w Fefferman's theorem, which is so famous I've even seen it on a T-shirt, affirms that BMO is the correct answer: more precisely, (/-/1), = BMOA, where "A" stands for "analytic." You can find a nice treatment of these matters, with references to the primary sources, in Chapters 4 and 5 of Donald Sarason's cult classic [4]. So The Monster created a n e w field of functional analysis, and a new classification of Beurling subspaces; and it foretold the famous /-/LBMO duality theorem that was to revolutionize Hardy space theory over the next decade. Nice work, I think, for a single paper. I hope I've been able to give you some idea of what a privilege it was to work as a student with Allen. These words can't convey everything this extraordinary man has meant to me over the years, but writing them d o w n has helped me realize how fortunate I am to have known him. He's gone now, but he'll always be there somewhere in my head, tilting back in his chair, offering suggestions, making wisecracks, and shouting encouragement.

3 He punctuates this remark with a gesture that always broke Allen up, but that I can't describe in a family magazine. [Editor's comment: The Mathematical Intelligencer is not a family magazine.]

Department of Mathematics Michigan State University East Lansing, MI 48824 USA

References 1. P. L. Duren, B. W. Romberg, and A. L. Shields, Linear functionals on lip spaces with 0 < p < 1, J. Reine Angew. Math. 238 (1969), 32-60. 2. N. J. Kalton, N. T. Peck, and J. W. Roberts, An F-Space Sampler, London Math. Soc. Lecture notes #89, Cambridge University Press (1984). 3. L. A. Rubel and A. L. Shields, The space of bounded analytic functions on a region, Annales Inst. Fourier (Grenoble), 14 (1966), 235-277. 4. D. Sarason, Function Theory in the Unit Circle, Blacksburg: Virginia Polytechnic Institute and State University, 1978.

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The Exact Answer to a Question of Shields Donald Sarason

Besides being a professional mathematician, Allen Shields was a mathematics buff, a person endlessly intrigued by all things mathematical. In his "Years Ago" column, he shared his fascination with how mathematical ideas originate and develop. The first "Years Ago" column in The Mathematical Intelligencer (vol. 9, no. 1, 1987) concerned algebraic topology. In it Allen mentioned that the concept of an exact sequence was first clearly recognized by Witold Hurewicz (who had been Allen's advisor at M.I.T.). Hurewicz's only relevant publication is a brief abstract in the 1941 Bulletin of the American Mathematical Society (vol. 47, p. 562), w h i c h Allen r e p r o d u c e d in his column. The term "'exact sequence" was not used by Hurewicz; it made its debut in a 1947 paper of John L. Kelley and Everett Pitcher (Ann. of Math., vol. 47, pp. 682-709). About a month before Allen's death I was fortunate to be able to visit him in a hospital room in Ann Arbor. Although Allen's body was rapidly dying, his mind retained its full vigor. I had always found him an inspiring person, now more than ever. The conversation was mainly about mathematics and mathematicians. Allen reminisced some about his student days, mentioning that his advisor had been the topologist Hurewicz. A more natural advisor, given Allen's interests, would have been Rapha61 Salem. However, Salem spent only half of each year at M.I.T., making it difficult to work under him. Allen recalled the information on the origin of exact sequences from his column. He

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admitted that, although the term "exact sequence" was first used in print by Kelley and Pitcher, he was uncertain whether they or someone else had invented it. It occurred to me at the time that I should be able to find the answer to this question, because Kelley is my colleague at Berkeley. John Kelley retired from teaching several years ago. He still lives in Berkeley though, and from time to time he can be spotted around the mathematics department. I reached him by phone a week or so after Allen died. Kelley attributed the term "exact sequence" to Samuel Eilenberg and Norman Steenrod. Before Kelley and Pitcher published their results, they communicated them at two American Mathematical Society meetings in late 1945 and early 1946. The abstracts of their talks, which appeared in the 1946 Bulletin of the American Mathematical Society (vol. 52, pp. 74 and 443), do not use the term "exact sequence" but rather "natural homomorphism sequence," following Hurewicz. At one of these meetings they encountered Eilenberg, who at the time, in collaboration with Steenrod, was writing the now famous book Foundations of Algebraic Topology (Princeton University Press, 1952). Eilenberg informed Kelley and Pitcher that he and Steenrod had settled u p o n the term "exact sequence" to use in their book. The term seemed so marvelously fitting that Kelley and Pitcher adopted it. Kelley describes Steenrod as being especially creative in devising terminology. Steenrod is responsible for the term "net" as it is now commonly used for gen-

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Donald Sarason, Allen Shields, and Sheldon Axler (left to right). Photograph taken by Smilka Zdravkovska in 1984 in the Lake District of England during a break in a conference on Operators and Function

Theory. eralized sequence (in the sense of Moore-Smith). The term first appeared in this role in a 1950 paper of Kelley (Duke Math. Journal, vol. 17, pp. 277-283). Kelley had been planning to use the term "way"; that would have resulted in what we now call a "subnet" being referred to as a "subway." Steenrod, when informed by Kelley of his plan, apparently regarded Kelley's choice of terminology as frivolous, and after being prodded by Kelley, he suggested the term "net" as an alternative. His judgment prevailed. A call to Pitcher confirmed Kelley's recollections. I wrote to Eilenberg to request additional information. A short time thereafter I made a brief visit to the University of Georgia and met there the topologist David A. Edwards. At the urging of my host, Douglas Clark, I asked Edwards what he knew about the origin of the term "exact sequence." Edwards was sure the term was invented by Eilenberg and/or Steenrod. He recalled reading or hearing that, as Eilenberg and Steenrod were writing their book but before they devised a satisfactory term, they left a blank everywhere the term "exact" would later appear. During the week of my return to Berkeley, Saunders Mac Lane turned up and gave a delightful colloquium talk on " M a t h e m a t i c s for sixty years: What has c h a n g e d ? " I c o r n e r e d h i m b e f o r e the talk a n d p u m p e d him for information. He told me the same story as Edwards and said he heard it directly from Eilenberg. Two days later Eilenberg phoned in response to my

letter. Indeed, he related, during about the first year he and Steenrod worked on their book, they wrote "blank sequence" everywhere for Hurewicz's concept, with the intention of replacing the word "blank" by the "right word" once they found it. They refrained from using a provisional term in fear that would distract them from their search for the "right" term. Once they hit u p o n the term "exact" they shared it with anyone interested. Eilenberg used it in a course at the University of Michigan in the spring of 1946. I did not press Eilenberg on whether it was he or Steenrod w h o originally dreamed up the term. At the time it seemed a rude thing to ask, and the question seemed unimportant. In some visible ways Allen was not a tidy p e r s o n - he was notorious, for example, for missing deadlines. His papers and his lectures, on the other hand, were always organized with extreme care. During his lectures he could transmit more information on the blackboard, in less space and with greater coherence, than anyone I know. So in some ways he was a tidy person, and I think I have not violated his spirit by tidying up his first "Years Ago" column with this small footnote. He would have enjoyed the story.

Department of Mathematics University of California Berkeley, CA 94720 USA

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Allen L. Shields Paul R. Halmos

The best seminar I ever belonged to consisted of Allen Shields and me. We met one afternoon a week, for a b o u t t w o h o u r s . We d i d not p r e p a r e for o u r meetings, and we certainly did not lecture at each other. We were interested in similar things, we got along well, a n d each of us liked to explain his thoughts and found the other a sympathetic and intelligent listener. We would exchange the elementary puzzles we heard during the week, the crazy questions we were asked in class, the half-baked problems that p o p p e d into our heads, the v a g u e ideas for solving last week's problems that occurred to us, the illuminating comments we heard at other seminars-we would shout excitedly, or stare together at the blackboard in bewildered silence--and, whatever we did, we both learned a lot from each other during the year the seminar lasted, and we both enjoyed it. From I Want to Be a Mathematician Department of Mathematics Santa Clara University Santa Clara, CA 95053 USA

Allen Shields at work in Paul Halmos's office, 1964. Photo by Paul Halmos.

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F. W. Gehring

The days of universalists are past, and it is quite unusual to find a mathematician w h o knows more than one field. Allen Shields was one of this rare breed. After finishing a thesis with the topologist W. Hurewicz, he made substantial contributions to semigroups, measure theory, complex functions, functional analysis, and operator theory. Allen's standards were high, his taste impeccable, and his ideas deep. He was one of the world's most versatile practitioners of the art of applying functional analysis to gain insight on and solve problems of classical function theory. He had a profound effect on the development of this subject through his research and his personal contacts with colleagues and students. Allen was a gregarious mathematician with complete dedication to his work, a vast store of knowledge, and an amazing eagerness to discuss mathematics in any circumstances. He had a contagious enthusiasm and the knack for coming up with interesting and important questions. A conversation with him always left one's head singing with new ideas. Allen's lectures were models of clarity and always highly informative. He was an engaging speaker and a constant source of inspiration for students and faculty alike. Many mathematicians have said that there is no person they would rather have as a colleague than Allen. He showed us by example what a mathematician ought to be. Allen is gone and colleagues, students, friends all mourn this great loss. But Allen is not and will not be forgotten. His personal charm and extraordinary generosity have changed the lives of all who knew him. His broad and penetrating contributions to mathematics have altered the face of the science he so loved. Allen's last legacy is a poignant lesson for all. For though his final days were very difficult, he showed us h o w to meet death with dignity and grace. Department of Mathematics University of Michigan Ann Arbor, MI 48109 USA

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Recollections of Allen L. Shields Ethel Rathbun

In the spring of 1967 I returned to the United States, fresh from a seven-year sojourn in Europe, seeking employment. I obtained a position in the Department of Mathematics at the University of Michigan. None of my previous positions had quite prepared me for the scene that was Ann Arbor in the late 1960s; this was my first acquaintance with the academic world as an employee. During my first two and one-half years I was the secretary in the Graduate Student office. The departmental copy machine was located in my office, and my office was also the pathway to the faculty mailboxes. So I saw most of the faculty quite often, but didn't really have the opportunity to get to know them. It w a s not until 1975 w h e n Allen Shields b e c a m e Chairman that I really got to know him as a boss and a friend. Very early on I remember warning him that I thought he should add a particular word to his vocabulary; he looked rather puzzled, and then I suggested the simple two-letter word "NO." It was very difficult, if not impossible, for Allen to decline any request from a family member, colleague, student, stranger, or friend, whether it be for money, a letter of reference, to referee a paper, etc. His intentions were of course always the best, but it was indeed impossible for him to actually carry out all his promises. To say that Allen wasn't overly fond of the administrative duties of the chairmanship is not to do him an injustice, for his sense of responsibility was very great: it simply didn't always extend to the bureaucratic paperwork. When it came to important department business, he would give generously of his time. It was this quality of being always ready to help that made m y job interesting on occasion. It was often a challenge for me to get him to read his daily mail and act on it. Occasionally he would ignore the "In" basket, preferring to talk with colleagues about their problems, to discuss mathematics, etc. There were times w h e n I would have important messages or important documents that needed immediate handling. If they were left in the "In" basket, he just wouldn't see them. To

draw his attention to something I found it necessary to place it on his desk chair. However, this wasn't always a solution either, as he just sat down on it. Some years later I had a discussion with an earlier colleague about Allen and the fact that Allen would spend so much time in our Commons Room. The colleague was shocked to hear this, saying that couldn't be true, as he remembered Allen was always too busy p r o v i n g t h e o r e m s to w a s t e time chatting in the Commons Room; he then said, you realize it was only Allen procrastinating on his administrative duties that kept him in the Commons Room where we have no telephone. It was here where he usually pored over the New York Times and talked with students and colleagues. Allen's predecessor as Chairman, Fred Gehring, kept a "Log" of all personal conversations made in his official capacity as Chairman. We encouraged Allen to follow suit. Those who know Fred will realize the tremendous contrast in their modus operandi! Allen kept the Chairman's Log with a faithfulness that sharply contrasted with his usual aversion to paperwork. He considered it very important that any commitments made by the Chairman be recorded: more evidence of his real concern for his fellows. Allen was irrepressibly curious. On one occasion we were at the home of a colleague where a visitor to the d e p a r t m e n t was being entertained. As we w e r e standing next to the door saying our thanks to the hostess, Allen noticed what appeared to be a doorbell inside the house, but at an unusual height--unusual for most people, but not Allen w h o was 6'4"---and Allen casually pushed the bell, saying "What is this for?" I had never seen the host move so quickly--the bell was part of the security system to notify the police of problems. We left the house immediately. Allen loved fresh vegetables. There is a Farmers' Market that Allen often visited on Saturdays. Early one fall w h e n I came to the office on Monday morning my colleague was quite disturbed because of an odor in the office; she thought there must be something

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mill [l'

I'

Allen Shields at his fiftieth birthday party, 1977.

dead (a mouse, or worse). Upon careful examination we discovered Allen had purchased a beautiful head of cauliflower on Saturday, placed it in a plastic bag, and put it in his desk drawer. We then asked that he please not use his desk for a vegetable bin. On another interesting evening, Allen was hosting a party for the entire department on the occasion of a special lecture series. Allen was living alone at the time in a house in the country and I offered to assist him. The party was progressing quite well and all were enjoying themselves. Allen rarely had more than one or two drinks in an evening, but at one point I heard Allen say, "I just realized I don't have to drive home tonight," and therefore he enjoyed his own party. Actually there was one part of the Chairmanship that Allen thoroughly enjoyed: the Chairman's Office with its large conference table to accommodate the Executive Committee. With his fondness for a catnap, the table was an ideal site.

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Ann Arbor has a large Street Art Fair in mid-July each summer. Many of the streets in the immediate vicinity of the campus are closed to traffic and filled with booths where artists set up shop. Locals complain about the traffic. The fair runs from Wednesday through Saturday. Because Allen's condition had already deteriorated significantly by mid-July 1989, I didn't expect him to come to campus during the fair. It was with astonishment that I saw him in Angell Hall on Friday, 21 July; he had ridden his bicycle to campus to avoid the parking hassle because he was to meet with his graduate students! That was the last time he was to come to his office. He listened, offered advice when asked, and was often just there. He was a good friend, one I will now miss. It was my pleasure to have known him and to have worked for him.

Department of Mathematics University of Michigan Ann Arbor, M148109 USA

The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to the editor-in-chief, Sheldon Axler.

Lie and K i l l i n g I read with great interest A. J. Coleman's paper on Wilhelm Killing's life and a c c o m p l i s h m e n t s [The greatest mathematical paper of all time, Mathematical Intelligencer, vol. 11 (1989), no. 3, 29-38]. I agree that even in mathematics, the most objective product of the human brain, not all outstanding contributions are adequately k n o w n and appreciated, and that even some of the greatest achievements are not properly understood, or even pass into oblivion. Although mathematical values can hardly be ordered, Coleman's paper will certainly be effective in reviving interest in, and appreciation of, the work of an outstanding mathematician. I disagree, however, with Coleman's conclusion on page 37 that Sophus Lie "was quite negative about Killing's work" and that this was due to "sour grapes" on Lie's part. The section about Killing in Lie and Engel's book about transformation groups (pages 768-778 in Coleman's reference [20]) concludes as follows:

As short as our report about these works by Killing is, nonetheless the reader will be able to recognize that the results of these works are extremely important. Even if Killing's presentation and proofs suffer from some deftciencies, no one can deny that Killing created new and fruitful methods. The 'private vote' of Lie in the discussion about his possible successor when he was leaving the University of Leipzig for the University of Christiania (Norway) in 1898 should also be noted (Teubner-Archiv zur Mathematik, Vol. 8, 1989, Leipzig, pages 230-231). Lie wrote: In my opinion the colleagues Scheibner and Neumann err when they designate Professor Engel as the absolutely best German representative of my theories. The professors Study in Greifswald and Killing in Mfinster have developed considerably greater original strength in their works in this area, although Engel may stand higher as a 24

teacher; also other mathematicians, especially the professors Scheffers, Schur, and Maurer could be taken into consideration next to Engel . . . . In my opinion, Professor Engel is a talented mathematician and good lecturer, who possesses solid and expansive knowledge. But in his publications up until now he has not shown originality in the higher sense of the word: I repeat my sincere appreciation for Coleman's paper, but I believe that the quotations above express Lie's evaluation of Killing's work. B~la SzSkefalvi-Nagy Bolyai Institute University of Szeged H-6720 Szeged, Hungary

9Riesz's Problem. I w o u l d like to add a historical note to the topic of fractal geometry recently discussed in the Mathematical Intelligencer. In 1952 F. Riesz proposed the following problem for solution in the Hungarian mathematical journal Matematikai Lapok [Problem 54, Matematikai Lapok 3 (1952), 286]: Given a complex number a, for what complex values of zl does the iteration

zn+

=

1 2 (zn + a)

yield a convergent sequence? If the proposer did not know the solution of a published problem, the editor noted it in a footnote. Problem 54 has no such footnote. However, in 1956, shortly after the death of F. Riesz, the Matematikai Lapok p u b l i s h e d , a m o n g others, a Russian and a French translation of F. Riesz's problem [Matematikai Lapok 7 (1956), p. 125 (Russian), p. 147 (French)]. The French translation contains the footnote, "The author submitted the problem without knowing its solution.

THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 2 9 1990 Springer-Veflag New York

Until now no solution has been received by the editor." If we use the notations z n = 2x, and a = 4c, we arrive at the equation x,+l = x2 + c which recently has been extensively studied. See, for example, H.-O. Peitgen and P. H. Richter: The Beauty of Fractals, Springer-Verlag, Berlin, 1986, and B. Branner: "The Mandelbrot set," in Chaos and Fractals, R. L. Devaney and L. Keen, editors, Proceedings of Symposia in Applied Mathematics, Vol. 39, American Mathematical Society, 1989, 75-105. If Professor F. Riesz were alive today, he would be very pleased to see these new results.

Lajos Talcs Department of Mathematics and Statistics Case Western Reserve University Cleveland, OH 44106 USA

What's Hot and What's Not

course, that calculus allows us to express the length of a circle by a definite integral; but that is hardly essential. To the question "What is the value of that integral?" we ordinarily answer just by giving a name, ~, to that value. Again, of course, the integral indicates a method for obtaining arbitrarily close numerical approximations to the value, and calculus helps us to find other expressions--e.g., infinite series--that can also be said to "answer the question." But before the development of calculus, Vi~te and Wallis had found two remarkable expressions for ~r in the form of infinite products, and such expressions stand both as answers to the question "What is the length of a circle?" and as means for calculating arbitrarily close rational approximations to that length. As far as Greek geometry is concerned, it ought to be noted that Archimedes, in his treatise On Spirals, proves a theorem equating the length of an arbitrary circular arc to the length of a certain straight line-segm e n t - - a segment that is determined by the tangent to the Archimedean spiral at a point suitably related to the arc. This really remarkable result, involving both the general tangent to a transcendental curve and the

Not long ago we were all abuzz about the potential of sets with fuzz, and Thom and Zeeman made us see how almost everything's a catastrophe. They showed us w h y the Market's lows and highs are determined by the shapes of butterflies, and h o w to tell if a barking dog will bite by placing it on a cusp, just right. Today this stuff's no longer hot. Thom's been replaced by Mandelbrot. So now we take z2 + c and iterate it endlessly, to learn about attractors strange and how nature makes a mountain range.

