Four

Tom Banchoff

Donald J. Albers

As an undergraduate, Tom Banchoff missed the class of one of his teachers, a distinguished mathematician, and made the mistake of apologizing in the hallway. His teacher was infuriated by his absence and, in a very loud voice, told young Banchoff, “You will never be a mathematician, never, never!” That teacher was wrong, for Tom Banchoff has gone on to a distinguished career as a researcher and teacher. He is now Professor of Mathematics at Brown University.

As a high school student, Banchoff, after reading a Captain Marvel comic book, became interested in the fourth dimension and Flatland, a book by Edwin Abbott Abbott that has had a huge influence on his work. In the early 1970s, he was producing pioneering films that illustrated the fourth dimension. His passion for and involvement with the fourth dimension has continued unabated for more than fifty years. During that time, as technology has evolved, his illustrations of higher dimensions have improved dramatically. Over that same period, his appreciation for Abbott has grown to the point that he and William Lindgren have written the quintessential guide to Flatland, published jointly in July 2010 by the Mathematical Association of America (MAA) and Cambridge University Press. He also has written Beyond the Third Dimension (Scientific American Library, 1990) and Linear Algebra through Geometry (second edition, Springer Verlag, 1991).

From an early age Professor Banchoff knew that he wanted to be a teacher. Even a brief conversation with him reveals that his dedication to teaching is extraordinary. Throughout his career at Brown he has involved students in research projects, many of which were centered on higher dimensions. He speaks with great pride about his students’ accomplishments, which stand them in good stead for graduate school and/or employment.

Banchoff’s commitment to students is reflected in his winning several teaching awards, including the MAA’s Award for Distinguished College or University Teaching (1996) and the National Science Foundation Director’s Award for Distinguished Teacher Scholar (2004).

Professor Banchoff served as President of the MAA from 1999 to 2000, and he is the recipient of two honorary doctorates.

Banchoff: My father never pushed me to become a scholar, although he impressed upon me the fact that learning English and arithmetic (I didn’t hear the word “mathematics” until I was in high school) were the sorts of things that are really important, and none of the other things are. He was also very conscious of the fact that I should be “a regular guy.” So I took on the additional responsibility of becoming a regular guy. All these other things that I did, such as reading, he couldn’t complain about.

MP: What did he mean by being “a regular guy”?

Banchoff: He meant that you shouldn’t be eccentric, like a book worm, studying all the time. He was actually rather suspicious of intellectuals, and he didn’t particularly like them. He knew people who read books and talked about books. He, himself, wasn’t a reader. My mother, on the other hand, read all the time, and she was very encouraging. There was a little bit of tension there, I suppose. A middle-aged reclusive neighbor was my father’s example. He said, “You don’t want to grow up like him.” He worked at a library and used to carry his laundry in a paper sack.

MP: Did you become an all-around guy?

Banchoff: Yes. I was on the tennis team, and I played soccer. I was in the school play, the school orchestra, the band, and I was on the debate team. I was editor of the school paper and a member of the yearbook staff. I was in the Third Order of St. Francis and president of the Latin Club.

Figure 4.1 Banchoff, on his fifth birthday in red boots, trying out his new toy fishing rod.

How to Get Perfect Grades

MP: Where did you attend school?

Banchoff: Trenton Catholic Boys’ High School in Trenton, New Jersey. I ran a 99.29 average for four years. Most people saw me totally involved with other activities all the time. I didn’t want them to think I was studying or something like that. It was one of these silly situations in which there was a very undemanding curriculum. At the same time, I apparently got the idea that the way you should approach school is to figure out how to get perfect grades in all the examinations and just do that much.

MP: How did you go about figuring that out?

Banchoff: That’s easy: pay attention.

MP: To your homework?

Banchoff: To your homework, yes. That never took any time. I liked it.

Figure 4.2 Banchoff, age 11, in suit, tie, and hat, about to take his first flight. He was off to Chicago to be a contestant on the “Joe Kelly & The Quiz Kids” radio program.

MP: What subjects did you particularly enjoy in high school?

Banchoff: Mathematics, of course, and languages. I was interested in the structure of languages rather than literature. There weren’t any subjects that I didn’t enjoy, although I didn’t care much for laboratory science. And I didn’t read enough to decide to go into English. I had already decided that, since I hadn’t read Thackeray, there was no sense being an English major.

MP: You were taking Latin.