David J. Sprows Department of Mathematical Sciences Villanova University Villanova, PA 19085 USA

Greek Geometry and .~ In John Stillwell's splendid review of a splendid book [Mathematical Intelligencer, vol. 11, no. 4 (Fall 1989), pp. 63 ft.--review of Geometries and Groups by V. V. Nikulin and I. R. Shafarevich] there is one nit that deserves to be picked. After confessing that he finds Euclidean geometry boring and referring to "the bland straight lines and planes of Euclid," Stillwell remarks in parentheses: "Admittedly Euclid studies the circle, but being without calculus he could not answer one of the simplest questions about i t - - w h a t is its length?" Now, this is a little hard to interpret. It is true, of THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 2, 1990 2 5

rectification of a curvilinear length, was achieved with the means of Euclid's geometry, just slightly extended (by a principle--an inequality concerning lengths of convex arcs--that can be regarded as in effect an axiomatic characterization of such lengths). As to the numerical determination of the length of a circle, in Measure of a Circle Archimedes (a) demonstrates the relationship between the area of a circle and its circumference; (b) shows how one can arrive at arbitrarily close approximations to the area of a circle by using inscribed and circumscribed polygons; and (c) obtains, by a particular application of the method he has described, the numerical estimate that would be expressed in our notation as 3 1/7 < ~ < 3 10/71. Clearly, then, the geometry of the G r e e k s - - t h e geometry of Euclid (at least as extended by Archimedes) - - h a d essentially complete mastery of the question Stillwell mentions (and, in fact, of considerably deeper questions), both from a theoretical and from a numerical point of view. Howard Stein Department of Philosophy University of Chicago Chicago, IL 60637 USA

9Z e r o - O n e Matrices.

I read K. Sutner's article on cellular automata (Mathematical Intelligencer, vol. 11, no. 2, pp. 49-53) with great interest. I find that the problems and techniques described in this article have much in common with the coding-theoretic approach to design theory. A l t h o u g h Sutner's Theorem 3.2 was stated and proved in terms of graphs, it could be rephrased as follows (F2 denotes the field {0,1}): T h e o r e m A: If A is a symmetric zero-one matrix with only ones on the main diagonal, then the all-one vector is in the row-span of A over F2. O n the other hand, T h e o r e m 3 in [B. Bagchi & N. S. N. Sastry, Even order inversive planes, generalized quadrangles and codes, Geometriae Dedicata 22 (1987), pp. 137-147] may be rephrased as: T h e o r e m B: If B is a symmetric zero-one matrix, then the main diagonal of B (regarded as a vector over F2) is in the row-span of B over F2. Thus stated, it is clear that Sutner's Theorem A is a special case of our Theorem B. Bhaskar Bagchi Indian Statistical Institute Calcutta 700 035 India

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THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 2, 1990

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Karen V. H. Parshall*

Who Was Otto Nikodym? Waclaw Szymanski

I dedicate this article to the memory of Allen Shields---a fine colleague and a highly respected mathematician. Doylestown is a small American city of about 9,000 inhabitants, 35 miles north of Philadelphia. About a mile west of Doylestown on PA Route 313 one sees a road sign "Cz~stochowa" pointing to the south. Following this sign for another mile or so on Ferry Road, one arrives at a spectacular site. On the right, on the top of a hill, there is a big, white church, with a 200-foot-high tower. This is Cz~stochowa, but to be precise, there is no city or village by this name here. There is, however, a city called Cz~stochowa in s o u t h e r n Poland, about 80 miles n o r t h w e s t of the royal city of Krak6w. Cz~stochowa in Poland is an important historic as well as religious place. In 1382 Wladystaw, a Polish prince of Opole, donated a Byzantine-style icon of the Virgin Mary and funded the Pauline monastery there. The cult of the Virgin Mary of Cz~stochowa started and developed rapidly after a big Polish-Swedish war, called the Deluge in Polish history, in the middle of the seventeenth century. It is believed the monastery was saved miraculously by the Holy Virgin. The day after Christmas of 1655 the Swedes abandoned not only the siege of the monastery, but also the whole idea of conquering Poland, and their troops withdrew from Poland shortly afterwards (see [1], p. 450-453). The icon, whose origin, creator, and exact age are unknown, is the object of numerous pilgrimages of devoted Polish Catholics.

Undoubtedly, Cz~stochowa is now the holiest place in Poland. The idea of an American Cz~stochowa was created by a Pauline Father, Micha! Zembrzuski, in 1951, w h e n he came from Rome to the United States to do missionary work among the Poles in America. Everything started from a small, wooden chapel, called The Barn, which still exists. Its interesting external architecture follows the pattern of n u m e r o u s w o o d e n churches in the mountains of southern Poland. The large, n e w church mentioned earlier, designed by the Polish-American architect George Szeptycki, is an impressive, modern structure built in 1965-1966 to com-

* Column Editor's address: Departments of Mathematics and History, U n i v e r s i t y of V i r g i n i a , C h a r l o t t e s v i l l e , V A 22903 USA. THE MATHEMATICALINTELLIGENCERVOL. 12, NO. 2 9 1990 Springer-Verlag New York

27

During the preparation of this article I learned from Father Zembrzuski and Wanda (Mrs. Adam) Styka, both of w h o m n o w live in N e w York City, that Dr. Stanistawa Nikodym died shortly before Easter of 1988 in Poland. To explain these personal connections let me m e n t i o n here only that the N i k o d y m s became friends with the Stykas shortly after the former came to the United States from Europe in 1948. They remained friends ever since. Father Zembrzuski joined the circle of the Nikodyms friends later on. Eventually, he led Professor Nikodym's funeral ceremony. The Radon-Nikodym Theorem

Czgstochowa, Pennsylvaniamthe main church. Picture reprinted with the permission of Schellmark, Inc. memorate the Polish Millenium. (In 966 the Polish prince, Mieszko, w a s baptized, and with him the whole nation that he ruled. This is considered historically as the moment of creation of the Polish State and of the incorporation of Poland into Western civilization; see [1], Ch. 1.) A faithful copy of the icon of the Holy Virgin from Cz~stochowa in Poland, called the Black Madonna, because the faces of Mary and Jesus are v e r y dark, occupies the central place on the w e s t e r n wall of the church. There are remarkable stained glass w i n d o w s in the church, designed by George Bialecki. They picture important moments of Polish and American history on the southern and northern walls, respectively. Pauline Fathers are curators of the shrine. Between the modern church and The Barn there is a large burial ground, which Father Zembrzuski calls the Polish Arlington. In the northern part there are simple, almost identical graves of Polish soldiers who fought in both World Wars. Distinguished Poles who contributed to Polish and American culture are buried next to the soldiers. The first notable Pole buried there was the famous painter Adam Styka, w h o was also a volunteer soldier in General J6zef Haller's "Blue A r m y " formed in France to s u p p o r t the Allies in Europe during World War I (see [2], p. 242). Near his grave there is a gravestone which does not even look like one, because it is decorated with a colorful mosaic full of flowers. At the bottom one reads: Prof. Dr. Otton Martin Nikodym, Mathematician--Matematyk, um. (i.e., died) 3. VI. 1974. Next to that inscription is another: Dr. S t a n i s h w a Nikodym, Mathematician, Painter--Matematyczka, Malarka. The dates of her birth and death are not given. From the inscription on the side of the stone one learns that the gravestone was designed by Dr. Stanistawa (Mrs. Otto) Nikodym. She had her burial place ready next to her husband. 28

THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 2, 1990

Every graduate student of mathematics knows the Radon-Nikodym theorem. This fundamental result in measure theory, in its contemporary version, reads as follows (see for example, [4], p. 128): Radon-Nikodym Theorem. Suppose S is a o--algebra of subsets of a set X. Let p, be a positive, c-finite measure on S. Let -r be a or-finite signed measure on S. If "r is absolutely continuous with respect to p, (i.e., p~(E) = 0 implies ,r(E) = 0), then there is an S-measurable function f on X such that

(*)

9(E) = J'E f d~ for each E in S.

Moreover, such f is unique p,-almost everywhere. The function f of this theorem is called the RadonNikodym derivative of ~- with respect to p,. The notations

f = d,r/dp~ or d'r = f dp~ or "r = fp~ are commonly used to express the relationship (*). The Radon-Nikodym Theorem is of essential importance in the theory of measure and in all areas of mathematics that " d e p e n d " u p o n measure properties, such as the theory of Le and HP spaces, or contemporary probability (which may be seen as a part of measure theory or vice versa). To give a natural probabilistic example, the concept of conditional expectation of a random variable, fundamental for the theory of stochastic processes, is possible only because of the Radon-Nikodym Theorem (see, for example, [5], p. 295). Any textbook that treats measure theory will surely q u o t e a n d give a p r o o f of the R a d o n - N i k o d y m Theorem. It was, indeed, Otto Nikodym who proved that theorem almost 60 years ago, in the paper Sur une g~n~ralisation des intdgrales de M. J. Radon published in Fundamenta Mathematicae, vol 15, p. 131-179, in 1930. According to [3], Stanislaw Saks suggested the name for this theorem, meaning perhaps, that it is Nikodym's theorem on Radon integrals. In French literature this theorem is sometimes called Lebesgue-Nikodym, in Italian, Vitali-Nikodym.

Otto Nikodym The year 1987 marked the one hundredth anniversary of Otto Nikodym's birth. He was born on 13 August 1887 in the small town of Zabtot6w, near Kotomyja, in eastern Poland. A historical comment is in order here. At the time of Nikodym's birth, the State of Poland did not exist formally, as a result of three consecutive partitions in 1772, 1793, and 1795 by Russia, Austria, and Prussia. The part of Poland where Nikodym was born was taken by the Czar's Russian Empire. When the Polish State was recreated after World War I, Zabtot6w, Kotomyja, and Lw6w, to which the Nikodyms moved about 1897, returned to Poland. After World War II these places were taken from Poland again, this time by the Soviet Union, as a result of the Roosevelt-Churchill-Stalin agreement in Yalta. Otto Nikodym received almost his entire education in Lw6w. At the Jan Kazimierz University he attended lectures in mathematics by Waclaw Sierpir~ski and Konstanty Puzyna, and in physics by Marian Smoluchowski. In 1911 he graduated in mathematics and also received the license to teach mathematics and physics in high schools. One rather widely popularized event related to the Polish School of Mathematics, described by one of the three persons involved, is Hugo Steinhaus's story of how he "discovered" Stefan Banach--one of the fathers of functional analysis--one summer evening in 1916. Steinhaus, walking in one of Krak6w's parks, called Planty, which surrounds the Old Town, overheard the words "Lebesgue integral," which struck him as certainly not the most common phrase that one hears in such places. Surprised and interested, he approached a bench where two young men discussed a problem in measure theory. The well-remembered part of the story goes on to say that Stefan Banach was one of the young men. The not so well-remembered one is that it was Otto Nikodym who was talking with Banach. N i k o d y m w e n t to Krak6w immediately after he graduated from the University in Lw6w, to take the teaching position offered to him by one of the high schools in Krak6w. The impressive list of his publications (see [3]) contains not only textbooks, but also several papers about teaching mathematics, for he was always a concerned teacher. One can u n d e r s t a n d the atmosphere of Krak6w only after living there for some time. Krak6w has been a major world center of letters and science for eight centuries now. Undoubtedly, the most important date is 1364, when the Polish king, Kazimierz Wielki (Casimir the Great), established Akademia Krakowska (the Academy of Krak6w), later called Jagiellonian University. Ever since then the atmosphere of the city has been overwhelmingly influenced by the life of the University. In 1918 independence came back to Po-

land. The combination of the ever-present historic atmosphere and the great patriotic enthusiasm caused by r e g a i n e d i n d e p e n d e n c e m u s t have c r e a t e d a unique, stimulating environment. Intellectual life was invigorated. Rapid growth of the phenomenon that is now called "The Polish School of Mathematics" was one of the clear signs of that new vigor and a highlight of that period. On 2 April 1919, the Polish Mathematical Society was f o u n d e d by 16 m a t h e m a t i c i a n s - among them Otto Nikodym. Five years later, under strong pressure from Sierpi~ski, Nikodym agreed to take his doctoral examination. It seems he did not care much for the title--his response to Sierpi~ski's persuasion was: "Am I going to be any wiser because of that?" (It is commonly known that Banach never had any regular education.) According to Wanda Styka, Nikodym had no advisor. His doctoral dissertation was entirely his work, with no guidance; Sierpir~ski's name appeared as a necessary formality. At the time his doctorate was bestowed, Nikodym was 37. He moved from teaching high school to a position at the Jagiellonian University only after he received the doctorate. His first publication Sur une propri~t~ de l"op&ation (A) appeared in Fundamenta Mathematicae only in 1925. He published nothing before the doctorate.

From the left: Wanda (Mrs. Adam) Styka, Dr. Stanislawa Nikodym, Professor Otto Nikodym. This picture was probably taken in Utica, NY, around 1970.

In the academic year 1926-1927 Nikodym visited the Sorbonne. There his path crossed with Jerzy Neyman's, who later became one of the founding fathers of mathematical statistics and who created in Berkeley the first statistical laboratory in the United States. Immediately after returning to Poland, Nikodym obtained the degree of doctor habilitus--an equivalent of the British D.Sc. In 1931 Nikodym presented a paper to the Polish Mathematical Society entitled Sur le principe de minimum dans le problhne de Dirichlet, in which he proved that each convex, norm-closed subset of a Hilbert space has a unique element of the least norm. Again, every graduate student of mathematics knows this theorem, fundamental for the theory of Hilbert spaces and with enormous consequences for other areas (including least-square approximation and prediction theory). This theorem, which can be found in any book that treats the basics of Hilbert spaces, is often attributed to Frigyes Riesz, who published it in 1934 in the paper Zur Theorie des Hilbertschen Raumes in Acta Sci. Math. (Szeged). A short abstract of Nikodym's paper appeared in Ann. Soc. Polon. Math. in 1931. Nikodym's full proof appeared in print only in 1935 in a Romanian journal Mathematica Cluj, which contained proceedings of the Second Congress of Romanian Mathematicians held in Turnu Severin in 1932, to which Nikodym presented his paper. From 1930 till 1945 the Nikodyms lived in Warszawa (Warsaw). Until the beginning of World War II he lectured at the University, and his wife Stanislawa was an assistant of Professor Franciszek Leja, famous for 30

THE MATHEMATICAL lNTELLIGENCER VOL. 12, NO. 2, 1990

his research in the theory of functions of complex variables. During this period Nikodym published 32 research papers and four textbooks, one of which was written jointly with his wife. In Poland during the war the Nazis suppressed all activities that they considered unnecessary. These included any education above the elementary, or, at most, vocational level. Most of the teachers and university professors who were not imprisoned or sent to concentration camps conducted secret classes in private apartments, perfectly aware of the punishments, including death, that they and their pupils were facing. The Nikodyms, who managed to survive in Warszawa, took part in these activities. Otto Nikodym even h a d a s t u d e n t w h o m he prepared for the master's examination in mathematics. This student was killed in the last days of the Warszawa Uprising in 1944. In the academic year 1945-1946 the Nikodyms moved back to Krak6w. This was their last year in Poland. In 1946 they left for Belgium, and shortly afterwards they came to France. It is now hard to get a definitive answer as to why they decided to leave Poland. Before the war several Polish mathematicians went abroad, in particular to the United States, and r e m a i n e d there (for example, Marek Kac, Jerzy Neyman, and Stanistaw Ulam). Their justification for leaving is generally well known. The Nikodyms left just after the war. Seemingly, the country was free again. Or was it? What a contrast with the year 1918. There are hints that they realized quickly what was really happening, and came to the conclusion that there was no place for them in the country under Soviet occupation. In France, Otto Nikodym, invited by the Institute of H. Poincar6, began his work on mathematical foundations of quantum mechanics. In 1948 he was offered, and accepted, a position in the United States at Kenyon College in Gambler, Ohio. He never changed his place of employment again. After he retired, the Nikodyms moved to Utica, New York, where he continued his research, sponsored in part by the Atomic Energy Commission. After 1947 he wrote about 50 research papers, most of them on the subject he started to research in France. Nikodym published mostly in French and Italian mathematical journals; a substantial number of his articles appeared in Comptes Rendus (Paris). Very few of his papers appeared in American journals. At a certain point he decided to collect a portion of his research and to write a book. On the recommendation of Professors Helmut Hasse and B. L. van der Waerden, Springer-Verlag published in 1966 this impressive monograph The Mathematical Apparatus for Quantum Theories in the series Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, volume 129. Clearly, it is impossible to summarize the contents of this work here. Generally speaking, the

book contains a detailed spectral analysis of unbounded normal operators in Hilbert spaces, with a special emphasis on Boolean lattices as the main tool. At the end of his active life Nikodym almost completed a second book, which contained results of his later work. He never finished or published it. From Kenyon College Nikodym went several times to Europe on the invitation of the most prestigious scientific institutions. In 1953, for instance, he lectured to the Belgian Mathematical Society, of which he was an h o n o r a r y m e m b e r ; at the Belgian M a t h e m a t i c a l Center; to the French Academy of Sciences; at the Institute of H. Poincar6; at the Sorbonne; and at the University of Heidelberg. In 1965 he was invited for a semester by the University of Naples, Italy, to lecture on measure theory. Eugene Nassar, a close friend of the Nikodyms and a Professor of English at Syracuse University, quotes Otto N i k o d y m as saying: "Work is a reward and pleasure for those w h o understand it." Indeed, Professor Nikodym w o r k e d until he suffered a serious stroke in 1972, from which he never recovered. He died two years later. Every Pole w h o lives in or visits the United States w a n t s to see the American Cz~stochowa. The Nik o d y m s often visited Cz~stochowa, w h e r e he now l i e s - - a brilliant Polish mathematician w h o spent 26 years of his life in the United States.

Acknowledgments I thank Father Michat Zembrzuski, Wanda Styka, and Professor Eugene Nassar for conversations that helped me to understand the life of Otto Nikodym. The phot o g r a p h of W a n d a Styka and the N i k o d y m s was kindly supplied by Wanda Styka from her private collection. I thank Professor Nassar for sending me some articles and newspaper clippings. Also, I thank Dr. Jan Janas from the Institute of Mathematics of the Polish Academy of Sciences for making a photocopy of [3] and sending it to me.

References 1. Norman Davies, God's Playground, vol I, New York: Columbia University Press (1982). 2. Norman Davies, Heart of Europe, New York: Oxford University Press (1986). 3. Alicja Derkowska--Otton Marcin Nikodym--Roczniki Polskiego Towarzystwa Matematycznego, Seria II, Wiadomodci Matematyczne 25 (1983), 74-88. 4. Paul R. Halmos, Measure Theory, New York: van NostTand (1950). 5. M. M. Rao, Measure Theory and Integration, New York: Wiley-Interscience (1987). Department of Mathematical Sciences West Chester University West Chester, PA 19383 USA THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 2, 1990

31

The Opinion column offers mathematicians the opportunity to write about any issue of interest to the international mathematical community. Disagreements and controversy are welcome. An Opinion should be submitted to the editor-in-chief, Sheldon Axler.

Tentative Affiliations: An Apology George Piranian

To Allen Shields: Nobody could have owned you; but because of who you chose to be and what you taught us about the basic principles of human existence, we'll treasure to the end of our time the memories of your warm companionship.

Apology: a sincere or hypocritical statement of contrition with an implicit admission of guilt. That is not what I offer here, for none of my errors or transgressions would astonish, delight, or even entertain the readers of the Mathematical Intelligencer. Besides, why should I request forgiveness? Ultimate absolution will envelop us when we enter a Black Hole, but not before. In an older sense, "apology" means defense of an action, of a w a y of life, or of an attitude. With moderate arrogance, I shall defend a position into which I have drifted in the course of my life. Had I taken the position deliberately, I w o u l d not hesitate to brag about my sound judgment; however, like a pedestrian whose path has brought him into pleasant places, I can only praise my fortune. I do not deserve credit for being born into an ethnically mixed household, for the frequent minor geographical dislocations that engendered my early taste for impermanence, or even for my westward crossing of the Atlantic in 1929; but I can assert that fifteen is an appropriate age for learning about the joys and difficulties of decisive emigration and immigration. In Salt Lake City, I was lucky enough to become a flunky in the sheet-metal shop of the Sugarhouse Lumber and Hardware Company. Also, I discovered the public library, from which I carried armloads of books ( I recall especially Romain Rolland's Jean-Chris32

tophe). The income from the sheet-metal shop was useful in the household, and surely the books did no harm. Sometime during the second year of my existence as a successful young immigrant, somebody tattled on me, and Mr. George Eaton, Assistant Superintendent of Schools, summoned me to his office in City Hall. To my astonishment, he was so reasonable that we were able to compromise; in return for protection from further molestation, I u n d e r t o o k to attend part-time school from one o'clock until five, every Monday afternoon.

THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 2 9 1990 Springer-Verlag New York

At the age of seventeen, recognizing the advisability of stronger educational measures, I voluntarily visited Mr. Eaton, and he enrolled me as a full-time highschool student. A few w e e k s after the opening of school, my mother informed me that in October we would return to Europe. Without wasting time on rational meditation, I declared "Ich n6d" (not I), and to this day I take joy in the romantic impulse with which I declared my geographical and cultural assimilation. The following year, thanks to Mr. Eaton's administrative and diplomatic skills, I entered the School of Forestry at Utah State University. Before the end of my freshman year, psychological passions caused me to transfer to the Department of Botany. Shortly before turning twenty-three, I tasted blood in mathematics. The conversion to mathematics was the last major shift in my professional loyalty. Through much of my adult life, I have diligently constructed Riemann surfaces and infinite series and products. Nevertheless, I still subscribe to the principle that we should honor our intellectual affiliations only as tentative commitments. By analytic continuation, the principle also applies to cultural allegiances. Infatuation with Wagner or Nietzsche does no harm, provided it is of short duration. Our deepest sympathy should go not to those w h o err, but to those who are stuck. Minor reservations aside, I have long regarded the United States as my home; when I consider the social, educational, and material benefits I owe to my impatriation, I feel gratitude and affection. As to the reservations, they reflect my chronological maladjustments as much as my spatial dislocations. I regard them as corollaries to the theorem that we do not owe absolute

The doctrine that we should recruit students

into the mathematical profession has given me indigestion. allegiance to any of the cultural groups w h o claim us as members. Proper men and women are not subject to ownership. The principle of tentative loyalties has educational implications for practitioners of nearly every discipline. Throughout my career, the doctrine that we should recruit students into the mathematical profession has given me indigestion. Is a mathematician more valuable than a physicist or biologist, and is a scientist better than a poet or a creative and honest expert on law? Has our national welfare suffered more from inadequate mathematical understanding than from insufficient study of history? Many of our formal and informal institutions devote a substantial portion of their resources to self-perpetu-

afion. Some maintain a class of priests whose primary duty is to praise the institutional practices and doctrines and to guard them against modification and dissent. I suspect that in allocating merit raises, some of our academic executive committees even count the converts we have recently led into the faith. Nevertheless, in the counselling office and in the classroom, our primary obligation is to our students rather than to our profession or university. When a student visits me because of difficulties in the choice of a career, I should let our conversation wander lovingly among the available options. If all goes well, I can give assistance; but under no circumstances must I let my professional sentiments, institutional loyalties, or personal tastes sway the student's decision. Thou shalt not pretend to speak with a divine voice; neither shalt thou wear the cloak of a fallen angel. Sometimes I ask students whether they are pleased with the w a y their university treats them. The answer is usually affirmative, perhaps because disproportionately many of my contacts are with participants in our Honors Program. The second question: If you could modify the way the university operates, what would you change? Regardless of what university the respondent has attended, and even if he or she has just graduated, the reply is predictable. Above everything else, students want more opportunity to become acquainted with faculty members. Allocation of blame is not one of our agenda; but we should consider how teachers and students might promote acquaintance. Faculty members living within easy reach of the campus can give supper parties. In some large departments, graduate students take the initiative in organizing picnics, or they manage a weekly precolloquial indulgence in bread and cheese, fibrous vegetables, fruit, and appropriate fluids. Playing volleyball and dipping broccoli into a communal bowl of yogurt with herbs diminish social segregation. Still, we can hardly expect such activities to generate the kind of professorial acquaintance that students seek. Would more generous office hours bring in the timid students? H o w many visitors w h o say they have just dropped in for a chat could we satisfy in one week? An opportunity for letting students know us better awaits us whenever we enter a classroom. Most of us accept o u r academic position with a tacit u n d e r standing that we shall not subject our captive audience to religious or political propaganda or to the promotion of business deals and that we shall scrupulously refrain from personal or social improprieties; but neither my university nor the AAUP has ever intimated that I should deliver pasteurized lectures packaged in cellophane. In calculus courses, I have usually assigned the problem of differentiating THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 2, 1990

33

emX(A cos px + B sin px) and then choosing the coefficients A and B so that the derivative reduces to e~ c o s p x

or e~sinpx.

I would then show how hindsight allows us to integrate the latter two functions by inspection. Later, in the first or second lesson on integration by parts, I would point out w h y in the case of these two functions integration by parts is loaded with invitations to computational error and that here integration by inspection is more efficient, reliable, and satisfying. I would add that m y preference reflects personal taste and that textbooks do not support it.

Above everything else, students w a n t more opportunity to become acquainted w i t h faculty members.

I would stray not merely from the textbook, but also from the subject. Fifty minutes of uninterrupted technicalities is too much for the attention span of most freshmen, and I would not hesitate to sacrifice bits of our valuable time for horseplay or references to music, painting, literature, language, history, or questions of ethics. A few weeks into a fall semester, I entered a classroom and saw on the central blackboard a nearly written, straight-laced, unsigned rebuke; its author wanted me to stick strictly to business. To one side and written by a different hand stood the opposition's

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THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 2, 1990

declaration "We like your style; continue," followed by many initials. Our social organization makes division of labor almost inevitable; but division of labor carried to excess is an apotheosis of stupidity. As mathematicians, we should provide diverse services. We should contribute to the solution of fundamental or technical problems in p u r e or applied mathematics. To s t u d e n t s we should impart insight into certain mathematical structures and theories, together with reasonable technical skill and some appreciation for the intellectual landscape in which we work. Without twisting arms, we should inform able students that mathematics is an attractive and accessible profession. Moreover, we should serve as possible role models for educated women and men of our time. Our students are entering the age of serious doubt and dissatisfication. Having discovered sundry hypocrisies in conventional doctrine, they experiment with skepticism. Some of them, distrusting even the identity they have cultivated through childhood and adolescence, look for alternatives. They seek nonauthoritarian authority. Cross that out. They seek friendship with older people whom they can respect and with w h o m they can discuss basic problems. Serve them, unless you can refer them to somebody who is better qualified than you. Such service may require a shift in loyalties. We can afford the shift, for the transience of our existence renders even our professional affiliations tentative. Long before becoming mathematicians, we w e r e human. I have apologized. Vide supra.

Departmentof Mathematics The Universityof Michigan Ann Arbor, M148109 USA

The Einstein-Wertheimer Correspondence on Geometric Proofs and Mathematical Puzzles Abraham S. Luchins and Edith H. Luchins

"Those were w o n d e r f u l days, beginning in 1916, when for hours and hours I was fortunate enough to sit with Einstein in his study [in Berlin], and hear from him the story of the dramatic developments which culminated in the theory of relativity" [22, p. 213]. Thus Max Wertheimer (1880-1943), the chief founder of Gestalt psychology, began his account of the thinking that led Albert Einstein (1879-1955) to relativity theory.

Readers of the Mathematical Intelligencer need no introduction to Einstein but may want some background information on Wertheimer and the basis of their long friendship. Wertheimer, who was intensely interested in the nature and processes of productive thinking, was a highly innovative thinker. Einstein said of him, "He is not a man of dusty erudition, but a man of independent, lively thinking" [11, p. 193]. Rebelling against the atomistic, reductionistic viewpoints pre-

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vailing in the early 1900s, Wertheimer advocated a method of analysis "from above," which began with and was guided by properties of the whole structure of the phenomena, rather than analysis "from below," beginning with the elements or parts [16]. The fundamental "formula" of Gestalt theory he expressed in this way: "There are wholes, the behavior of which is not determined by that of their individual elements, but where the part-processes are themselves determined by the intrinsic nature of the whole" [17, p. 2]. Instead of arbitrary associations, the Gestalt psychological approach looked for intrinsic relations and laws of organization of psychological phenomena. Similarly, Einstein had "a deep faith . . . that there are laws of Nature to be discovered" [14, p. 154] and "faith in the simplicity of fundamental laws" [2, p. 167]. But whereas Einstein was eminently successful in achieving mathematical formulations of physical laws, Wertheimer had little success in his life-long efforts to find mathematical formulations of the laws of psychological structures. Wertheimer applied the Gestalt psychological approach to many fields, including perception, music, logic, thinking, learning, and teaching. Often his illustrations involved problem solving in mathematics, especially in arithmetic, algebra, and geometry [22]. He focused on such problems in his lectures at the N e w School for Social Research [8] and discussed them with his mathematician friends, among them Richard Courant and Kurt Friedrichs, his neighbors in N e w Rochelle. One of us, ASL, was Wertheimer's research assistant beginning in 1937. When ASL introduced the other author, EHL, to Wertheimer and commented that she was a mathematics major, Wertheimer told her pensively, "Sometimes mathematics is my only consolation." Wertheimer applied the Gestalt theoretical approach also to issues traditionally neglected by psychology, such as issues of truth [18], ethics [19], democracy [20], and freedom [21]. Einstein, who shared concerns about these issues, wrote that he and Wertheimer were in contact for many years because "of common interests in scientific problems and common human concerns . . . I know few people who seek truth and understanding as passionately as he does" [11, p. 193]. In a hitherto unpublished letter of 26 September 1940, Einstein wrote to Wertheimer [in German]: "Today I have read your essay on freedom and am really quite enthusiastic about it. I really believe that there are few to w h o m erudition has done so little damage as to you."

The Einstein-Wertheimer Friendship The two men may have met during the period (March 1911-August 1912) that Einstein held a professorship at the German university in Prague, the Karl-Fer-

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dinand University. Prague was Wertheimer's native city, which he visited often [6]. Their friendship flourished when Wertheimer served during World War I together with Einstein's close friend, Max Born, in the Prussian Army's artillery testing department, which was located near Einstein's home [2]. In 1916 Wertheimer had a Lehrauftrag (assignment to lecture) at the University of Berlin [9], where Einstein was a professor of physics from 1914 to 1933. Both Einstein and Wertheimer were Jews who felt the sting of anti-Semitism even before the Nazi era. Despite his fame, Einstein worried that he would not get the appointment in Prague because he was Jewish; in Berlin he found anti-Semitism strong [13, p. 192, p. 314]. Although Wertheimer was the senior member of the trio w h o founded Gestalt psychology, for years he had a lower academic rank than the other two, Wolfgang K6hler and Kurt Koffka [10], and was less well known in America [7]. Indeed, Wertheimer's professorship was at the University of Frankfurt, the socalled "Jewish" university [10] that was founded in 1914, largely with the donations of rich Jewish businessman, and had racial non-discrimination written into its charter [15, p. 135]. Appointed in 1929 when he was 49 years old, Wertheimer held the position only until 1933, when the Nazis terminated his professorship because he was a non-Aryan, despite an open letter on his behalf by K6hler. Linking Einstein and Wertheimer were their love of music (they played chamber music together), their enjoyment of sailing, and their mutual friends. Among them was Born, a Nobel Laureate in Physics, w h o wrote that Wertheimer "became a great friend of ours. He was a deep thinker, but of a different type from any I had known before" [2, p. 173]. Born was astounded that Wertheimer "did not even accept logic and arithmetic, and he tried to construct a meta-logic and a meta-arithmetic b a s e d on different axioms" ]ibid.]. Although the materials available at that time did n o t s h o w the fertility of these ideas, L. E. J. Brouwer's subsequent attack on the "law of the excluded third" and the efforts in the foundations of q u a n t u m mechanics to establish a non-Aristotelian logic led Born to think that Wertheimer might have been right [2], Born acknowledged the role that the concept of Gestalten played in his own work and in field theory, as shown in his autobiography [2] and in his correspondence with Wertheimer and with Einstein [1]. For example, Born's letter to Einstein of 31 March 1948 referred to observational invariants as "descendants of Wertheimer's Gestalt in a new form" [1, p. 166]. The close parallel between Einstein's theory and Gestalt psychology was also described by the psychologist George Humphrey [5]. Einstein had invited Born and Wertheimer to join him on a political adventure in November 1918 that eventually resulted in their obtaining the release of the

A photomontage of Wertheimer, Einstein, and Born, sent to Einstein by the Borns as a postcard on 28 July 1918. Reproduced with the permission of Otto Nathan.

rector and other professors of the University of Berlin who had been imprisoned by the students. The incident is described in [4], which contains a copy of a photomontage of Born, Einstein, and Wertheimer that was sent by the Borns to Einstein in July 1918 and later referred to by Einstein as a clover-leaf [4, p. 13]. On 17 October 1933, the Einsteins and Helen Dukas (his secretary since 1928, w h o was considered a member of the family) arrived in New York. That same day they were driven to the Peacock Inn in Princeton and a few days later they moved into a rented home at 2 Liberty Place. Only three days later, a letter with the name and address of Peacock Inn crossed out, and the 2 Liberty Place a d d r e s s a d d e d , w a s w r i t t e n [in German] by Einstein's wife Elsa to Wertheimer, who had arrived in N e w York together with his family one month earlier. Even allowing for old-world courtliness, this previously unpublished letter reveals the closeness of the friendship: "First I must make a confession. In all this terribly large [schrecklich grossen] America, there is no person w h o m Albert looks forward to seeing as much as you. If I have not written to you up to now, the reason is that we moved in only three days ago, and before that, the time was filled with all possible kinds of things which had to be settled." Mrs. Einstein invited Wertheimer and his wife to meet them at the home of Henry Morgenthau on Fifth Avenue in New York City or to come to their home in Princeton. The Einsteins preferred the latter plan. "Nowhere else," Mrs. Einstein pointed out, "could my husband be together with you in such peace of mind [Gemutsruhe]." The close relationship between Einstein and Wertheimer is evident in their published correspondence [11], which started in 1920. It continued through August 1943, two months before Wertheimer's death,

when he sent Einstein the manuscript of the chapter describing the thinking that led to relativity theory and asked for his comments. Miss Dukas recalled that Einstein a n n o t a t e d the chapter. If the a n n o t a t e d manuscript could be found, it might resolve a controversy over Wertheimer's account of the development of relativity theory [12]. On 12 October 1943, Wertheimer died suddenly of a coronary thrombosis. The memorial service a month later at the N e w School for Social Research was attended by Einstein. One of the authors, EHL, recalls the sadness on his face throughout the service, as he huddled near his bundled up overcoat, and how afterwards he graciously shook hands with her and other admirers. The New

Letters

Almost all of the correspondence between Einstein and Wertheimer was in German. They were more comfortable writing in German, even when they had been in the United States for ten years and could speak and read English fluently. "But I cannot write in English," Einstein explained to Born, in a letter of 7 September 1944, '%ecause of the treacherous spelling. When I am reading, I only hear it and am unable to remember what the written w o r d looks like" [1, p. 148].

Some of the letters to which Wertheimer referred in the published correspondence [11] were not in the Einstein Archives. Miss Dukas suggested that this was probably because they were handwritten. Indeed, this turned out to be the case for most of the missing letters that have since been found in the many boxes of Wertheimer's papers.

Continued on p. 40 THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 2, 1990

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THE MATHEMATICAL1NTELLIGENCERVOL. 12, NO. 2, 1990 ~ 9

Continued from page 37 Three letters that may be of particular interest to the readers of the Mathematical Intelligencer concern geometric proofs and mathematical puzzles. Two of these are hitherto unpublished letters by Einstein. Translations of these letters are presented in this article together with a previously published letter by Wertheimer [11, p. 186] that is relevant to them. Although all the letters were originally undated, they are now believed to have been written in the fall of 1937. Miss Dukas informed us that Einstein spent the summer and early fall of 1937 in Huntington, Long Island, and that Wertheimer's visit there was in September. That month Wertheimer wrote to Einstein thanking him for the hours in Huntington that they had spent discussing the "'problem of axioms.'" Wertheimer sought to illustrate the problem with two procedures for describing a particular planar network. 40

THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 2, 1990

He classified one procedure as blind to properties of the whole structure and the other procedure as guided by these properties. The reply by Einstein, which is translated on page 38, dismissed Wertheimer's examples as irrelevant to the problem under discussion. Einstein noted that whereas the choice between two axiom systems might be only a matter of taste, it was different when we compared the value of two proofs, based on the same system of concepts and axioms. As a "pretty example," Einstein offered two proofs of Menelaus's Theorem in plane geometry, one more "elegant" and satisfying than the other. Wertheimer thanked Einstein and posed two mathematical "~orainteasers"; see above. In the third and final letter, translated opposite, Einstein expressed uneasiness even about the "better" proof of Menelaus's Theorem and, indeed, about every proof. He also gave his reactions to the mathematical puzzles.

1nk1n9

Wertheimer's Gestalt psychology with its stress on intrinsic requirements. Einstein continued to think about the proofs, admitting that he felt uneasy even about the "better" proof, and adding the interesting belief that every proof leaves a certain residue of doubt. A long letter by Wertheimer, which was not presented here, further analyzed the two proofs and related them to the two friends' mutual interest in thinking processes. Perhaps it was this interest (or just fascination with "brain-teasers") that motivated Wertheimer to send Einstein the two puzzles. We saw the human side of Einstein w h e n he gleefully admitted that he and a friend were fooled by the first puzzle but only the friend by the second puzzle. His delight in the puzzles, and his comments that they show how stupid we are, further revealed his down-to-earth character. This characteristic was also apparent when Einstein called himself a stupid beast for having used the formal instead of the informal mode of address. Such informality was also found in Wertheimer's drawing of a sailor hat, with his initials in it, as his signature. They obviously enjoyed seeing and writing to each other. To use Einstein's words, the correspondence helped us to see each of them as "a real person."

Painting of Albert Einstein by Betty Frechette. Reproduced with the kind permission of the artist. Conclusions The new letters fill gaps in what was known about the Einstein-Wertheimer correspondence. A previously published letter by Wertheimer, which was not presented here and which Miss Dukas believed was written in the winter of 1937, thanked Einstein for the beautiful example of the " u g l y " and "beautiful" proofs concerning the theorem of Menelaus [11, p. 191]. Now it is d e a r that the reference was to the first new Einstein letter (Letter 1). The new letters also help to date the Wertheimer letter that was presented. Originally believed to have been written in 1934, it now seems clear that it was written in 1937, the same year as the new letters. The letters also cast light on the relationship between the two friends and their mutual interests. Wertheimer's letter followed up on their hours of conversation in H u n t i n g t o n about " t h e problem of axioms." Einstein took the time to write a long letter rejecting Wertheimer's example as irrelevant to the problem and giving the detailed proofs of an ancient geometric theorem that is not commonly taught today. Einstein contended that we could be completely satisfied with a proof only if we felt that each intermediate concept was intrinsically related to the proposition to be proved. This thesis reflected the influence of 42

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Acknowledgments Copies of the correspondence between Albert Einstein and Max Wertheimer that were in the Einstein Archives in Princeton were obtained by us in 1974 from the co-trustees of the Einstein estate, Otto Nathan and Helen Dukas, who kindly sent along brief explanations and gave us permission to publish. We also obtained letters in the correspondence from the Wertheimer Archives, as well as from Michael Wertheimer and other members of the Wertheimer family, to all of w h o m we extend our appreciation. After preliminary editing and translation, some of the letters were published [11]. The present artide includes translations of letters that were found subsequently. We thank Professor Emeritus Kurt Bing of Rensselaer Polytechnic Institute and Professor Peter Heath of the University of Virginia for help with translations of portions of the correspondence. If phrases that are less than felicitous remain, it is our responsibility. We also thank Rensselaer Polytechnic Institute and Lorraine Pisarczyk, Senior Secretary of the Department of Mathematical Sciences, for typing of the manuscript.

References 1. Max Born, The Born-Einstein Letters: The Correspondencebetween Albert Einstein and Max and Hedwig Born, 1916-1955 (translated by Irene Born), New York: Walker (1971). 2. Max Born, My Life: Recollections of a Nobel Laureate, New York: Scribner (1978). 3. Howard Eves, A Survey of Geometry (Revised ed), Boston: Allyn and Bacon (1972).