Banchoff: And German, and I studied French on my own.

MP: On your own?

Banchoff: One of my teachers told me that French was better than German, so when I didn’t have a job one summer, I did all the exercises in my teacher’s French text.

MP: That’s fairly uncommon. Did you know any other students doing that?

Banchoff: No, but I knew a couple of the students who were studying French. I used to work with them. I never learned to speak French correctly. It’s a total embarrassment. I also was very interested in philosophy and theology. But I had been counseled by people that you don’t do that as an undergraduate; you do that later in life.

MP: Who provided that counseling?

Banchoff: Well, my teachers. I was taught by Franciscan priests. And they were right after all.

MP: They should know.

Banchoff: Yes. They never pushed me to become a priest. And I think I’m really better doing what I’m doing. The idea that I would go off to a seminary was never pushed, which was nice because in a certain sense in those days good students were encouraged to become priests.

MP: Do any high school teachers stand out in your memory?

Banchoff: I knew one special teacher, Father Ronald Schultz, when I was a freshman. He believed the best of you, even some of these guys who were questionable in my estimation. He really was a spiritual father, although he never taught me. He was also a math teacher, and he listened to people. He actually listened to me. He was the head of the Mathematics Department, and he was also the moderator of the Mass Servers Society, which was a kind of fraternity. It was an all-around-guys kind of fraternity. He’d get up at five in the morning and pick us up, and we’d go and serve mass for the twenty-eight priests. Then we’d sit around playing cards or doing homework or whatever, waiting for school to start. It was a real fraternity and a really good group of guys. Some of the toughest kids in the school belonged to it. It was really a very interesting experience, and everybody loved Father Ronald, who was the kind of person who would just go out of his way to stick up for you.

The Fourth Dimension

I proved my first geometry theorem when I was a freshman because we used to have to go to mass every Friday morning, and being in homeroom 1-1, we filed in first, so we were way up in the first pew. You’d have to wait until the other 800 kids came in. So you had a lot of time to pray or to contemplate or to watch shadows move across the tiles. There were square tiles set obliquely across the altar rail—we used to have altar rails in those days—which would cast a shadow. By the beginning of the mass, it was very thin on one side, and by the end, it would cover almost an entire triangle so I posed to myself the question: when was the shadow exactly half way across? Not half way in terms of distance, but half way in terms of area. I hadn’t had geometry yet, of course, but I figured it out. When it’s half way across, you get an isosceles right triangle, which is half of what you get if you cut it the other way, so just cut it in half, and you’ve got it. I remember once explaining this to one of my friends at an evening meeting of the Knights of the Blessed Sacrament, and he said, “Father Ronald, the new kid here has a theorem.” “Well, what is it?” “Here’s the problem.” He said, “Well, you’d have to set up a relation and a proportion and solve some algebraic equation.” I said, “No, I didn’t do it that way.” He said, “If you already knew the answer, why did you ask me?” I said “But I didn’t ask you.” And then, fortunately, something happened to distract him, but we both remembered that later on. Anyway, he was the only one who listened to me because I was really very much into the fourth dimension at the time.

MP: As a high school student?

Banchoff: Yes. When I was in junior high, I was into it. I was really interested in dimensional analogies and theological mysteries. How you could understand time and how you could understand how a circle could be a straight line, etc. I was into the Flatland analogy, not really through “Flatland” so much but comic book versions of it. I’d like to think Father Ronald was the one who gave me “Flatland.” He never told me that I was asking dumb questions. At the end of my freshman year, he was transferred, so I never had him as a teacher. I remember one thing, though. Once my freshman algebra teacher gave us a number of things that we were supposed to memorize. One of them was every number multiplied or divided by zero equals zero. I couldn’t wait. I ran up to where the teachers used to smoke between classes. Father Ronald said to me, “Tommy, what’s the matter?” I said, “Father Noel just said that we had to memorize that every number multiplied or divided by zero equals zero.” He said, “Well, since you know that’s not true and I know that’s not true, let’s leave it at that.” The funny thing was that Father Noel actually explained what he meant by it. He said he read somewhere that division by zero is undefined. So he defined it.

MP: Did you have any other important influences in high school?