4. Banesh H o f f m a n (with the collaboration of Helen Dukas), Albert Einstein: Creator and Rebel, New York: The Viking Press (1972). 5. George Humphrey, The theory of Einstein and the Gestalt Psychologie: A parallel, American Journal of Psychology 35 (1924), 353-359. 6. Abraham S. Luchins, Max Wertheimer, International Encyclopedia of the Social Sciences, Vol. 16, New York: Macmillan (1968), 522-527. 7. Abraham S. Luchins, The place of gestalt psychology in American psychology: A case study. Gestalttheorie in der modernen Psychologie (Suitbert Ertel, Lilly Kemmler, and Michael Stadler, eds.). Darmstadt, Germany: Steinkopff Verlag (1975), 21-44. 8. A b r a h a m S. Luchins and Edith H. Luchins, Wertheimer's Seminars Revisited: Problem Solving and Thinking, Vols. I, II, & I I I , New York: SUNY-Albany Faculty Student Association. Distributed by Laurence Erlbaum Publishers (1970). 9. Abraham S. Luchins and Edith H. Luchins, Max Wertheimer: His life and work during 1912-1919, Gestalt Theory 7 (1985), 3-28. 10. Abraham S. Luchins and Edith H. Luchins, Max Wertheimer: 1919-1929, Gestalt Theory 8 (1986), 4-30. 11. Edith H. Luchins and Abraham S. Luchins, Introduction to the Einstein-Wertheimer correspondence, Methodology and Science 12 (1979), 165-201 (Special Einstein Edition). 12. Arthur Miller, Albert Einstein and Max Wertheimer: A Gestalt psychologist's view of the genesis of special relativity theory, History of Science XIV (1975), 75-103. 13. Abraham Pals, "Subtle is the L o r d . . . ' The Science and the Life of Albert Einstein, N e w York: Oxford University Press (1982). 14. Abraham Pais, Knowledge and belief: The impact of Einstein's relativity theory, American Scientist 76 (2) (March-April 1988), 154-158. 15. Max Pinl and Lux Furtmiiller, Mathematicians under Hitler, Leo Baeck Institute Yearbook, XVIII, London: Secker & Warburg (1973), 129-182. 16. Max Wertheimer, Untersuchungen zur Lehre v o n d e r Gestalt, I, Psychologische Forschung 1 (1922), 47-58. Abridged translation in Willis D. Ellis, A Source Book of Gestalt Psychology, N e w York: Harcourt, Brace (1938), 12-16. 17. Max Wertheimer, Uber Gestalttheorie, an address before the Kant Society, Berlin, 17 December 1924. Abridged translation in Willis D. Ellis, A Source Book of Gestalt Psychology, New York: Harcourt, Brace (1938), 1-11. 18. Max Wertheimer, O n truth, Social Research 1 (1934), 135-146. 19. Max Wertheimer, Some problems in the theory of ethics, Social Research 2 (1935), 353-367. 20. Max Wertheimer, On the concept of democracy, Political and Economic Democracy (M. Ascoli & F. Lehmann eds.), New York: Norton (1937), 271-273. 21. Max Wertheimer, A story of three days, Freedom: Its Meaning (Ruth N. Anshen ed.), New York: Harcourt, Brace (1940), 555-569. 22. Max Wertheimer, Productive Thinking (enlarged ed., Michael Wertheimer, Ed.), New York: Harper (1959).

Department of Psychology Rensselaer Polytechnic Institute Troy, NY 12180 USA Department of Mathematical Sciences Rensselaer Polytechnic Institute Troy, NY 12180 USA

Excellent Titles for Mathematicians G. Graflhoff, University of Hamburg, FRG

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The Story of the Higher Dimensional Poincar6 Conjecture (What Actually Happened on the Beaches of Rio)* Steve Smale

This paper is dedicated to the memory of Allen Shields.

Although these pages tell mainly a personal story, let us start with a description of the "n-dimensional Poincar6 Conjecture." It asserts: A compact n-dimensional manifold M n that has the homotopy type of the n-dimensional sphere Sn = {x ~ Rn+ll [[xl[ = 1} is homeomorphic to S". A "compact n-dimensional manifold" could be taken as a closed and bounded n-dimensional surface (differentiable and non-singular) in some Euclidean space. The homotopy condition could be alternatively defined by saying there is a continuous map f:M" -~S n inducing an isomorphism on the homotopy groups; or that every continuous map g:S k ---* M", k < n (or just k n/2) can be deformed to a point. One could equivalently demand that M " be simply connected and have the homology groups of S n. Henri Poincar6 studied this problem in his pioneering papers in topology. In [13], 1900, he announced a proof of the general n-dimensional case. A counter-example to his method is exhibited in a subsequent paper [14] 1904, where he limits himself this time to 3 dimensions. In this paper he states his famous problem, but not as a conjecture. The traditional description of the problem as "Poincar6's Conjecture" is inaccurate in this respect. Many other mathematicians after Poincar6 have

claimed proofs of the 3-dimensional case. See, for example, [18] for a popular account of some of these attempts. On the other hand, there has been a solidly developing b o d y of theorems and techniques of topology since Poincar6. In 1960 1 showed that the assertion is true for all n > 4, and this is an account of that discovery. The story here is complemented by two articles [15] and [16], but the overlap is minimal. I first heard of the Poincar6 conjecture in 1955 in Ann Arbor at the time I was writing a thesis on a problem of topology. Just a short time later, I felt that I had found a proof (3 dimensions). Hans Samelson was

* This article is an e x p a n d e d version of a talk given at the 1989 annual joint meeting of the American Mathematical Society, and the Mathematical Association of America. 44 THE MATHEMATICALINTELLIGENCERVOL. 12, NO. 2 9 1990 Springer-VerlagNew York

in his office, and very excitedly I sketched m y ideas to him. First triangulate the 3-manifold and remove one 3-dimensional simplex. It is sufficient to show the remaining manifold is homeomorphic to a 3-simplex. Then remove one 3-simplex at a time. This process doesn't change the homeomorphism type and finally one is left with a single 3-simplex. Q.E.D. Samelson didn't say much. After leaving his office, I realized that my "proof" hadn't used any hypothesis on the 3-manifold. Less than 5 years later in Rio de Janeiro, I found a counterexample to the 3-dimensional Poincar6 Conjecture. The "proof" used an invariant of the Leningrad mathematician Rohlin, and I wrote out the mathematics in detail. It would complement nicely the proof I had just found that for dimensions larger than 4, the result was true. Luckily, in reviewing the counterexample, I noticed a fatal mistake. Let us go back in time. I was born into the "Golden Age of Topology." Today it is easy to forget h o w much topology in that era dominated the frontiers of mathematics. It has been said that half of all the Sloan Postdoctoral Fellowships awarded in those years went to topologists. Today it would be hard to conceive of such a lopsided distribution. Topology of that time, in fact, had revolutionizing effects in algebra (K-theory, algebraic geometry) and analysis (dynamical systems, the global study of partial differential equations). In 1954 Thom's cobordism paper was published. That theory was used by Hirzebruch to prove his "signature theorem" (as part of his development of Riemann-Roch). In turn, already by 1956, Milnor used the signature theorem to prove the existence of exotic 7dimensional spheres. I followed these results closely. Also as a student I learned from Raoul Bott about Serre's use of spectral sequences and Morse theory to find information on the homotopy groups of spheres. A little later, Bott himself was proving his periodicity theorems, also with the use of Morse Theory. I received my doctorate in Ann Arbor in 1956 with Bott. My first encounter with the mathematical world at large occurred that summer with the Mexico City meeting in Algebraic Topology. I had never been to any conference before. And this conference was a historic event in mathematics by any standard. I have not seen that concentration of creative mathematics matched. My wife Clara and I took a bus from Ann Arbor to Mexico City, originally knowing only Bott and Samelson. Before leaving Mexico, I had met most of the stars of topology. There I also met two graduate students from the University of Chicago, Moe Hirsch and Elon Lima, w h o were to become part of our story. That fall I took up my first regular position as an instructor in the college at the University of Chicago, primarily teaching set theory to humanities students. I had good relations with the mathematics department and went to the lectures of visiting professor Ren6

Steve Smale with his son Nat, Chicago, 1958.

Thorn (whom I had met in Mexico City) on transversality theory. I also pursued work in topology showing that one could "turn a sphere inside out." Chicago was a leading mathematics center at that time, before Weil, Chern, and a number of other important mathematicians left. An important part of the environment was created by the younger mathematicians, especially Moe Hirsch, Elon Lima, Dick Lashof, Dick Palais, and Shlomo Sternberg. I was lucky to be there during that period. A two-year National Science Foundation (NSF) postdoctoral fellowship enabled me to go to the Institute for Advanced Study in the fall of 1958. Topology was very active in Princeton then. I shared an office with Moe Hirsch and we attended Milnor's crowded lectures on characteristic classes and Borel's seminar on transformation groups. I frequently e n c o u n t e r e d Deane Montgomery, Marston Morse, and Hassler Whitney, played go (with handicap) with Ralph Fox a n d m e t his s t u d e n t s Lee N e u w i r t h and J o h n Stallings. Moreover, in the summer of 1958, Lima introduced me to Mauricio Peixoto, w h o sparked my interest in structural stability. That interest led to an invitation to spend the last six months of my NSF in Rio de Janeiro at I.M.P.A. (Instituto de Matematica, Pura e Aplicada). Thus, at the beginning of January 1960, with Clara and our children, Laura and N a t / I arrived in Rio to meet our Brazilian friends. We arrived in Brazil just after a coup had been attempted by an Air Force colonel. He fled the country to take refuge in Argentina, and we were able to rent his apartment! It was an 11room luxurious place, and we also hired his two maids. The U.S. dollar went a long way in those days. THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 2, 1990

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Most of our immediate neighbors were in the U.S. or Brazilian military. We could sit in our highly elevated gardened patio and look across to the hill of the favela (Babylonia) where Black Orpheus was filmed. In the hot, humid evenings preceding carnaval, we would watch hundreds of the favela dwellers winding down their path to dance the samba in the streets. Sometimes I would join their wild dancing, which paraded for many miles. Just one block in the opposite direction from the hill lay the famous beaches of Copacabana (the Leme end). I would spend the mornings on that wide, beautiful, s a n d y beach sometimes s w i m m i n g , or, depending on the height of the waves, body surfing. Also I took a pen and pad of paper and would work on mathematics. Afternoons I would spend at I.M.P.A. discussing differential equations with Peixoto and topology with Lima. At that time I.M.P.A. was located in a small old building on a busy street. The next time I was to visit 46

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Rio, I.M.P.A. was in a much bigger building in a much busier street. I.M.P.A. has n o w moved to an enorm o u s m o d e r n palace s u r r o u n d e d by jungle in the suburbs of Rio. Returning to the story, m y mathematical attention was at first directed towards dynamical systems and I constructed the " h o r s e s h o e " [15]. As I continued working in gradient dynamical systems, I noticed how the dynamics led to a n e w way of decomposing a manifold simply into cells. The possibilities of using this decomposition to attack the Poincar~ conjecture soon developed, and before long all my work focused on that problem. With apparent success in dimensions greater than 4, I reviewed m y proof carefully; then I went through the details with Lima. Gaining confidence, I wrote Hirsch in Princeton and sent off a research announcement to S a m m y Eilenberg. The box titled "Dynamics and Manifold Decomposition" contains a mathematical description of what was happening.

I was already planning to leave Rio for three weeks in Europe during June of that year, 1960. There was the famous Arbeitstagung, an annual mathematics event organized by Hirzebruch. This was to be followed by a topology conference in Z~irich to which I had been invited. The two meetings provided a good

M y b e s t - k n o w n w o r k w a s done on the

beaches of Rio de Janeiro. opportunity to present m y results. For a change of pace and with his consent, I have put two recent letters of Stallings to Zeeman in a box later in this article. These letters describe well the events that took place in Europe. While m y memory in general is consistent with Stallings', I don't believe Hirsch helped me as StaUings conjectured. I do recall spending some relaxed days in St. Moritz with Moe Hirsch and Raoul Bott, after a more dramatic and traumatic week in Bonn. Sometimes I have become upset at what I feel are inaccuracies in historical accounts of the discovery of the higher dimensional Poincar4 conjecture. For example, Andy Gleason wrote in 1964 [1]: " . . . It was a great surprise, therefore, when Stallings in 1960 (4) proved that the generalized Poincar4 conjecture is true for dimensions 7 and up. His result was extended by Zeeman (10) shortly thereafter to cover dimensions 5 and 6." (I wasn't mentioned in his article.) Paul Halmos in his autobiography ([8], page 398) writes of my anger with him. I am sorry that I was angry with Paul, and I wish I could be more relaxed about this subject in general. My recent correspondence with Jack Milnor illustrates the issue.

Milnor wrote back (as mildly modified and expanded by him):

I wrote him, thanking him, saying that I appreciated his letter. THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 2, 1990

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After the Zfirich conference, I returned to join my family in Rio and shortly thereafter took up my position in Berkeley. During the next year I wrote several papers extending the results, culminating with a paper proving the "h-cobordism theorem" in June 1961. A mathematical survey of all these matters, with references, is given in [17]. Because of Serge Lang and an irresistible offer from Columbia, we sold our house and left Berkeley in the THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 2, 1990

summer of 1961. After three years of studying various aspects of global analysis, we returned from New York to Berkeley because of the prospect of better working conditions. That was the fall of 1964, and the Free Speech Movement (FSM) caught my attention. After the big sit-in, I helped obtain the release from jail of mathematics graduate students David Frank and Mike Shub. The Vietnam war was drastically escalated early in 1965. I felt that the U.S. heavy bombing of Vietnam

was indefensible and threatened world peace. My involvement in the protest increased, and included organizing teach-ins and militant troop train demonstrations. I became cochairman with Jerry Rubin of the VDC, or Vietnam Day Committee. (Our headquarters near campus was later destroyed by a bomb.) Already during the fall of 1965, I was becoming disillusioned with the VDC and returned to proving theorems. The subsequent part of this story is described in [16]. In Moscow, August 1966, I criticized

the U.S. in Vietnam (and Russia as well) to return home under a storm of criticism. The University of California s t o p p e d p a y m e n t of m y NSF s u m m e r salary. Dan Greenberg [2] gives a full account of what happened next. Upon being informed of the agitation surrounding him, and the withholding of his check, Smale sent to Connick an account of his summer researches--an account which Continued on page 51 THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 2, 1990

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50

THE MATHEMATICALINTELLIGENCERVOL. 12, NO. 2, 1990

Continued from page 49 will probably be a classic document in the literature of science and government. He quickly established that he had satisfied the requirement of 2 months of research for 2 months of salary. Connick was Vice-Chancellor of Academic Affairs at Berkeley. Greenberg's "classic d o c u m e n t " surprised me, b u t m y letter did contain m y most f a m o u s quote. As r e c o u n t e d b y Greenberg, I wrote to Connick: However, during the remainder of this time I was also doing mathematics, e.g., in campgrounds, hotel rooms, or on a steamship. On the S.S. France, for example, I discussed problems with top mathematicians and worked on mathematics in the lounge of the boat. (My best-known work was done on the beaches of Rio de Janeiro, 1960!) I would like to repeat that I resent your stopping of my NSF support money for superficial technicalities. The reason goes back to my being issued a subpoena by the House Un-American Activities Committee and the subsequent congressional and newspaper attacks on me. Before long, I did receive the NSF f u n d s a n d things quieted down. H o w e v e r , within a year, a m u c h bigger explosion took place w h e n the NSF r e t u r n e d to me m y n e w proposal amidst Congressional pressures. T h e articles [ 3 - 7 ] in Science b y D a n G r e e n b e r g chronicle these events. See also [11] and [12]. Eventually as the situation was settling d o w n once m o r e , m y s t a t e m e n t a b o u t the beaches of Rio surfaced. This time it was the science adviser to President Johnson, Donald H o r n i g [9], w h o wrote in Science: This blithe spirit leads mathematicians to seriously propose that the common man who pays the taxes ought to feel that mathematical creation should be supported with public funds on the beaches of Rio de Janeiro or in the Aegean Islands. (I also had visited the Greek islands in A u g u s t 1966 but, of course, not with NSF money.) I was very h a p p y at the response of the Council and P r e s i d e n t C. M o r r e y of the American Mathematical Society. Morrey's letter [10] in Science started: The Council of the American Mathematical Society at its meeting on 28 August asked me to forward the following comments to Science: Many mathematicians were dismayed and shocked by the excerpts of the speech by Donald Hornig, the Presidential Science Adviser (19 July, p. 248). His . . . comments about mathematics and mathematicians are . . . uncalled for. Implicit in Hornig's remarks about vacations on the beaches of Rio or the Aegean Islands was a thinly veiled attack on Stephen Smale. The allegations against Smale were adequately disproved by Daniel S. Greenberg in his articles in Science on the Smale-NSF controversy. At the same time a letter to the Notices w o u n d up: 9 . . the policy of creating a major scandal involving the implied application of political criteria in grant administrat-ion, apparently for the purpose of placating and warding off the demagogic attack of a single Congressman, is not a policy that will cause anyone (and least of all Congress) to have any great respect for the principles and integrity of

those who adopt it. The conspicuous public silence of the whole class of Federal science administrators with regard to the future effects of current draft policy and the Vietnam war upon American science has been positively deafening. In this context, Hornig's apparent attempt to turn the discussion of the current crisis in Federal support for basic science into a hunt for scapegoats in the form of "mathematicians on the beaches" would be ludicrous if i t were not so destructive. Hyman Bass E. R. Kolchin F. E. Browder S. Lang William Browder M. Lo~ve S. S. Chern R . S . Palais Robert A. Herrmann M. H. Protter I. N. Herstein G. Washnitzer It was especially gratifying to see such support from m y friends. Let m e end, as I did in Phoenix: "Thanks very m u c h for listening to m y story." References

1. Gleason, A., Evolution of an active mathematical theory, Science 145 (July 31, 1964) 451-457. 2. Greenberg, Dan, The Smale case: NSF and Berkeley p a s s through a case of jitters, Science 154 (Oct. 7, 1966) 130-133. 3. - - , Smale and NSF: A new dispute erupts, Science 157 (Sept. 15, 1967) 1285. 4. - - , Handler statements on Smale case, Science 157 (Sept. 22, 1967) 1411. 5. , The Smale case: Tracing the path that led to NSF's decision, Science 157 (Sept. 29, 1967) 1536-1539. 6. , Smale: NSF shifts position, Science 158 (Oct. 6, 1967) 98. 7. , Smale: NSF's records do not support the charges, Science 158 (Nov. 3, 1967) 618-619. 8. Halmos, P., I Want to be a Mathematician, an Automathography. New York: Springer-Veflag (1985). 9. Hornig, D., A point of view, Science 161 (July 19, 1968) 248. 10. Morrey, C., Letter to the editor, Science 162 (Nov. 1, 1968) 514-515. 11. , The case of Stephen Smale, Notices of the American Math. Soc. 14 (Oct. 1967) 778-782 12. , The case of Stephen Smale: Conclusion, Notices of the American Math. Soc. 15 (Jan. 1968) 49-52 and 16 (Feb. 1968) 297 (by Serge Lang). 13. Poincard, H., Oeuvres, VI, Gauthier-Villars, Paris 1953, Deuxi~me Compldment a L' Analysis Situs, 338-370. 14. - - , Cinqui~me Compldment a L'Analysis Situs, 435-498. 15. Smale, S., On How I Got Started in Dynamical Systems, The Mathematics of Time. New York: Springer-Verlag (1980). 16. , On the Steps of Moscow University, Math. Intelligencer 6 no. 2 (1984) 21-27. 17. - - , A survey of some recent developments in differential topology, Bull. Amer. Math. Soc. 69 (1963) 131-145. 18. Taubes, G., What happens when Hubris meets Nemesis, Discover, July 1987. Department of Mathematics University of California Berkeley, CA 94720 USA THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 2, 1990 5 1

Ian Stewart* The catapult that Archimedes built, the gambling-houses that Descartes frequented in his dissolute youth, the field where Galois fought his duel, the bridge where Hamilton carved quaternions-not all of these monuments to mathematical history survive today, but the mathematician on vacation can still find many reminders of our subject's glorious and inglorious past: statues, plaques, graves, the cafg where the famous conjecture was made, the desk where the

famous initials are scratched, birthplaces, houses, memorials. Does your hometown have a mathematical tourist attraction? Have you encountered a mathematical sight on your travels? If so, we invite you to submit to this column a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks. Please send all submissions to the European Editor, Ian Stewart.

Historical Associations of Fermat in Beaumont and Toulouse, France J.-B. Hiriart-Urruty Pierre de Fermat was born in Beaumont-de-Lomagne in August 1601 and died in Castres in January 1665. He spent most of his professional life in Toulouse as ConseiUer du Roi au Parlement de Toulouse and Commissaire des Requites du Palais (a public lawyer in brief). Fermat is known for his variational principle, for his contributions in laying the foundations of analytical geometry (along with Descartes), of probability theory (together with Pascal) a n d - - t h e subject which has made him notorious--number theory. Every mathematician has thought at least once about what is referred to as "Fermat's Last Theorem" (but still a conjecture): given an integer n/> 3, the equation x n + yn = Zn has no strictly positive integer solutions x, y, z. Beaumont-de-Lomagne is a small town of 4000 inhabitants, 60 kilometers northwest from Toulouse**. The role of market town played by Beaumont (especially for garlic) can easily be guessed w h e n gazing at the large covered market located in the central part of the town. A statue of Fermat is placed on the side of this covered market (see Figure 1). It was erected in

* Column Editor's address: M a t h e m a t i c s Institute, U n i v e r s i t y of Warwick, C o v e n t r y CV4 7AL England. ** It is n o t so easy to get to B e a u m o n t from Toulouse. Take the seco n d a r y road called D2 a n d t h e n D3 t h r o u g h t h e countryside; the road is c h a r m i n g b u t s o m e w h a t narrow. 52 THE MATHEMATICALINTELLIGENCERVOL. 12, NO. 2 9 1990Springer-VerlagNew York

From Springer-Verlag 1882 a n d on its p l i n t h are e n g r a v e d the c e l e b r a t e d conjecture a n d q u o t a t i o n s f r o m Laplace, Pascal, a n d Cauchy. Postcards s h o w i n g the statue a n d the covered m a r k e t are sold in a small s h o p just in front of the statue. The h o u s e w h e r e Fermat was b o r n (Figure 2), located just 100 m e t e r s f r o m the m a r k e t place, h a s b e e n b o u g h t a n d r e n o v a t e d quite recently b y the t o w n (Fermat has no descendants); it n o w a c c o m m o d a t e s the tourist office a n d a lending library. For the last few years, a r o o m in the library has b e e n d e v o t e d to a perm a n e n t e x h i b i t i o n o n F e r m a t , i n c l u d i n g c o p i e s of s o m e of his m a n u s c r i p t s . A m a t h e m a t i c i a n will e v e n find a booklet b y s o m e obscure writer entitled: "Proof a n d generalization of F e r m a t ' s Last T h e o r e m " ! A n o t h e r statue of F e r m a t can be seen in the "salle d e s illustres" of the t o w n h a l l (called " C a p i t o l e " ) of Toulouse. Located on the first floor of the " C a p i t o l e , " this large r o o m , m u c h visited b y tourists, contains statues of several great figures of the region. A m o n g t h e m is the statue of F e r m a t with the following (somewhat overstated) quotation: "Fermat, inventeur du Calcul Diff6rentiel" (see Figure 3). The l e g e n d of this statue is: " F e r m a t a n d a m u s e , " but a g r o u p of m a t h e maticians I b r o u g h t w i t h m e for a visit w h i s p e r e d mischieviously: " F e r m a t a n d his Ph.D. s t u d e n t . . . . " A h i g h s c h o o l in T o u l o u s e b e a r s t h e n a m e of Fermat, as does an a m p h i t h e a t r e in the Paul Sabatier University. Department of Mathematics University Paul Sabatier 118, route de Narbonne 31062 Toulouse Cedex, France

Stephen Wiggins, California Institute of Technology, Pasadena, CA

Introduction to Applied Nonlinear Dynamical Systems and Chaos This significant volume is intended for advanced undergraduate or first year graduate students as an introduction to applied nonlinear dynamics and chaos.Wiggins has placed emphasis on teaching the techniques and ideas which will enable students to take specific dynamical systems and obtain some quantitative information about the behavior of these systems. He has included the basic core material that is necessary for higher levels of study and research. Thus, people who do not necessarily have an extensive mathematical background, such as students in engineering, physics, chemistry and biology, should also use this text. 1990/app. 704 pp./291 illus./Hardcover $49.95 ISBN 0-387-97003-7 Texts in Applied Mathematics, Volume 2

G. looss, Universit6 de Nice, France, and D.D. Joseph, University of Minnesota, Minneapolis, MN

Elementary Stability and Bifurcation Theory Second Edition

From the reviews of the first edition: "I think that this book is a useful contribution to the textbook literature. I can easily imagine giving a course built around its exposition, and I recommend it as a text or reference for anybody wishing to give a course on bifurcation." mMathematical Reviews

This new edition has been substantially revised. Its purpose is to teach the theory of bifurcation of asymptotic solutions of evolution problems governed by nonlinear differential equations. 1989/324 pp., 60 illus./Hardcover $49.95 ISBN 0-387-97068-1 Undergraduate Texts in Mathematics

Order Today! Call Toll-Free: 1-800-SPRINGER (in NJ call 201-348-4033). For mail orders, send payment plus $2.50 for postage and handling to: Springer-Verlag New York, Inc., Atm: S. Klamkin Dept. $423. 175 Fifth Avenue, New York, NY 10010. We accept Visa, MasterCard and American Express charges (with signature and exp. date noted) as well as personal checks and money orders. NY, NJ and CA residents please include state sales tax. Or visit your local technical bookstore! Please write or call for details on textbook examination copies.

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Quasicrystals: The View from Les Houches Marjorie Senechal and Jean Taylor

1. I n t r o d u c t i o n Soon after the announcement of their discovery in 1984 [1], quasicrystals hit the headlines. Here was a substance--an alloy of aluminum and manganese-whose electron diffraction patterns exhibited clear and unmistakable icosahedral symmetry (a view along a five-fold axis is shown in Figure 1). A clear and unmistakable diffraction pattern of any sort is evidence of "long-range order": the diffraction pattern is a picture of a Fourier transform. Long-range order is usually s y n o n y m o u s with periodicity, and every periodic structure has a translation lattice. But a simple argument shows that five-fold rotational symmetry is in-

54

compatible with lattices in R2 and R3" every lattice has a minimum distance d between its points, but if two points at this distance are centers of five-fold rotation about parallel axes, the rotations will generate an orbit with smaller distances between them (Figure 2). By this chain of reasoning, it appeared that the impossible had occurred. For the past five years, quasicrystals have been studied intensively by metallurgists, solid-state physicists, and mathematicians (few crystallographers have shown much interest in them). The problem has gradually been resolved into three more or less separate questions, not necessarily according to the field of the researcher:

THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 2 9 1990 Springer-Veflag New York

1. Crystallography. How are the atoms of real quasicrystals arranged in three-dimensional space? 2. Physics. What are the physical properties of substances with long-range order but no translational symmetry? 3. Mathematics. What kind of order is necessary and sufficient for a pattern of points to have a diffraction pattern with bright spots? As Cahn and Taylor pointed out in 1985 [2], to answer Question 3 we must draw on a variety of techniques from many branches of mathematics, including tilting theory, almost periodic functions, generalized functions, Fourier analysis, algebraic number theory, ergodic theory and spectral measures, representations of GL(n), and symbolic dynamics and dynamical systems. This article is a report on the current status of the problem. We began our discussions while attending a conference on Number Theory and Physics, at the Centre de Physique, Les Houches, France in March 1989. One of the foci of that conference was quasiperiodicity and quasicrystals, and during our ten days there we enjoyed extended discussions with a variety of observers and practitioners of this field. But we warn the reader that the view we present here may not be widely shared; in particular, Q u e s t i o n 3 is u s u a l l y not phrased in such generality (see Section 6). And like the view of the Mont Blanc massif from the conference center (Figure 3), the general outline and size of the problem is rather clear, but features that are prominent from our perspective may mask others, including the summit. 2. W h a t Is a Crystal?

The discovery in 1912 that crystals diffract X-rays lent overwhelming experimental support to the hypothesis that crystalline structure is periodic. What could be a better example of Pierre Curie's banal but widely admired Principle of Symmetry: "'When certain causes produce certain effects, the elements of symmetry in the causes ought to reappear in the effects produced"? Since then, the lattice has been taken as the definition of the crystalline state. For the first year or two after their discovery, the question most hotly debated among solid-state scientists was: are quasicrystals crystals? By this was meant, can the structure of these alloys be interpreted within the framework of periodicity (for example, as a mosaic or intergrowth), or is it something truly new? Now, nearly five years later, quasicrystals of varying compositions (aluminum-lithium-copper, uraniumpalladium-silicon, and many others) and high perfection have been grown in laboratories all over the world and have been analyzed in great detail, and the mosaic

Figure 1. A diffraction pattern of an aluminum-manganese quasicrystal. Its five-fold rotational symmetry produced shock-waves in the world of solid-state science. Photograph courtesy of John Cahn.

Figure 2. Five-fold symmetry is incompatible with periodicity, because it violates discreteness. Two five-fold centers at the minimum distance d generate points whose distance is less than d.

Figure 3. The Mont Blanc massif, seen from the Centre de Physique, Les Houches, France. Photo by Pierrette CassouNouges. THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 2, 1990 5 5

and intergrowth models have been discarded by essentially everyone but Linus Pauling [3]. It is clear that the question should be interpreted differently. To ask whether quasicrystals are crystals really means to ask what we mean by "crystal." We cannot see inside the solid state; we know it only through the images provided by diffraction, electron microscopy, and other modern techniques. It might be more appropriate to define a "crystal" to be a structure with sufficient long-range order to exhibit images with properties associated with those we call crystalline, such as a diffraction pattern with sharp spots. A diffraction pattern for a material is essentially a two-dimensional slice of the square modulus of the Fourier transform of its density distribution; it faithfully records the amplitudes of the transform, but gives no direct record of the phases. Geometrically, we can think of a periodic crystal as an orbit of its symmetry group, which is an infinite discrete group with compact fundamental region. It can be shown that every orbit of such a group is a union of a finite

Finding a periodic framework on which to hang these structures can be likened to the

pre-Keplerian problem in astronomy of trying to explain planetary orbits by decorating the circle with the right number and arrangement of epicycles. number of congruent lattices. In the first approximation, we can take the density distribution of a periodic crystal to be a sum of weighted delta functions located at the nodes of each of these lattices (with the same weight for each point of a particular lattice). The Poisson summation formula implies that the Fourier transform of a lattice sum of deltas is again such a sum. Thus the diffraction pattern of a lattice is a lattice; it is in fact its dual lattice. The original lattices and their weights can be recovered from the diffraction pattern with the help of some techniques for recovering the "phase factor"; this is the experimental and theoretical framework for crystal structure analysis. However, the lattice hypothesis is not without its problems. There are crystals with extremely large repeat units, with thousands of atoms in the unit cell. There are crystals that are more or less r a n d o m stackings of two-dimensional periodic structures. There are crystals in which the lattice is disturbed by a modulation. Finding a periodic framework on which to hang these structures can be likened to the preKeplerian problem in astronomy of trying to explain planetary orbits by decorating the circle with the right number and arrangement of epicycles. No number or arrangement will be correct for the quasicrystals! The quasicrystal phenomenon shows us that a diffraction 56

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pattern can theoretically show sharp spots even if there is a single nonperiodic but well-defined geometrical pattern that gives rise to it. And although it is widely assumed that the crystal lattice is a global consequence of the play of local interatomic forces, from the standpoint of physics or mathematics this is an o p e n problem. Indeed, Miekisz and Radin have shown that generically one would expect local forces to generate nonperiodic structures [4]. In fact, one can even find crystals almost arbitrarily close to "ideal q u a s i c r y s t a l s , " in the same w a y that irrational numbers can be approximated by rational numbers. Thus the deeper question is: what local ordering properties are necessary and sufficient to produce orderly images? Two minimal properties that we might require of a locally ordered point set L C R n are discreteness and relative density: there is a minimum distance d between any pair of points of L, and a number ~ > 0 such that every sphere of radius 8 contains at least one point of L; such an L is sometimes called a Delone system. (Incidentally, Delone's name is sometimes spelled Delaunay, reflecting the fact that his forebears went to Russia with Napoleon and stayed on.) A finite region of a Delone system with no additional structure roughly describes the centers of the atoms in a monatomic gas in a closed container. Increasing the structure increases the order. Sufficient local order implies periodicity: Delone and his colleagues proved [5] that there is a number k = k(8,n), where n is the dimension of the space in which L lies, such that if the sets {x E Rn: Ix - u I ~ k} M L are congruent for each u E L, then L is an orbit of a crystallographic group. The patterns we are interested in have order somewhere between amorphous and periodic. The question is, what intermediate conditions are necessary and sufficient to ensure that L can produce a diffraction pattern with "bright spots"? For example, one condition might be local isomorphism: every finite configuration of L occurs in every bounded region of sufficient size. But although local isomorphism is present in all the examples that we know about, there is no proof that it is either necessary or sufficient. Other local ordering conditions can of course be proposed, but not much is known about their effect either. The question remains open. One obvious difficulty is that "bright spot" is not well defined. We can simplify matters by defining it to mean that there are Dirac deltas in the Fourier spectrum, weighted so that some peaks appear isolated. Then there are two cases to consider: either the entire Fourier spectrum is a set of deltas (possibly dense), or else the spectrum also contains a continuous component. But from the experimental point of view a bright spot need not represent a "real" delta; it might be due to features of the transform that closely approximate delta functions. Point sets with this property are of

theoretical as well as experimental interest (see Section 4). We can formulate these conditions more precisely. Any Delone system D has a density distribution that is a countable infinite sum of weighted Dirac deltas on the points of D; we can assume as a first approximation that all the weights are equal to 1. Then the distribution p(x) can be written Ev~o 8(x - v), where x Rn; a distribution of this form is sometimes called a Dirac comb. We are looking for Dirac combs whose Fourier transforms ~(t) = Ev~o exp(2"rrit 9 v) are closely related to Dirac combs, where by "closely related" we mean one of the following: (a) The Fourier transform is precisely a Dirac comb; such a comb is also called a Poisson comb. (An important special case is w h e n the frequencies v at which the delta functions of the Fourier transform occur have a finite basis over the integers; the original density is then said to be quasiperiodic.) (b) The Fourier transform contains a Dirac comb together with a continuous part. (c) The Fourier transform "looks like" it contains a Dirac comb but does not in fact do so; this can happen when the spectrum has a singular continuous component. Characterizing the order properties of Dirac combs satisfying (a), (b), or (c), and classifying these combs, is a central problem of quasicrystallography, and involves all of the branches of mathematics listed above. In the absence of a complete answer to our question, we study examples. The two classes of combs that have been studied in most detail are those obtained by projection, and one-dimensional combs. We will also discuss some of the relations b e t w e e n combs and tilings; some interesting recent work is discussed in Section 5.

3. Point Sets O b t a i n e d b y Projection Three years before the discovery of quasicrystals, the English crystallographer Alan Mackay [6], long an advocate for a more general crystallography, devised an ingenious experiment. He arranged for an optical diffraction mask to be constructed whose holes were located at the vertices of a tiling by Penrose rhombs (Figure 4). These filings, which are constructed by juxtaposing the r h o m b s according to strict matching rules, are nonperiodic. Yet they aren't "'disordered": one can discern a great deal of local order. Local configurations with 5-fold symmetry not only occur, they occur all over the place. Indeed, the pattern of vertices has the local isomorphism property discussed above. Moreover, the filings are self-similar. (These and other properties of the Penrose tilings will be discussed in

Figure 4. Part of a Penrose tiling by rhombs. Vertex colors and edge arrows must be matched. more detail in Section 5.) As Mackay suspected, the diffraction pattern obtained with this mask was clear and sharp, almost crystalline. And it had the crystallographically forbidden five-fold symmetry. De Bruijn's construction. Since the quasicrystal in question, i.e., the set of the vertices of the Penrose tiles, is not a lattice, how can we explain Mackay's experimental results? The necessary insight was supplied by N. G. de Bruijn, in a remarkable set of papers published in 1982 [7]. In these papers, de Bruijn gave a global method for constructing the Penrose tilings that allows us to index the vertices of the rhombs with five integer coordinates (x0. . . . . x4). Thus the vertices can be identified with a subset S of the points of the integer lattice in R5. Moreover, de Bruijn showed that 4

1~ ~,xi~4forall~

= (x0. . . . .

x4) E S .

(1)

k=O

Since this sum is also the scalar product (x0. . . . . x4) 9 (1 . . . . . 1), the points of S lie in a region M C R s which projects to a bounded interval on the line containing ~ = (1,1,1,1,1); note that this vector is the body diagonal of the unit 5-cube ~/s. The vertices of the Penrose rhombs are the projections of S onto a plane 11, one of the two invariant planes of the five-fold rotation about that diagonal, which cyclically permutes the five coordinate axes. Both of these planes are irrational: their intersections with the lattice are just {0}. The tile vertices are integral linear combinations of the projections of the five orthonormal unit coordinate vectors of R s onto 1-[. Not all vectors satisfying (1) are in S; S is the set of points M' of M whose projection onto FI• lies in the projection of ~/5 onto that subspace. (Any projection of an n-cube is a zonohedron; in this case it is a rhombic icosahedron.) Katz and Duneau [8], among others, have shown that the projection formalism greatly facilitates the THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 2, 1990

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Figure 5. A plane tiling by three kinds of rhombs, projected from R7. Courtesy of Andr4 Katz.

computation of Fourier transforms. The density function of the set S is the product of the density function of the integer lattice in Rs and the characteristic function of M'. Thus, since the Fourier transform of a product of two functions is the convolution of the Fourier transforms of the individual functions, and since the Fourier transform of the projection of the density function of S is the restriction of the Fourier transform of that function to H, we can compute the diffraction pattern of the Penrose vertices. (That all of this can be made rigorous has been shown, by rather different arguments, by de Bruijn [9] and by Porter [10].) The Fourier transform is a countable sum of delta functions at a dense set of points in the plane; thus the set of vertices is a Poisson comb. Although the delta functions are dense, we see bright spots in the diffraction pattern, because the amplitude of the transform attains local maxima at isolated points, and at most other points is very small. The general case. The Penrose tilings are of course very special. To what extent does the property of being a Poisson comb depend on their remarkable properties? Curiously, this dependence is not very strong. For example, while it is easy to construct Poisson combs by projection, as far as we know most of them are not self-similar. (If we translate M in II3_, the projected pattern will include local vertex configurations that are forbidden by the matching rules of the 58

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Penrose tiles.) Or, we can carry out the analogous construction in R', producing plane point sets that are the vertices of tilings with local n-fold symmetry for which no matching rules are k n o w n (see Figure 5). All of these patterns have the local isomorphism property, however. More generally, let A be a lattice in R n, and let IIk be any irrational k-dimensional subspace of R" (again, irrational means that IIk N A = {0}). First, let us see under what conditions we can obtain a Delone system in Hk by projection. Since IIk is irrational, the orthogonal projection of all of A onto IIk will be nonperiodic, but it will also be dense. We need to find a subset S of A that projects to a discrete set (relative density is guaranteed by the fact that A is a lattice). We know that there is a minimum distance d between points of A: if -~ and ~ are two vectors of A, then I ; - ~J t d. We can d e c o m p o s e w = x - y m t o l t s l l k a n a t l k 3_ . . 3_ components w k and w k. If we resist that twkl < d - e for some e > 0, then IWkl >! d. This means that we can obtain a discrete set of points in IIk by requiring that 9 . . .J. the image of the set S C R" under projection to IIk lie 9 . 3_ . m an appropriately chosen compact set T C Flk; T Is sometimes called the window for the projection. Thus S lies in the cylinder M = T • I] k C R n. The computation of the Fourier transform then proceeds as in the examples above. The projected set always turns out to be a Poisson comb9 There are many variations on the projection theme 9 The window need not lie in the orthogonal complement of IIk; the vector space need not be Euclidean. One technique used quite extensively at the moment is to try to replace A by a periodic set of connected surfaces in R', and to consider the intersection of H k with these surfaces9 (Again, de Bruijn has provided a firm mathematical basis for much of this, in a different language [11].) It is true that if one has a Poisson comb and its Fourier transform's delta functions have a finite basis, and if one knows the full complex amplitudes of these deltas, then it is possible to reconstruct a periodic density in R" and a plane IIk such that the restriction of the density in R" to IIk will give the delta functions of the density in R k. However, it is not at all obvious that there are densities in R" that consist of "surfaces" of any particular smoothness. Also, although using arbitrary surfaces, rather than those from projecting a lattice, gives a broader class of Poisson combs, it does not always give a noticeably broader class of diffraction patterns, because these combs may differ from the lattice-projection ones only in their intensities and phases, as de Bruijn has noted