Herbie Knew Everything

Banchoff: One of my big influences was Herbie Lavine. Herbie was three years older than I was. He was the son of a grocer up the street, and he was really smart. He knew everything. When I was in sixth grade he already knew how to factor polynomials, and he would teach me while working for his father. His father would say, “Herbie, you’re supposed to be unloading that packing crate, not doing algebra on it.” Herbie would always teach me this stuff he was learning in high school when I was in junior high. When I was in seventh grade, he told me about this neat problem involving twelve billiard balls in which one of them was either heavier or lighter than all the rest, and you were to find the “odd ball” in three weighings. I thought about it for a while. Herbie went off to college, and I went off to high school. After several years of studying the stained glass windows while in church, I worked out the solution and wrote a letter to Herbie who was at Michigan at the time. He wrote back and said it was right. He gave me another problem which was trivial, and I solved that immediately. Herbie went on to become an actuary and a professional bridge player. Herbie and his son and wife actually came and visited Brown a couple of years ago, and I took them out to lunch. I told this story to the wife and a son who couldn’t believe he was the guy I idolized more than anybody else. He knew everything and taught me. He could have been a mathematician. He likes actuarial work, and he likes playing bridge. In any case, I convinced myself at that point that I could solve hard problems. Of course, that’s not such a hard problem. Although I wasn’t working constantly on it, it took me three years. The fact that I had been able to hold a problem in my mind for a long time and finally solve it was better in my mind than being able to solve it immediately. So by the time I graduated from high school, I had solved a tough problem and come up with a theorem and written a little paper.

I also discovered that 1³ + 2³ + 3³ + 4³ + . . . + n³ equals (1 + 2 + 3 + . . . + n)², although I didn’t know how to prove it. I just noticed it. I remember at Notre Dame, when Dr. Taliaferro was lecturing about mathematical induction, he gave that identity as a problem. I said, “I discovered that.” He said, “Now you can prove it.” The thing was that I was right, and it was wonderful to discover that mathematics was something you could prove. I wasn’t discouraged in high school. The teaching wasn’t good, but the mathematics was good. And I was able to do anything I felt like because I was a good guy and got 100s on every test. I paid my dues in all these other clubs. I really did decide that mathematics was something that I liked and that I could do and where I could ask questions that were different from the questions that my classmates and my teachers were asking. When I got to college, I realized that was still true. That’s what I liked about it. I could have insights that were not what everybody else was having. I saw that it might get to the point that maybe you have some insight that nobody else would have. It wasn’t just a question of proving things that others had already proved, but maybe you’d end up proving something that no one else would ever have proved. And that was very appealing to me. I loved the creative aspect of mathematics. I was lucky enough to realize something about the creative aspect of mathematics when I was young. These are almost trivial things, but I remember them very clearly. Each of them has later turned out to be kind of important, not that I ever did anything with the number theory, but I certainly proved theorems about cutting triangles in half, and I certainly have done things about visualizing complex functions.

Figure 4.3 At Notre Dame in 1957, Banchoff won the Borden Prize for the highest grade point average among all freshmen.

Communicating—I Figured That’s What Teaching Was

MP: How did you choose mathematics and teaching as a career?

Banchoff: My father was always a little bit uncomfortable about the fact that I seemed to be interested in teaching. But I knew I wanted to teach before I knew what I wanted to teach.

MP: How did you know that?

Banchoff: Teaching was big in our family. My mother was a kindergarten teacher; my aunt was a teacher; my father’s mother had been a teacher. It was a respected occupation, and I liked the teachers at school. So, when I was in grammar school, I thought I’d be a high school teacher; when I was in college, I thought I’d be a college teacher. I never knew there was such a thing as a mathematician. I never knew there was such a profession. We didn’t know any professionals, except for our family doctor and a couple of small town lawyers in our neighborhood. But we really didn’t know anybody who had gone to college, and I didn’t know about college until I went there. But I knew I wanted to be a teacher. I liked learning things, and I liked telling people. And that’s what it was for me—communicating. I figured that’s what teaching was. My experiences of trying to tell other people, and having some success doing that, made me feel that I was going in the right direction. I certainly got encouragement from my mother and my aunt. My father thought I should be a banker or an engineer. He said: “You’re smart enough to be a banker or an engineer; what do you want to be a teacher for?”

MP: You could have made more money as a banker.

Banchoff: Well, actually, he asked nicely. He asked whether I could make the kind of money as a teacher that would enable me to support a family in a comfortable way. This was a serious question.