[111. A word about symmetry. The most striking thing about the diffraction pattern of the Penrose vertices is its five-fold rotational symmetry; quasicrystals might never have been noticed if this symmetry had not

been observed. Indeed, successful crystal structure determination d e p e n d s on finding directions of high s y m m e t r y so that bright spots will appear in the diffraction pattern. The symmetry we observe in the Fourier transform of a projected p a t t e r n d e p e n d s on the s y m m e t r y group G of the lattice A and on the choice of II k. G is a semidirect product of Z" and a finite subgroup P C O(n), where P is the stabilizer of 0 E A. If Ilk is invariant under P, or u n d e r a subgroup of P, the Fourier transform will reflect this. This leads us to the important and interesting problem of determining which lattices in R n have invariant subspaces of whatever dimension, and h o w crystallographic groups built on these lattices act on those subspaces. In short, the projection method has o p e n e d an interesting chapter in n-dimensional crystallography. To date, those lattices for which G is or contains the icosahedral group have been studied in the most detail (see, e.g., [12]), because they arise in the theory of the three-dimensional Penrose tiles (see Section 5) and in the interpretation of diffraction patterns of real quasicrystals. But from our point of view, it is the bright spots that are f u n d a m e n t a l , not s y m m e t r y per se. Since bright spots are theoretically present in every projected pattern, we k n o w that their occurrence is not d e p e n d e n t on rotational symmetry. Indeed, it seems that rotational symmetry has nothing a priori to do with our problem, except t h a t w h e n we find noncrystallographic rotational s y m m e t r y in a diffraction pattern, we k n o w that it was produced by a nonperiodic Dirac comb. On the other h a n d , bright spots in a diffraction pattern indicate some sort of long-range order or generalized symmetry. This brings us back to the questions raised in Section 2. 4. Order on the Line The one-dimensional case is the most tractable; here we find examples of all three types of ordering for

n o n p e r i o d i c point sets. We will describe a few of them. Sequences with average lattices. The standard example of a one-dimensional quasicrystal is the "Fibonacci" sequence of points u, = n + (~ - 1)[n&],

where "r = (1 + V5)/2 is the golden n u m b e r of classical and m o d e r n fame, and [x] is the greatest integer function. (We will explain the relation of this sequence to the classical Fibonacci sequence below.) This sequence can be obtained by projection from R 2 to a line 11 with slope 1/'r. Let us consider the more general case in which the line has slope % where o~ is an irrational number. Using as our w i n d o w the projection of a unit 3_ square of the integer lattice A onto 11 , the cyhnder M is the strip b o u n d e d by the lines y = c~x and y = o~x + + 1 (Figure 6) and M' = M n A. The points in this strip have coordinates (x,y) = ([n/(oL + 1)], n - [n/(o~ + 1)]), and project onto the points (x + o~y)-ff,where -ff is the vector (1,~)/(1 + c~2) along II. T h u s the projected points form a sequence p, = om - (~ - 1

9

9

9

9

9

9

9

9

D

9

9

9

9

9

9

9

9

e

9

9

i

9

9

9

! 9

9

9

8

-

9

o

9

9

9

9

9

e

9

9

9

9

9

9

9

9

,

(3)

and p , + 1 - P, = 1 or c~. W h e n c~ = 1/'r, we obtain the sequence above (if we multiply everything by 1/'r). The methods of the preceding section can be used to show that all of these sequences are Poisson combs. The sequences obtained in this w a y are often called " o n e - d i m e n s i o n a l q u a s i c r y s t a l s . " But t h e y do n o t really illustrate the quasicrystal p h e n o m e n o n , because in fact these sequences are one-dimensional modulated lattices. Modulated crystals were k n o w n long before the quasicrystals were discovered, and have been intensively studied for m a n y years.

~

9

(2)

9

y=Coc+~+l

a

9

9

9

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Figure 6. A tiling of the line obtained by projection from R2. All filings of this type have average lattices. (Adapted from Ref. [8].) INTELLIGENCER

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In what sense are these sequences modulated lattices? Using the equality [x] = x - {x}, where {x} is the fractional part of x, we have pn = ~n - (~ - 1)

---.n ot+l

n

+ (a - 1

~+1

+ (or - 1)

or in the case of the Fibonacci sequence (2), u n = (2 - 1/'r)n + (~ - 1){n/,r}. Thus we see that {Pn} is built u p o n the one-dimensional lattice of points of the form n(ot2 + 1)/(or + 1) for n E Z, deviating from it by an a m o u n t which is at most [or - 1[. This property, of having a limiting average spacing and a b o u n d e d modulation away from the lattice with this spacing, is called having an "average lattice." O n e can c o m p u t e the Fourier t r a n s f o r m s of sequences of type (4), or indeed of any sequence of the form v n = c~n +

[3{~/n}

(5)

in a straightforward m a n n e r [13]; the sequences are always Poisson combs. In fact, a n y sequence w h o s e elements are of the form a n + g ( n ) , where g ( n ) is periodic or almost periodic, is also a Poisson comb. For appropriate choices of parameters, these sequences will be nonperiodic; it is not k n o w n w h e t h e r they can be obtained by projection. It is not k n o w n which of the sets obtained by projection onto subspaces of dimension greater than 1, considered in Section 2, have average lattices. However, some of them do. Duneau and Oguey have recently s h o w n [14] that the set of vertices of a tiling obtained by projection from R 8 to R 2 has an average lattice; the construction applies to certain other tilings as well. S e q u e n c e s o b t a i n e d b y s u b s t i t u t i o n . If we interpret the letters a and b to be line segments of lengths ~"and 1, respectively, t h e n the sequence u n discussed above is the limit lim Tn(w0), n-~.o0 where w0 is a w o r d of the two-letter alphabet {a,b} and T is the map, or substitution rule, T(a) = ab,

T(b) = a.

W h e n w0 = b, t h e n the length of the w o r d Tn(w0) is where F n is the n th Fibonacci n u m b e r (F0 = 0, F1 = 1, Fn+ 1 = Fn + F n _ l ) ; Un is sometimes called a Fibonacci sequence, although the classical Fibonacci sequence is Tn(a). It is not k n o w n which of the more general sequences v n discussed above can be produced by substitution rules, but obviously, more general substiFn+ 1,

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tution rules can be used to produce sequences on the line. W h a t can be said about the Fourier transforms of substitution sequences? To prove that a Dirac comb is a Poisson comb requires k n o w i n g the whole Fourier transform; the only way this has been accomplished so far is to show that the density p is the sum of delta functions on a lattice m o d u l a t e d by a periodic or almost periodic function, or that p is obtained by slicing t h r o u g h a periodic density in a higher-dimensional space. On the other hand, one can show that a density h a s p r o p e r t y (b) as follows: The d e n s i t y p(x) = Y'n=0,1.... ~ ( x - Vn) h a s F o u r i e r t r a n s f o r m 15(q) = E n e x p ( 2 ~ r i q 9 Vn). For any frequency q, the sequence of partial sums {Enn=0exp(2"rriq 9 Vn)} is b o u n d e d by N + 1. If we can show that for some q the sequence grows like c N for some positive c, t h e n asymptotically the s u m is a Dirac delta. It is possible to use this technique for some substitution sequences. A n y composition rule T acting on an alphabet of n letters can be represented by an n x n matrix M with nonnegative integer entries. If there is a k ( Z such that all the entries of M k are positive, then the PerronFrobenius theorem tells us that M has a simple eigenvalue 0 that is greater in absolute value than all the others. Bombieri and Taylor [13] showed that if [0[ > 1, while all its conjugates have m o d u l u s less than one (i.e., if 0 is a Pisot-Vijayaraghavan, or P-V, number) t h e n the Fourier transform of the sequence can be c o m p u t e d as a limit of the Fourier transforms of the words Tn(w0). The transform contains a Dirac comb, because there are frequencies (forming a dense set) for which the sequence of transforms grows like N. In fact, all of these sequences lie in sets that can be obtained by projection. Every substitution T on a finite alphabet gives rise to a topological dynamical system. By assigning appropriate lengths to the letters of the alphabet, a fixed point of a sequence Tn(w0) can be interpreted as the list of the successive differences of an increasing sequence of real numbers, and we can s t u d y the order properties of such sequences. The dynamical systems associated with substitutions of constant length, and their spectra, have been studied by Queff61ec [15]. (Note, however, that the Fibonacci sequence is n o t of constant length.) Other one-dimensional Poisson combs. Aubry, Godr~che, a n d Luck [18] studied a family of sequences that, for some choices of parameters, appear to be Dirac combs of type (c). Let h be a subinterval of (0,1) and 00 be any positive real number. Consider the sequence Wn of O's and l's obtained by setting w n = 0 if {too} ( zi and w n = 1 otherwise. There are two ways to build a sequence of points on the line from the sequence Wn. One can start with a one-dimensional lattice w h o s e nodes are 1o-

cated at the points no~, n ~ Z and then omit those nodes for which the corresponding wn is equal to 0. In this way we obtain a lattice with vacancies, which can be shown to be a Poisson comb. Kesten's theorem [17] asserts that this sequence has an average lattice, in the sense defined above, if and only if ~ -= r00 rood(l) for some r ~ Z. Thus there exist Poisson combs with no average lattice! The second way to build a sequence is to choose two unequal lengths l1 and 12 and let u 0 = 0, Un+ 1 - un = 11 + (12 - ll)Wn. In this case, it may happen that the Fourier transform has a singular continuous component, i.e., the sequence u n is a Dirac comb of type (c). As far as we know, property (c) has never been completely established for any example. However, there are cases [18] where the possibility of Dirac peaks can be eliminated using the procedure appropriate to case (b), and the possibility of the transform being absolutely continuous can be essentially eliminated by numerical calculation. The spectrum should therefore contain a singular continuous part; numerical calculations then show it "looks like" a Dirac comb. 5. T h e T i l i n g C o n n e c t i o n

Crystallographers have used tilings as models for crystal structures for centuries. For example, the lattice can be regarded as a framework for the partition of space into congruent parallelepidal tiles or "unit cells." These fictitious boxes in turn contain congruent, real, atomic configurations. The diffraction patterns of the first quasicrystals looked remarkably like the one obtained earlier by Mackay. Thus it was natural to ask whether the threedimensional analogue of the Penrose filings might be a model for quasicrystals, with the two kinds of tiles playing roles analogous to the unit cells. Further experimental work has shown that this is not the case (see, e.g., [19]). In any case, the tiles in a nonperiodic tiling are not analogous to the unit cells of periodic patterns, although it is frequently asserted that they are. There are infinitely many ways to choose the shape and position of the unit cell for a periodic crystal, all equally valid from the abstract point of view. In contrast, in the few cases in which nonlattice point sets can be associated with tilings by copies of one or a few shapes, the choice of cells is usually unique, and it is by no means clear what the relation between the transforms is when masses are placed at vertices or in the tile interiors. In fact, it is not dear what aspects of a real structure the tiles in a nonperiodic tiling might represent. Like the Big Dipper and other stellar constellations that one learns to identify as a child, the tiles sometimes appear to be highly artificial from a physical point of view, even w h e n they are convex. For example, the minimum distance between vertices in a tiling by Penrose

Figure 7. A self-similar tiling for which no matching rules are known to exist. (From

Ref. [20].) rhombs is the short diagonal of a thin rhomb; in a reasonable structure model one w o u l d expect nearest neighbors to be linked in some way. Still, the possible connection between tilings and quasicrystal structure continues to be studied, partly because the things help us to visualize some kinds of nonperiodic order (this is w h y four of our eight illustrations are filings.). It is easy to produce nonperiodic tilings by the projection method. In the special case where T is the projection of the unit n-cube %, the projected points are the vertices of nonoverlapping projections of the k-dimensional faces of %. For suitably chosen subspaces, the number of distinct tile shapes (prototiles) will be small (O(n)). Thus we can construct many interesting examples. What do nonperiodic tilings have to teach us? The Penrose tilings have three important properties: (1) they have matching rules that force nonperiodicity, (2) they can be obtained by a substitution process and they are self-similar, (3) they have strong local order (in fact they are quasiperiodic). Surprisingly, it appears now that these properties are independent to some extent. There are substitution-produced tiings with matching rules that are not self-similar (several examples are s h o w n in [20]). Figure 7 shows a tiling that is self-similar but for which no matching rules seem to exist; recently it has been shown that this tiling is not quasiperiodic (see below). THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 2, 1990 6 1

Tilings built with the tiles shown in Figure 8, using matching rules, are quasiperiodic but no substitution rule has been found for them.

Matching rules. Why are matching rules of interest in the study of quasicrystals? Evidently they are not needed in order for a tiling to be a Poisson comb. Their importance lies instead in our feeling about what features a good m o d e l should have. The projection method says nothing about h o w the quasicrystal g r o w s - - w h y the atoms order themselves in such a pattern. Some sort of local forcing rules would seem to be an important part of a good model for quasicrystals, since they are an analogue of the local bonding rules that presumably determine the structure. The matching rules discovered by Penrose and by Ammann ([20]) were found by trial and error. Is there a more systematic way to do this? De Bruijn showed that his indexing system for the Penrose vertices leads to an u n a m b i g u o u s reconstruction of the Penrose rules, but his arguments do not apply if the set M is translated in R 5. Neither has it proved possible to apply it to any of the other plane tilings projected from R ". This does not mean that no matching rules exist in t h e s e cases. For e x a m p l e , A m m a n n has f o u n d matching rules for certain tilings of the plane by squares and rhombs, projected from R s (again, see It seems to us no more appropriate to define quasicrystals at this stage of our knowledge than to cling to the definition of a crystal as a periodic structure. [20]). But as de Bruijn points out [21], although Ammann's rules are expressed locally, the property of an unmarked tiling to be Ammann-markable is not a local property. More recently, some progress has been made. Using homologous arguments, Katz has developed a method for decorating the tiles of certain projected tilings [22]. He applied it to the "three-dimensional Penrose tiles," thus proving that these tiles can be equipped with matching rules that force nonperiodicity (Figure 8). However, the construction is not a simple one: when decorations are taken into account the two rhombohedra fall into twenty-two classes. Recently, Danzer has announced the discovery of a set of four marked tetrahedra [23] that tile R3 only nonperiodically. Although the method by which he found them appears to be less systematic than Katz's, it is of interest because the number of prototiles is small.

Self-similarity. The self-similarity of the Penrose tilings is one of their most remarkable features. But 62

THE MATHEMATICAL 1NTELLIGENCER VOL. 12, NO. 2, 1990

until very recently self-similar tilings have been almost as hard to find as matching rules. In the first place, to be self-similar, a tiling must be a geometric realization of a "fixed point" of a substitution map. Any primitive matrix defines a substitution map, but we do not know of any theory that tells us which substitutions can be realized as tilings. Even when such a realization exists, the tiling need not be self-similar in the sense that the larger configurations into which the tiles are grouped by the action of the substitution map are geometrically similar to the original tiles. Conversely, given a tiling (such as that in Figure 5) it may be very difficult if not impossible to determine whether it is invariant under some substitution map T. Recently, Thurston has developed a method for associating selfsimilar tilings with fractal tile boundaries to a class of algebraic integers [24]. Substitution rules are implicit in the method, but it is not yet clear to us how to e x tract them. Nevertheless, tilings invariant under primitive substitution maps are of interest in our context because they necessarily have the local isomorphism property. Moreover, we can sometimes use the substitution map to prove that a tiling is nonperiodic, a property that may not be obvious. Notice, for example, that the use of matching rules does not guarantee that a tiling is nonperiodic; some other argument must be invoked. Two different arguments can be used to establish the nonperiodicity of a tiling with the substitution property. First, if the grouping of tiles into larger ones is u n i q u e , the tiling has a hierarchical structure that must be preserved by any translation. But this is impossible, since repeated iteration of this grouping implies that at some hierarchical level the inradius of the tiles will be larger than any specified translation length. The other argument might be called a "ratio test" for nonperiodicity. It involves the eigenvectors of the substitution map. Let T be any primitive, integer n x n matrix, and let {a1. . . . . an} be any finite alphabet. A word w of this alphabet contains x i copies of the letter a i. We can think of -~ = (xv . . . , xn) as a vector of the integer lattice in "configuration space." Then -~T is another vector in this space; its components are the numbers of copies of each of the letters after one application of T. The components of a left eigenvector corresponding to its leading eigenvalue 0 are the relative numbers of the different letters in the infinite word produced by iterating T. We can now state the ratio test. If T has an eigenvalue 0 which is a P-V number, then for any initial configuration vector T0, the sequence 0-n~0 T" will converge to a left eigenvector of 0. If the components of this eigenvector have irrational ratios, then the tiling will be nonperiodic, since in a periodic tiling the relative numbers of kinds of prototiles is given by the numbers in single repeat unit.

If T acts on a tiling, then the prototiles of the tiling play the role of the letters of an alphabet. We assume that they are arranged in such a way that each application of T effects a grouping of the tiles into larger tiles. These tiles need not be similar to the original ones, but if they are, then the relative volumes of the n prototiles after each application of T are the components of a right eigenvector of 0. This gives us a way to decide whether a tiling produced by substitution is self-similar; the Penrose tiles pass the test. Which substitution-invariant tilings produce diffraction patterns with bright spots? There is no definitive answer yet. We have seen that if the tiling can be obtained by projection, then its set of vertices is a Poisson comb. Recently Godr6che and Luck [25] have extended the Bombieri-Taylor method for computing the Fourier transforms of substitution sequences to tilings of the plane by assigning masses to the tiles themselves and expanding the definition of T to take into account the geometry of configurations as well as the numbers of tiles in them. They then showed that Fourier transforms of this density distribution contain Dirac combs even w h e n the matching rules are relaxed. During the Les Houches conference, Godr~che succeeded in showing that the tiling of Figure 7 fits into case (b) (but the possibility that the spectrum also contains a continuous component has not been ruled out). It is especially interesting that in this case there is no finite basis for the frequencies of the delta functions of the Fourier transform, so that the tiling is not quasiperiodic [25]. The local ordering properties of the Penrose tiles are discussed in [20], so we will not go into detail here. They include local isomorphism, and the fact that the number of different configurations within any finite radius is bounded and grows slowly as the radius increases. These properties hold for all projected and substitution tilings. But it remains an open question to what extent these properties, independently of projection and substitution, can account for the tilings' Fourier transforms. Local order.