I remember when I got my first job. My salary was $7,100 at Harvard, which was a considerable fraction of what my father was making after thirty-nine years as a payroll accountant. There I was, married with a child. At that point, he relaxed and thought it was probably all right to be a teacher.

Figure 4.4 Banchoff, the young father, with his son Tommy, wife Lynore, brother Richard, mother Ann, and father Thomas.

MP: He must have known that Harvard was tops if you’re going to be a professor.

Banchoff: No, he wasn’t impressed by the Harvard thing, but he was impressed by the $7,100. To him it meant that I could have a good life. He wanted me to be happy and to get a job that I liked. That was the important thing—a job that I enjoyed. This was from a man who didn’t enjoy his job at all!

MP: Wow!

Banchoff: He had a terrible job. He was responsible for the payroll of about 125 people at the local gas manufacturing plant. And it was a job that nowadays could be done in about two hours a week by a semi-trained person using a program written by a high school student. But in those days, there were no computers; it had to be done by a person. Just keeping track of the week-to-week, day-to-day variations of these 125 people for a whole week entailed using big adding machines with fifteen registers. It’s a familiar enough story, and unfortunately, he died before personal computers became popular. I think he would be happy if nobody else had to do what he did. He knew that it was a miserable situation. In any case, I was convinced that I wanted to be in a job that I loved. So I chose one. He was very happy about that.

MP: Let’s back up for a minute to high school, where you accumulated a rather sensational grade-point average. This all-around-guy stuff is really good in terms of gaining admission to good places. How did you decide where to go?

Banchoff: I decided to go to a Catholic college of course. Among other things, I also was interested in theology and philosophy. I narrowed it down quickly to Georgetown and Notre Dame—Georgetown for languages and Notre Dame for mathematics. I never even considered applying to any other schools. I had visited Georgetown a few times, but I had never visited Notre Dame. I had a chance to go out to Notre Dame in my senior year because I went to the National Science Fair in Oklahoma, having won the local science fair with a math project. I had a free train trip to Oklahoma, so I went home to Trenton via Notre Dame. I walked right up the middle of the campus into the first building I saw and into the office of the only person on campus who knew who I was, the person who was in charge of scholarships. Luckily enough, I had won a National Merit Scholarship. That was the first year of Merit Scholarships, so they knew who the winners were. So I got introduced to a few people, the Dean, Fr. Charles Sheedy, and Mr. Frank O’Malley, the legendary professor of rhetoric and composition. The Dean introduced me to the mathematician Ky Fan. I said I was a high school student who had just come back from the National Science Fair with a math project. He said, “What’s the project?” I said I was working on three-dimensional graphs for complex functions of a real variable. I said, “I have some models here,” and he said, “You should be learning mathematics; all wrong; you shouldn’t be trying to do mathematics. You should be learning mathematics.” He said, “Here, read Irrationalzahlen. Do you read German?” I said, “Yes.” He said, “Read this. Just study. Don’t try to do mathematics. You have to learn mathematics.” Then I was ushered into the office of the Department Chairman, Arnold Ross. He said to me, “I hear you are interested in mathematics.” I said, “I was until five minutes ago.” He said, “Oh, tell me about it,” in his very continental way. He listened, and then I said, “I was interested in the fourth roots of −1, and so forth, and I found a formula for it in a book. But the book was wrong. This is what it said.” “So, fix it,” he said, “Go to the blackboard.” I said, “You mean, go to the blackboard now and do it?” He said, “Yes.” So I went to the blackboard, and I figured out the correct answer. Then he smiled, and I smiled. I was a mathematician again. I didn’t take calculus until I was a sophomore. I took number theory and analytic geometry as a freshman. Then modern algebra and calculus as a sophomore. This was the old days.

The Only C I Ever Got

MP: That was very uncommon, though, wasn’t it?

Banchoff: Not at Notre Dame.

MP: For a freshman to be taking number theory?

Banchoff: No. That was the course. Dr. R. Catesby Taliaferro (pronounced “Tolliver”) was the teacher. It wasn’t a course that required algebra; it was the honors math course. If you came to Notre Dame and had the misfortune of having good SATs, you were put into the honors course whether you wanted it or not. There were four sections of it. Donald Lewis taught one; Ky Fan taught one; Dr. Taliaferro taught one; and Richard Otter taught the other. They killed a number of the best students by having them take the course, but I thought it was absolutely wonderful. Dr. Taliaferro was a genius as a teacher—he was a mathematician, a philosopher, and a historian. He taught a marvelous course in number theory, starting with the Peano axioms and going right through Euler’s inversion formulas. Then we did vector geometry and then construction of the real numbers. So it was a very fine honors course. Later I took his signature course, rational mechanics, which was given simultaneously for junior math majors and incoming graduate students.