6. A W o r d a b o u t D e f i n i t i o n s

We mentioned at the beginning that we have posed Question 3 more generally than is usually the case. We did not mention, but many readers will have observed, that we have offered no definition of "quasicrystal." In fact, most other writers define quasicrystals to be projected (or sliced) structures. There may be some justification for this. As we have shown, the projection/slicing method does produce an extremely large class of Poisson combs. Moreover, the models based on this approach are in very good agreement with experiment. But then, experimentally, it may be impossible to distinguish Poisson combs

49

Figure 8. The three-dimensional analogue of the Penrose rhombs are two rhombohedra. When decorated with matching rules according to Katz's scheme, the rhombohedra fall into twenty-two classes. Nets for eight of the rhombohedra are shown here; the others can be generated from this set (see Ref. [22]).

from the other two cases discussed in Section 2. It seems to us no more appropriate to define quasicrystals at this stage of our knowledge than to cling to the definition of a crystal as a periodic structure. Question 3 is nontrivial mathematically, and it is also nontrivial philosophically. The high-dimensional formalism is only a stop-gap to be used until we understand h o w quasicrystals grow. String theory notwithstanding, it is reasonable to assume that real quasicrystals, like real periodic ones, grow in R3, not in Rn. We need a theory that explains how the patterns that we are interested in can be generated at the local level; it is not clear to what extent the deterministic models we have described are physically meaningful. Modeling g r o w t h may require a combination of matching rules, modulations, understanding "the sociological behavior of large groups of atoms" [21], and possibly other ideas. It is too early to know what class of patterns will achieve this. In our view, the definition of quasicrystal should be left open until the fundamental questions have been answered.

Acknowledgments: We would like to thank H. S. M. Coxeter, N. G. de Bruijn, and C. Godr~che for helpful comments on this article. Also, the second author would like to acknowledge the support of NSF and AFOSR. THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 2, 1990 6 3

References 1. D. Shechtman, I. Blech, D. Gratias, and J. Cahn, Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett. 53 (1984), 1951-1954. 2. J. Cahn and J. Taylor, An introduction to quasicrystals, Contemporary Mathematics 64 (1987), 265-286. 3. L. Pauling, So-called icosahedral and decagonal quasicrystals are twins of an 820-atom cubic crystal, Phys. Rev. Lett. 58 (1987), 365-368. 4. J. Miekisz and C. Radin, Are solids really crystalline?, Phys. Rev. B 39 (1989), 1950-1952. 5. B. Delone, N. Dolbilin, M. Shtogrin, and R. Galiulin, A local criterion for the regularity of a system of points, Reports of the Academy of Sciences of the USSR (in Russian) 227 (1976). (English translation: Soviet Math. Dokl. 17 (1976), 319-322.)

6. A. Mackay, Crystallography and the Penrose pattern, Physica 114A (1982), 609-613. 7. N. G. de Bruijn, Algebraic Theory of Penrose's nonperiodic tilings of the plane, Kon. Nederl. Akad. Wetensch. Proc. Ser. A 84 (Indagationes Mathematicae 43) (1981), 38-66. 8. A. Katz and M. Duneau, Quasiperiodic patterns and icosahedral symmetry, Journal de Physique 47 (1986), 181-196. 9. N. G. de Bruijn, Quasicrystals and their Fourier transform, Kon. Nederl. Akad. Wetensch. Proc. Ser. A 89 (Indagationes Mathematicae 48) (1986), 123-152. 10. R. Porter, The applications of the properties of Fourier transforms to quasicrystals, M. Sc. Thesis, Rutgers University, 1988. 11. N. G. de Bruijn, Modulated quasicrystals, Kon. Nederl. Akad. Wetensch. Proc. Ser. A 90 (Indagationes Mathematicae 49) (1987), 121-132. 12. L. Levitov and J. Rhyner, Crystallography of quasicrystals; applications to icosahedral symmetry, J. Phys. France 49 (1988), 1835-1849. 13. E. Bombieri and J. Taylor, Quasicrystals, tilings, and algebraic number theory: some preliminary connections, Contemporary Mathematics 64 (1987), 241-264. 14. M. Duneau and C. Oguey, Displacive transformations and quasicrystalline symmetries, J. Physique (to appear, Jan. 1990). 15. M. Queff61ec, Substitution Dynamical Systems--Spectral Analysis, Lecture Notes in Mathematics 1294, New York: Springer-Verlag, 1987. 16. S. Aubry, C. Godr~che, and F. Vallet, Incommensurate structure with no average lattice: an example of a onedimensional quasicrystal, J. Physique 48 (1987), 327-334. 17. H. Kesten, On a conjecture of ErdOs and Sziisz related to uniform distribution mod 1, Acta Arithmetica 12 (1966), 193-212. 18. S. Aubry, C. Godr6che, and J. M. Luck, Scaling properties of a structure intermediate between quasiperiodic and random, J. Stat. Phys. 51 (1988), 1033-1075. 19. M. La Brecque, Opening the door to forbidden symmetries, Mosaic (National Science Foundation) 18 (Winter 1987/8), 2-23. 20. B. Grfinbaum and G. Shephard, Tilings and Patterns, San Francisco: W. Freeman, 1987. 21. N. G. de Bruijn, private communication; see also his preprint "Remarks on Beenker patterns." 22. A. Katz, Theory of matching rules for the 3-dimensional Penrose filings, Commun. Math. Phys. 118 (1988), 263288. 23. G. Danzer, Three-dimensional analogs of the planar Penrose tilings and quasicrystals, to appear in Discrete

Mathematics. 24. W. Thurston, Groups, filings and finite state automata, Summer 1989 AMS Colloquium Lectures (preprint). 25. C. Godr~che and J. M. Luck, Quasiperiodicity and randomness in tilings of the plane, J. Stat. Phys. 55 (1989), 1-28. 26. C. Godr~che, The Sphinx: a limit-periodic tiling of the plane, preprint.

Department of Mathematics (Senechal) Smith College Northampton, MA 01063 USA Department of Mathematics (Taylor) Rutgers University New Brunswick, NJ 08903 USA 64

THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 2, 1990

Steven H. Weintraub* For the general philosophy of this section see Vol. 9, No. 1 (1987). A bullet (e) placed beside a problem indicates a submission without solution; a dagger (t) indicates that it is not new. Contributors to this column who wish an acknowledgement of their contribution should enclose a self-addressed postcard. Problem solutions should

66

THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 2 9

be received by 1 August 1990. This column will have a new editor as of Vol. 13, No. 1 (I991). Problem solutions and other correspondence should be directed to David Gale, Department of Mathematics, University of California, Berkeley, CA 94720 USA.

1990 Springer-Verlag New York

THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 2, 1990

67

Chandler Davis*

Mathematica, a Program for Various Work Stations and Personal Computers Wolfram Research, Champaign, IL

Mathematica: A System for Doing Mathematics b y Computer b y Stephan Wolfram

framers of Mathematica thereby ensure that all Mathematica's computations will be machine independent while allowing users to take advantage of whatever computer/operating system pair they are running under. Here's a typical prompt: In[23]: =

Redwood City, CA: Addison-Wesley, 1989; xviii + 749 pp.; US $44.25

Reviewed by Alan Hoenig "The ultimate goal of mathematics is to eliminate all need for intelligent thought."** Do you agree with this thought? Whether yea or nay, you will find the program Mathematica an intriguing one, as it is a system for doing mathematics; you will find yourself able to exercise many of its talents without much need for thought. It is sufficiently easy to use, robust, and so all-embracing that it seems a significant step towards this ultimate goal. Let us look further.

Mathematica numbers and remembers your input; this prompt is the twenty-third one in this session. We use a natural one-dimensional input language to enter some m a t h e m a t i c s for evaluation. Following this prompt, we might enter Integrate[x^2 Sqrt[ 1 - x^2], x] as instructions to evaluate the indefinite integral fx2V1 - x2 dx. Mathematica responds with 0ut[23] -

How Mathematica Works Once Mathematica is started, it operates by prompting you for a line of mathematics, which it examines (for syntax) and evaluates. It returns the results of the evaluation to you. Mathematica accepts a rich variety of mathematics at the prompt. Mathematica will work on a wide variety of computers and to do that it is organized in two p a r t s - - t h e kernel and front end. The * Column editor's address: M a t h e m a t i c s D e p a r t m e n t , University of Toronto, Toronto, Ontario MSS 1A1 C a n a d a ** Ronald L. G r a h a m , D o n a l d E. K n u t h , a n d O r e n Patashnik, Concrete Mathematics, Reading, MA: A d d i s o n - W e s l e y P u b l i s h i n g Co. (1989), 56.

ArcSin[x] x( - 1 + 2 x 2) Sqrt[ 1 - x 2] 8 + 8

These two exchanges illustrate major aspects of Mathematica syntax. All reserved keywords begin with capitals (so as to distinguish them from variables and procedure names we might create) and are always written out in full. (We may define abbreviations if we wish.) Square brackets are reserved for arguments, and multiple arguments are listed in a sequence. Round parentheses are used only for grouping and for multiplication. The two arguments of Integrate specify the integ r a n d a n d variable of integration. We indicate multiplication by means of a space between adjacent factors, a refreshing change from the usual asterisk (although Mathematica does accept that as well). If we respond with

THE MATHEMATICALINTELLIGENCERVOL. 12, NO. 2 9 1990Springer-VerlagNew York 69

Plot[ I/Sin[x], {x,-5,5}]

_J -4.

9' '

-5

2.

-10

Figure 1. A Mathematica plot of y = 1/sin(x) for x( I-5,51.

In[24]: -- N[%,40]/. x---> .5 then Mathematica evaluates this expression at x = .5 to 40 places, and displays it to us. The percent symbol is a convenient reference to the immediately preceding result. There are several other ways we can ask Mathematica to display output. We can request that Mathematica use the linear input form that Mathematica reads. We can request that Mathematica format the result so that it can be used by programs in C or Fortran. (Mathematica can save this result in files that we can incorporate in program sources.) Finally, Mathematica knows enough of the typesetting language TEX to deliver its output in a form that TEX can use_ You'll be disappointed if you expected the result to involve genuine square roots, integral signs, and the like. Personally, I'm glad that Mathematica doesn't waste its time with WYSIWYG displays. Converting Sq~[ 1 - x] into Vi- - x is one of the few things left that I can do with little trouble. Furthermore, such effort is non-trivial. Don Knuth's TEX program, approximately 20,000 lines of well-crafted code, addresses just this problem. TEX is available on virtually any computer that runs Mathematica, and since we can feed Mathematica output into TEX, it makes sense not to add this overhead to Mathematica's burden. Where did this program come from? Stephen Wolfram spearheaded the team of eight developers who wrote and debugged the 150,000 lines of C code that comprise Mathematica's kernel. The front end for the Macintosh required an additional 50,000 lines. Wolfram is a recipient of a Macarthur "genius" award. At the time he received it (in 1981), he was the youngest person to be so honored. One eagerly explores Mathematica; can one discern the stamp of a certified genius in this product? Mathematica is a significant achievement, and Wolfram attracts so much of the limelight that his seven co-developers rarely receive the credit 70

THE MATHEMATICALINTELLIGENCERVOL. 12, NO. 2, 1990

they deserve. (For the record, these others are Daniel Grayson, Roman Maeder, H e n r y Cejtin, Theodore Gray, Stephen Omohundro, David BaUman, and Jerry Keiper.) The developers explicitly acknowledge influence from several different software sources, and ideas from them appear somewhere in Mathematica: interactive numerical languages such as BASIC; interactive numerical systems such as MathCAD, MATLAB, and TK!Solver; algebra systems such as Macsyma, Maple, Schoonschip, Scratchpad, and SMP; interpreted graphics languages such as PostScript; numerical and symbolic list manipulation languages such as APL and LISP; and structured programming languages such as C and Pascal.

What Else D o e s It Do? The show-stopper in any discussion of Mathematica is its graphing ability. Not only is two-dimensional plotting possible, but a wide variety of 3D plots--contour plots and true 3D plots--are available. It's possible to choose the mesh density, the "lighting," the viewpoint for three-dimensional plots, and many other parameters. Mathematica knows how to graph across most singularities, so that naive users who ask Mathematica to Plot[ 1/SinEx], {x, - 5, 5}] will get their graph (as in Figure 1), together with a warning (ignorable) about the problems Mathematica has crossing singularities. (Here's another syntax lesson. Curly brackets are always used to define lists. Most plot commands expect a 3-element list as the second argument. The first item is the variable to plot, and the final two are the endpoints of the domain.) Wolfram Research, Mathematica's parent company, has chosen the POSTSCRIPT page description language for these graphic constructions, and this ensures their display on a wide variety of readily available printers. The graphs in this article were produced by Mathema-

Plot[ Floor[x], {x,-6,6}]

I _~.

_~.

_~.

I

,

F4'.

Figure 2. A plot of y = [x]--a continuous function?

~.

tica and printed on an Apple LaserWriter printer. (Each figure includes the Mathematica command line to generate that plot.) Graphing, though, is only a small portion of Mathematica's repertoire. Here, for the record, is a reasonably complete listing of its abilities: Arithmetic and Numerical Calculations. Mathematica can compute numerical results to arbitrary precision. It recognizes indeterminacies and directed infinities. It can generate pseudorandom numbers. It possesses number-theoretic functions to yield k modulo n, the quotient of m and n, the greatest common divisor and the least common multiple; it factors integers, searches for the kth prime number, and works with the power modulus, the Euler totient function, the M6bius )~function, the divisor function crk, the Jacobi symbol, the extended GCD, and the reduced lattice basis for a set of integer vectors. Combinatorial functions include the factorial, the double factorial, the binomial and multinomial coefficients, Bernoulli numbers and polynomials, Euler n u m b e r s and polynomials, Stirling numbers of the first and second kinds, and two partition functions. Transcendental and Special Functions. All the usual elementary functions--exponential, trigonometric, hyperbolic functions, and their i n v e r s e s - - a r e built into Mathematica. (Mathematica knows about branch cuts in complex-valued functions.) It recognizes Legendre polynomials, spherical harmonic functions, Gegenbauer polynomials, Chebyshev polynomials, Hermite polynomials, Laguerre polynomials, and Jacobi polynomials, as well as Airy functions, Bessel functions, beta functions, error functions, exponential integrals, gamma functions, all kinds of hypergeometric functions, Legendre functions, Lerch's transcendent 9 functions, the logarithmic integral, the Pochhammer symbol, the digamma function, the nth derivative of the digamma function, the polylogarithmic function, and the Riemann zeta function and generalized zeta function. Mathematica knows all about elliptic functions and elliptic integrals (including the arithmetic-geometric mean of two numbers).

A simultaneous plot of 2 Bessel functions. We may use different levels of gray to enhance the graph. Mathematica Does Calculus. Mathematica can take all derivatives (partial, multiple, total . . . . ) symbolically. If possible, Mathematica will integrate a function and give the result in closed form, but numerical values of definite integrals can always be determined. The program will find the power series expansion of functions to a specified order, can manipulate series algebraically, and can invert series. Linear Algebra. Mathematica will construct matrices and knows matrix multiplication and inversion. It can find determinants and list the minors of a matrix. It can solve linear matrix equations. It will return the eigenvalues and eigenvectors of a system. Mathematica knows tensors. Numerical Operations on Data and on Functions. Mathematica can fit curves to lists of data. It can take Fourier transforms of data or invert a transform. In addition to this repertoire, Mathematica can recognize and pay attention to patterns and sets of rules. As a result, it's likely that you can create rules for your favorite mathematics and ask Mathematica to read them in w h e n needed. These collections of rules are called packages, and m a n y come with the program. One package, for example, enables Mathematica to work with Laplace transforms. (There are surprising gaps, though. Mathematica has no ready-made facility for analytic geometry, for example, although these rules are very easily created.)

Polynomial and Rational Functions. Mathematica can expand or factor any polynomial expression, and can pick out particular coefficients of these expressions. The program does polynomial division and a wide variety of additional operations on rational functions.

Mathematica's P r o g r a m m i n g L a n g u a g e

Manipulating Equations. Mathematica can solve one equation or a system of equations. If possible, the solutions will be in closed form. Otherwise, they are numerical to whatever precision is demanded. Mathematica can often solve an equation system subject to some side condition.

My favorite part of Mathematica is its programming language. It contains all the usual aspects of any highlevel programming language--decision statements, looping statments, and extensive input-output operations. It is sufficiently rich so that it's fun to solve problems with it. Here's one project that I was sucTHE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 2, 1990 7 1

h(x) = ~cos(x), x 1> 0

ContourPlot[ Sln[x] Sln[y], {x,-2,2}, {y,-2,2}, ContourLevels->40]

[x + 1, x < 0.

Mathematica will graph and evaluate such a function

1.5

properly but has to be told explicitly how to differentiate and integrate it. Mathematica rules for patterns make the program seem almost capable of learning. Examine this exchange. If you tell Mathematica that a function called log satisfies the pattern

i. 0.5 O.

-1.

log[x_y_] = log[x] + log[y]

-1.5 -2.

-2. -1.5 -i. -0.5

O.

0.5

I.

1.5

2.

then it will "know" that log[a ~2b ^3] = log[a ^2] + log[b A2].

A contour plot of z = sin(x) sin(y).

If you further explain to Mathematica that cessfully able to tackle. Given an expression and a

Mathematica rule (or procedure) for evaluating that expression, Mathematica will set the expression for me in

log[x_^r_] : = r log[x]

TEX, compute the answer, and set the answer for me in TEX as well. I will subsequently instruct TEX to typeset the questions and answers as different documents. Test-making will never be a chore again. You expect a mathematical program to be able to accept functions that you define, and Mathematica provides an exceptional set of rules by which to do this. One may define a simple function by means of a statement such as

then it will expand the above expression completely to be 2log(a) + 31og(b). Mathematica has an extensive and impressive set of rules for imposing patterns on functions. Groups of special definitions may be saved in files, which Mathematica can read in later sessions. Mathematica comes with a wide selection of packages created by the team at Wolfram Research. Beside their utilitarian value, these packages are useful examples of proper programming style within Mathematica.

fix__]: = Sqrt[1 - x^2] Is

the underscore character on the left of the assignment emphasizes the " d u m m y " nature of the single variable x. With this definition, Mathematica correctly evaluates f(a), f(89 and so on. (Without the underscore in the definition, Mathematica would only evaluate this function correctly for the argument x. In this case, f(1), for example, w o u l d be undefined from Mathematica's point of view.) Certain keywords to the right of the underscore character further restrict the pattern, such as

fact[n_Integer]: = n fact[n - 1] fact[O]: = 1 which defines the factorial function for integers. (This is just an example; Mathematica recognizes the traditional notation n! for factorials.) We m a y also use Mathematica's "such that" or condition operator, /;. With it, one might define a function appropriate for first-year calculus as follows: h[x_]: = Cos[x]/; x > = 0 h[x_]: = x + 1 / ; x < O equivalent to 72

THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 2, 1990

Mathematica W o r t h It?

Mathematica fascinates, as most readers who play with it will discover. Many of the evaluations of this program that I've read focus on the flashy special effects that Mathematica is so good a t - - t h e great graphing, the symbolic integration and differentiation, and the ability to work with as many significant digits as req u i r e d - a n d these are certainly reasons for rejoicing, But as I use Mathematica, I begin to feel that these are almost surface issues. I appreciate Mathematica's ease of use, and the natural feel of its syntax and programming language, which lie comfortably in one's hand like a well-broken-in baseball glove. All aspects of the user interface seem exceptionally well designed. Its ease of u s e - - e a s y in particular to do flashy things right a w a y - - m a y obscure the careful study that Mathematica repays. For example, when graphing a function such as y = csc(x) on the interval [ - 5 , 5] one rejoices in Mathematica's ability to recognize singularities and to deal with them in an intelligent manner. (Mathematica displays a warning message, although it does draw the graph.) Mathematica draws vertical lines at the singularities that do resemble vertical asymptotes. When we graph a nonsingular, discontinuous

function such as y = [x], we get no such warnings, together with a graph that may well be misleading to students, particularly beginners--the graph is continuous! (See Figure 2.) I can imagine h o w to write a Mathematica routine to generate a disconnected graph for this function (although I have not spent enough time to know if it is reasonable to expect Mathematica to draw the little circles at the "open" end of each segment), but this refinement is not something that users can expect to be able to do without some period of heavy exposure to the program. Mathematica integrates and differentiates m a n y equations with ease. But what about differentiating a function such as X

f(x) = x,

q- X, X > 0 ?

x <~O

A function defined by such a set of rules seems to frustrate Mathematica's ability to differentiate it. It's a nice and challenging problem in the Mathematica programming language to get Mathematica to perform this differentiation properly. I long for finer control of the program's algebraic manipulation. Mathematica insists on writing - 1 + x + x2 instead of x2 + x - 1, for example. Again, suppose we ask Mathematica to integrate x2X/1 - x2, and then to differentiate the result. I get a convoluted expression that I can Simplify and Cancel. The result is (x2 - x4)/V1 - x2. I f ~ a p p l y the operation Factor to get i x 2 V - 1 + xV1 + x, which is mathematically but not visually equivalent to the original function.

simple ones to exercise the basic skills and syntax of Mathematica, and more difficult ones calling for and challenging one's understanding of the Mathematica programming language. In the meantime, listings of Mathematica's packages help meet this need. I like the sophisticated yet clear coding of each package. They are good examples of using Mathematica's programruing language. Nevertheless, 750 pages is still a lot of territory, and one other area of slight disappointment is the index-not all special Mathematica symbols appear in it. The equal sign appears as a c o m p o n e n t of many such ,: ,f =, ^ - , + - , s y m bols - ,*=,>=, and so o n - - b u t it will be impossible to use the index to determine their meanings. (That is, unless you remember the Mathematica term that describes each. For example, := is the Mathematica operation 8etDelay, which is well indexed. However, all such special input forms are summarized in tables beginning on page 555. Mark this page well.) Also, items are indexed only as to the pages on which they are discussed. I would like to see index entries include references to Mathematica keywords as they are used in examples. If you notice a relevant instance of blocking in a discussion on Mathematica output, you'd like to be sure that the index entries under Block will steer you to that section. The listing of built-in Mathematica objects at the end of the volume is indispensable. The syntax, usage, and meaning of each Mathematica object (keyword) is clearly and succinctly displayed, together with telling examples of usage, the pages on which fuller discus-

The Mathematica B o o k Documentation for the Mathematica program appears in a book sold separately from the software. (The software includes only a written explanation of the front end for your system. That this explanation is surprisingly and gratifyingly short is a reflection of care in the design of the user interface--longer manuals are unnecessary.) You'll definitely need this book; don't forget to buy it. Mathematica: A System for Doing Mathematics by Computer is by Stephen Wolfram. For a book which is, after all, documentation, Wolfram deserves credit for fine writing, clear examples, and (essentially) complete coverage. The book is about 750 pages long, and that seems light to me. After all, the TEX typesetting system, the equivalent of about 20,000 lines of code, requires a 500-page manual. Although these are not linear relations, one expects that a program more than seven times TEX's length should need documentation more than 1.5 times as long as the TEXbook. What's missing? I would like to have seen many more examples in the body of the text, and I desperately miss the presence of problems. One needs a generous mix of exercises--

A 3-dimensional plot of cos(x a + ya) for x ( [ - 5 , 5 ] , y s [-5,5]. 2

v ] ( ~ +~)

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sions appear, and a suggested list of other objects which do similar things. I've spent many hours playing with two Mathematica implementations--for the Mac II and for a generic MS-DOS386 machine. The Mac implementation is the sexier of the two but ultimately the less satisfying. Mathematica, you should understand, is a memoryintensive application. On the Mac II, Mathematica needs at least 21/2 megabytes, but that will hardly suffice. In this minimum configuration, the computer's memory fills up rapidly as you use the program, and every fifteen minutes or so you will have to quit and restart Mathematica, or grit your teeth at the inevitable machine crash. (Apparently, there is no other way to clear Mathematica's memory. With more than this minimum memory, Mathematica works very well.) Two and one-half megabytes may not let you load in the symbolic integration routines. The front end is quite nice, though. Mathematica is able to display some very pretty images, courtesy of the Macintosh operating system and its graphic capabilities. These images are drawn from the fields of cellular automata (incidentally, Wolfram's specialty) and classical physics. (These d i s p l a y s are distinct from Mathematica's graphs.) Furthermore, Mathematica commands and comments can be grouped together in structures called notebooks. N o t e b o o k s are hierarchical structures in which some of the items can be suppressed from display. Under this scheme, it will be possible to generate c o u r s e w a r e with remedial or special topics lying hidden until summoned by the student. Despite these special features, I prefer the DOS version, even though it (currently) lacks the ability to create notebooks. I have come to feel dissatisfied in general with the Macintosh environment. It is sluggish going no matter whose program is executing. Furthermore, the extra memory that Mathematica needs is an expensive enhancement to an already overpriced machine. The lean, plain vanilla DOS v e r s i o n is more pleasing, even without any pretty displays and in the absence of notebooks. (These are in the works for future versions of the DOS front end.) You do need at least 1 megabyte of extended m e m o r y (not the expanded memory used by Lotus 1-2-3 and some other programs), but Mathematica is quite robust with this memory since it uses a virtual memory manager to pretend that space on the hard disk is equivalent to random access memory. (You need to ensure that your hard disk has at least 5 or 6 meg of free space. Please note, you cannot run Mathematica on anything less than a 386-class machine.) If your 386 machine has several megabytes of extended memory, Mathematica really flies. Subjectively, I get the feeling that Mathematica operates faster on my 386 clone than on the Mac. Special versions are available to take advantage of 80387 and Weitek math coprocessor chips. 74

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Mathematica is not cheap, although when the price is expressed in software units (1 unit = price of Lotus 1-2-3, say), then it really is a bargain--it does so much more than any commercial program. Nevertheless, potential purchasers may hedge their bets a bit by sampling a v i d e o t a p e of Wolfram d e m o n s t r a t i n g Mathematica at the Macintosh. His enthusiasm comes through clearly. The tape is available from the Science Television Company (165 Bennett Avenue, Suite 1G, N e w York, NY 10040; 212/569-8079) for $80 (but call them for quantity pricing). None of my critical quibbles is serious. Mathematica is an exciting new tool for students of all ages. I urge readers to seek it out. Department of Mathematics John Jay College of Criminal Justice City University of New York New York, NY 10019 USA

Plane Algebraic Curves by Egbert Brieskorn and Horst KnOrrer Birkhauser-Verlag, Basel, 1986, vi + 721 pp. Computational Geometry for Design and Manufacture by I. D. Faux and M. J. Pratt Ellis Horwood, Chichester, 1983, 331 pp.

The Mathematics of Surfaces edited by J. A. Gregory Oxford University Press, Oxford, 1986, 304 pp.

The Mathematical Description of Shape and Form by E. A. Lord and C. B. Wilson Ellis Horwood, Chichester, 1986, 260 pp.

Reviewed by Art Schwartz Euphonia, the heroine of Euphonia and the Flood [2], is fond of saying: "If a thing is worth doing, it's worth doing well." No doubt this has a nice ring to it, but in some cases it's misleading at best. Many such cases occur in the field of geometry applied to industrial problems via computer graphics, numerical control, robotics and the like, hereafter referred to as Computer Aided Engineering or just CAE. Oftentimes there is no readily available method to solve a problem really well. At the same time, even though an algorithm being used may be truly hideous, only the computer "knows" and the results may be elegant in spite of it all. Thus we have Schwartz's aphorism (SA): "If a thing is worth doing, it's w o r t h doing poorly." I hasten to add something of a converse: "If a thing can

be done well, this may be reason enough for doing it." I think most mathematicians recognize this latter principle. Sometimes, of course, we are fortunate enough to find something we really need that can be done really well. The field of computer-aided engineering covers a lot of ground. Part of it is concerned with the representation and production of shapes: car bodies, air frames, boats, and shoes, for example. One aspect of this endeavor is to construct mathematical models of curves and surfaces that satisfy the designer and can be manipulated rapidly by computers. This means we want to be able to find intersections of curves and surfaces and to construct offset or parallel surfaces. We want to calculate curvature and determine the reflective character of the surfaces that are designed. We do this so that engineers can determine the mechanical, structural, and aerodynamic characteristics of an object being designed, so that designers can see what it looks like, and so that dies and molds can be fashioned to produce it. Presently much of this work is being done b y representing geometric objects as collections of simple elements: straight line intervals, parabolas, triangles, etc. The processing of these elements is done by iteration and divide-and-conquer methods. Excellent results have been obtained this way but with great expenditure of programming effort and computer time. There is little doubt that the techniques and insights of algebraic and differential geometry can be used to provide more efficient, sophisticated methods. The dissemination of more modern mathematics among engineers and programmers as well as the familiarization with CAE problems b y mathematicians seems to be in order. The question that we address here is: what books are available to aid these efforts? It would be difficult at best to name one central and outstanding book in the field of applied geometry, or at least that segment of it referred to as geometric modelling, that by itself would provide a basis for a review article and that could represent the mathematical aspect of current work from a differential and algebraic point of view. I should say that the application of combinatorial geometry and of interpolation and approximation theory (in particular, splines) to computer-aided engineering seems to be better organized into mathematical disciplines with books, articles, and recognizable experts. But sophisticated methods of differential and algebraic geometry, which in my view are of great potential value, are still not available to most people working in CAE. M. Spivak, in the excellent five-volume A Comprehensive Introduction to Differential Geometry, remarked that " . . . one quickly senses that Differential Geometry is a field of overwhelming extent, beyond the comprehension of any mortal. I suppose such lucubrations ought to buoy up one's spirit with admiration for the achievements of man

[sic], but I confess that they usually lead me instead to a state of brooding melancholy." The situation in algebraic geometry is probably worse. Therefore, instead of reviewing a "central" book in applied geometry, I am going to review four noncentral books and mention a few others along the way. Nevertheless, the reader should not assume that this is a comprehensive survey of "the literature." In other words SA applies very much to this review as well as to the books being reviewed. Before getting down to business I have one more prefatory remark: I do not think the situation is hopeless. It may well be possible to develop readable material in the field of applicable algebraic and differential geometry that is accessible to (say) master's level mathematicians and engineers, if the subject is limited to curves and surfaces in three-space. Some very old books are often used profitably by some workers in the field; for example, books by Salmon [6], written in the nineteenth century. Such books emphasize manipulation, which can be helpful in practice. Unfortunately, they rarely take advantage of modern notation or rigor. If they mention matrices at all, they treat them as determinants waiting to be born. Such books are extremely difficult to read. I do not think all we have to do is translate these dusty tomes; almost certainly they need a healthy transfusion of modern mathematics. N o w let us consider the books under review. I will try first to describe briefly h o w they "fit" into the "big" picture, and what Euphonia and I think of them. 1. Faux and Pratt is a very good "entry'-level book. Many (not all) people agree with me that it is a good way for a mathematician to find out what applied geometry is all about. Something worth doing has been well done. The problem is (am I never satisfied?) that it is an entry-level book and thus does not go very deeply into anything.

2. The Mathematical Description of Shape and Form is an intriguing book. It discusses a wide variety of mathematical topics including combinatorial geometry and catastrophe theory. The message of the book is that all kinds of m o d e r n mathematics can have a role i n "Shape Science." It reads easily, like a collection of magazine articles. There seems to be no attempt at depth. 3. The Mathematics of Surfaces, edited by Gregory, is a conference proceedings and thus is not polished. Some articles are more or less progress reports. On the other hand, some articles get us a little deeper into technical details than the two books above. 4. Brieskorn and Kn6rrer's Plane Algebraic Curves is not really in the mainstream of applied geometry. No doubt, however, it would be at least indirectly useful. Frankly, I include it in this review because it is an exTHE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 2, 1990 7 5

Figure 1. Faux and Pratt discuss (among other things) the problem of finding the path of the center of a milling machine cutter; i.e., constructing a parallel or offset surface (p. 269).

Figure 2. The scope of Lord and Wilson's book is wide. They include some remarks on catastrophe theory in the potpourri (p. 104).

ample of extremely good writing (and illustrating) and

a tour de force of a lot of good mathematics. It is a beautiful combination of classical and m o d e r n mathematics. Let us look at these books in greater detail. Faux and Pratt begin with a brief review of elementary differential geometry and a smidgin of projective geometry and the theory of conics. It is presented pleasantly. The next part of the book discusses methods of constructing curve segments and surface patches and composing these into large geometric objects. The reader can quickly and painlessly learn about many basic techniques of geometric modelling 76

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and their lingos. The last part of the book deals with computer manipulation of surfaces: offsets, intersection, and cutter paths. Here the brevity is felt, because these subjects require more detail if they are to be usefully discussed. See Figure 1. This book is not easy to use as a textbook since it has no problems and few examples. Although this last remark is negative, I like the book very much. It helped me enter the field. Despite its age I include it in the review because of its high quality. I should note here that a very usable textbook is Penna and Patterson's Projective Geometry and its Applications to Computer Graphics [5]. As its title indicates, its subject matter is limited. It is designed for mathematically unsophisticated students. Lord and Wilson's book is a great advertisement for the applicability of "advanced" classical and modern mathematics in the study of shape and form (their definition of form includes structure and function). It should be read for impressions. If you look for hard information or detail, you will probably be disappointed. The authors claim to address a relatively unsophisticated audience, yet they fearlessly take on fairly technical and/or difficult topics--the equations of Gauss and Codazzi, singularities of mappings, catastrophe theory, and the classification of surfaces, to name a few. See Figure 2. They have some chapters on discrete geometry in addition to their discussion of continuous geometry. There is also some discussion of physical and biological topics--fluid flows, crystallography, and botany. The treatment is very brief. On the whole this book is quite entertaining. The Mathematics of Surfaces gets us a little deeper in some areas. The first article tries to prepare us for the rest of the book. Because some of the articles plunge right into technical details, this is not a bad idea. Unfortunately,

the brief introduction by Sabin and Martin is not really enough. Many of the topics in the book are not familiar to any wide group that I know of; nonetheless, they should be interesting and useful to many in CAE if made more accessible. The next articles discuss constructing and fitting together surface patches. Then come three articles that discuss differential geometry. A m o n g these articles, t w o by L. M. W o o d w a r d present a strong case for using the language of differential forms. Pratt and Geison then review progress in developing computer solutions to the problem of finding the intersection between surfaces. The large list of references indicates the practical as well as the theoretical difficulty of this problem. The next two articles apply differential geometry to structures under loads. We are reminded here that, historically, differential geometry is concerned with structure as well as shape. The next two articles are again concerned with composing patches to form larger surfaces. This is a subject that quickly becomes technically involved. As far as I know, it has resisted any effort to streamline or simplify it. It is an area where modern mathematics, particularly algebra and combinatorics, could make useful contributions to CAE. The following two articles return to differential geometry and apply it to composing cyclide patches. I think this is an excellent idea worth discussing at some length. See Figure 3. Unfortunately, the artides seem to assume more familiarity with cyclide surfaces than exists in the late

twentieth century. The only mention of them from a m o d e m point of view that I have seen is a very brief mention by Berger in Geometry, Volume II [1]. One gets (Dupin) cyclides by applying spherical inversions to tori. In the introduction to the book Sabin and Martin state, "By some magic of algebraic geometry it turns out that if a surface has the same (low) order of equation in both [point and tangent representation forms], it becomes simple in many other aspects too. The nice properties of the c y c l i d e . . , can be viewed as stemming from the fact that the dual of a cydide is another cyclide." I thank them for mentioning this. I sincerely wish someone in the book had elaborated this point. In the article "Cyclide Surfaces in Computer Aided Design" by Martin, de Pont, and Sharrock, the only descriptions of cyclide that we are given are (a) a family of rational parametrizations for them and (b) that their lines of curvature are circles. These are very useful technical facts but not the most illuminating for the neophyte cyclidist. The main concern of the article is the usability of cyclide surface patches with circular boundaries. The article is not intended to be a general introduction to cyclides, so should not be criticized for not being one. However, s o m e w h e r e in the book some better descriptions and definitions should have been given. In the last half of the nineteenth century and the beginning of the twentieth, cyclides were commonly discussed and an enormous amount of literature on the subject was produced. For some reason the subject

Figure 3. Cyclides. Depicted in Analytische Geometrie spezieller Fli~chen und Raumkurven by Fladt and Baur (pp. 364--365). THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 2, 1990 7 7

Figure 4. Brieskorn and Kn6rrer investigate the topology of knots associated with algebraic curves (in C x C) (p. 436). then vanished or nearly so until quite recently. Now, we seem to be in the midst of a cyclide revival, as the above authors and others have realized the applicability of these surfaces. There is still a need for introductory and theoretical material on this subject. Cyclides are objects of inversive geometry, which is also a subject that should be revived, see [9]. The final article of the book discusses the manipulation of geometric objects via recursive subdivision algorithms. It seems to me that it ought to have been placed alongside articles on patch composition. The last b o o k to be r e v i e w e d is Brieskorn and Kn6rrer's Plane Algebraic Curves. There are already at least three fairly well-known, well-written books that introduce this subject, namely those by Walker [8], S e i d e n b e r g [7], a n d F u l t o n [4]. B r i e s k o r n a n d Kn6rrer's book merits our attention because it is beautifully written and illustrated, introduces a lot of modern analysis and topology, and can be read in pieces. The treatment of Bezout's theorem is excellent. The proof of the Weierstrass Preparation Theorem is very instructive and uses an approach that depends on some difficult technical results, which are clearly stated, intuitively believable, but not proven. I don't like this sort of omission as a rule, but I am convinced that this is the best w a y to handle the theorem in question. Quite often, in fact, details are pointedly left out. The authors skillfully convey a sense that some things are long and difficult but still humanly understandable. The aim of the book is to expose the topological theory of curves using some algebraic topology applied to braids. See Figure 4. I recommend this book to anyone even vaguely interested in these topics. Finally I mention but will not review two books of 78

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great interest: Fladt and Baur's Analytische Geometrie spezieller Fl~ichen und Raumkurven [3] and M. Berger's Geometry (two volumes) [1]. Both are dedicated to advanced analytic geometry. Both have excellent illustrations. The former is somewhat old-fashioned, with a huge number of examples and little mention of application (although it is no doubt very applicable). The latter strives for great modernity (maybe too much); it illustrates many mechanical applications. Where does this all leave us? At the present time there seems to be a healthy a m o u n t of activity in mathematical science w h e r e exotic mathematics is used to model and design exotic physical systems. The use of analytic geometry to model geometric entities such as shoes, automobile fenders, and the like may seem old hat. In fact, there is much progress possible in this area. It is a widely accepted folk lemma of the auto industry that in designing a new car (and the tools to make it) the part of the process that requires the most lead time is the design of the front quarter panel (the piece of the body between the hood and the wheel well). Because much of the difficulty is geometric in nature, it is not difficult to believe that the study of applied geometry can lead to a significant improvement in this situation. There is a place for rigor, elegance, and clarity in this activity, and a role for mathematicians in providing them. What we need are books written with the p o w e r and clarity of Brieskorn and Kn6rrer in the area of real three-dimensional algebraic/differential geometry. This may require original research but there is a lot of classical material that can provide raw material. The books under review leave a lot to be done, although they all make positive contributions for people entering and working in applied geometry.

References 1. M. Berger, Geometry, Vols. I and II, Berlin: SpringerVerlag (1987). 2. M. Calhoun, Euphonia and the Flood, New York: Parents Magazine Press, (1976). 3. K. Fladt and A. Baur, Analytische Geometrie spezieller Flfichen und Raumkurven, Braunschweig: Friedr. Vieweg & Sohn (1975), 4. W. Fulton, Algebraic Curves, New York: Benjamin (1981). 5. M. Penna and R. Patterson, Projective Geometry and its Applications to Computer Graphics, Englewood Cliffs, NJ: Prentice-Hall (1985). 6. G. Salmon, Analytic Geometry of Three Dimensions, Vols. I and II, London: Longmans, Green and Co. (Sth ed. 1912). 7. A. Seidenberg, Elements of the Theory of Algebraic Curves, Reading: Addison Wesley (1968). 8. R. Walker, Algebraic Curves, New York: Dover (1962). 9. J. Wilker, Inversive Geometry, The Geometric Vein, New York: Springer-Verlag (1981), 379-442.

Department of Mathematics University of Michigan Ann Arbor, MI 48109 USA

* C o l u m n editor's a d d r e s s : Faculty of M a t h e m a t i c s , T h e O p e n University, Milton K e y n e s MK7 6AA E n g l a n d 80 THE MATHEMATICALINTELLIGENCERVOL. 12, NO. 2 9 1990Springer-VerlagNew York