As a sophomore I took a course in modern algebra for a year, then a graduate course in algebra as a junior, and then a third year of algebra with Hans Zassenhaus. I had a lot of algebra by the time I graduated. My analysis wasn’t too strong, although I did have a full year of complex analysis with Vladimir Seidel. I also took various other courses, but the premiere course was general topology. I had heard about topology and decided I wanted to take it. My roommate, a National Merit Scholar from Dallas, Texas, Claiborne Johnson, took topology when he was a junior, and I couldn’t take it when I was a junior for various reasons. He got an “A.” He was the darling of Ky Fan. I took it when I was a senior. I was not only not a darling of Ky Fan, I was the butt of the class. It was made up mostly of second-year graduate students and a couple of seniors. But I didn’t understand it. I didn’t have the background. I didn’t really know what a compact set was. I didn’t do very well, and I got a C—the only C I ever got in my life. I stayed with it and got a B at the end of the second semester. But somewhere about midway in the second semester, I infuriated Ky Fan for some reason.

Every once in a while, he would turn around and ask a question. Students knew there was something that he wasn’t very happy about when he did that. One day he turned around, and he said, “Banchoff, what is a set of second category?” I said, “Well, a set of the second category . . .” Well, I didn’t get any further than that. “Well,” he said. “Well. What do you mean ‘well’? I asked you a mathematical question; ‘well’ is not mathematical; ‘well’ is English. This is not an English course, this is a mathematics course. Give a mathematical answer. What is a set of second category?” I replied, “A set of the second category is a set that cannot be expressed as a countable union of nowhere dense sets.” “Right. Why didn’t you say that the first time? Well!” Then he turned around and continued his lecture. So everybody in the class concluded that there was something going on between Fan and me, but it wasn’t all my fault.

I took Seidel’s complex analysis when I was a senior. We read Knopp’s books. I didn’t like the problem books because I didn’t understand them, but I liked the geometry of it. At one point, I went up to Seidel and said, “These complex functions are great things—when are we going to graph them? He said, “Graph them?” I was into the fourth dimension by that time. He responded, “But you need four dimensions.” I said, “Yes.” He said, “Well, people tried a hundred years ago to graph these things and made models, but it didn’t help.” He dismissed the whole idea, but he was wrong, of course. It does help!

MP: What were your other big academic interests at Notre Dame?

Banchoff: I really was very much into courses in literature. I had a minor in literature and writing. I did a lot of writing in Frank O’Malley’s rhetoric course. He very much wanted me to finish in three years and get into Stanford’s creative writing program. I thanked him very much, but I had decided that I wasn’t good enough. In my writing and literature courses, I always got the top grades, and if I didn’t get top grades, I was very argumentative because I knew I was right; whereas, in mathematics I didn’t always get the top grades; I got a couple of A minuses. But people explained to me exactly why I was wrong when I was wrong.

You Will Never Be a Mathematician!

MP: Well, it’s easier to see when one isn’t right in mathematics.

Banchoff: Sure. Nowadays, I look at it and realize that it’s a lesser stage of psychological and moral development to want to know when you’re right and when you’re wrong. I didn’t have the tolerance necessary to become a real writer. I knew I could be a mathematician. I did find a challenge in it, and it wasn’t easy by that time. I wasn’t getting top grades in mathematics, particularly in Ky Fan’s class. But in my second semester with Ky Fan, I was doing a little better, having sort of caught on to what Bourbaki is about. For some reason or other, I missed a class. I had never missed a class up to that time. I already knew the notes, but I went to him to find out something about the homework. I said, “First of all, I’d like to say that I am sorry I missed class, but I have already made up the notes.” I made the mistake of telling him this in the hallway at Notre Dame. And this totally infuriated him that somehow or other I just wasn’t a serious mathematics student in his mind. His summary evaluation was, spoken at the top of his lungs: “Mr. Banchoff, you will never be a mathematician, never, never!” It was loud enough that everybody was sort of leaning out their doors to see what was going on. Some of them noted that it was only Ky Fan going after Banchoff. So, that was that, and I went off to Berkeley for graduate school. I didn’t get accepted at Harvard. Lucky for me. I’m a geometer, and I never would have found that out if I had gone to Harvard.

Figure 4.5 Ky Fan.

Figure 4.6 Banchoff taught at Notre Dame in the summers of 1960 and 1961. Here Sister Cecile Rickert is asking a question about a geometry theorem.

MP: How did you end up in differential geometry at Berkeley?

Banchoff: The way I really got into differential geometry was by taking good notes. Since I already knew point set topology very well by my Notre Dame experience, I placed out of the point set semester and went into algebraic topology. By that time I realized that you shouldn’t just study the notes of the course, you should read other books. So I started reading lots of books. But then, second semester, I sat in on Professor Ed Spanier’s course, which was absolutely marvelous. I couldn’t get over how wonderful it was because I already knew a lot of material. Here was somebody presenting it in such a clear way. I took very good notes, and Spanier knew that. At a certain stage, he asked me what I was going to do that next summer. I said I thought I’d probably go back and teach at Notre Dame again. (I had stayed on as a TA for Dr. Arnold Ross the first two summers after I graduated from Notre Dame.) He said, “Well, you should start doing some research. Would you like to be on my research contract?” It was a clear invitation to join him and work on some things.

Chern

That summer I attended the 1962 AMS Summer Institute on Global Geometry and Relativity at UC Santa Barbara. It was an idea of Charles Misner, who felt that all you had to do was get a bunch of physicists and a bunch of mathematicians together, and they would realize that they were talking the same language, make the translation, and discover they were talking about the same subjects. There would be a new birth of freedom, etc. It didn’t work, of course. There were marvelous lectures at the beginning of the conference by Shiing-shen Chern and Professor John Archibald Wheeler. But Chern had gone from the sum of the angles of a triangle to the Chern-Weil homomorphism in one hour. Then, the physicists had the idea that, if they could see Chern’s notes for just another hour or so, they’d be able to understand it completely. He didn’t have any notes. But somebody said, “I was sitting next to somebody who was taking what looked like really good notes.” I took notes; I always do. Chern said, “Bring them around tomorrow morning; I’ll take a look at them.” At least he gave me until the morning. I worked very hard. Chern looked at them and only made a few corrections. They became the notes of his lecture. Chern’s notes by T. Banchoff—wow! Chern asked if I would be taking his course in the fall and take notes. I said, “I’d be happy to.” He said, “You’re on Spanier’s contract? You should be on my contract. I’ll tell Spanier that you’re my assistant now.” I said, “Okay.” Chern and Spanier had just come from Chicago. They were not overloaded with students at that time. A couple of years later a friend of mine went into Spanier’s office to make an appointment, and Spanier said, “Before we even start talking, are you going to ask whether you can be on my contract and maybe be my student?” He said, “I was kind of hoping to lead up to that.” Spanier said, “I already have had five people today who asked me. I can’t take on anybody else, so save your time.” When I was there as a student, he wasn’t overloaded. We had marvelous opportunities. The seminars were small, and there was a lot of interchange. So I started working with Chern. He was wonderful. No matter what topic he would suggest, I sort of got into the combinatorial or polyhedral aspect of it where he wasn’t at all. However, he was totally supportive and encouraging.

Figure 4.7 Banchoff came under the influence of famed geometer Shiing-shen Chern in 1962, who soon became his thesis advisor.

After a few months, Chern told me that somebody was coming to visit whom I really ought to meet. “He’s like you, he thinks like you—Nicholas Kuiper.” He introduced me to Kuiper and said that Kuiper had done some work in totally absolute curvature and that I should show him the model I had made in a failed thesis project. Kuiper liked it. He said, “If you like that, you might like to read some of these papers that I did and try to find some polyhedral analogs.” I started looking at them. Kuiper was in town for a couple of weeks. I worked and worked and worked. I really found his work fascinating.

Figure 4.8 Nicholas Kuiper (left, with Banchoff). Chern told Banchoff to meet Kuiper because “he thinks like you.”

One day, I was folding laundry in the local laundromat. I was married by that time, and I was sketching as I was doing it. Then I said, “I’m not going to be able to prove you can’t solve the problem in 5-space; you can do it in 5-space. Here it is!” By the time I finished folding the laundry, I had the two basic examples. I went in the next day and showed them to Kuiper who really was astounded. He looked at me and said, “You know what you have here. You have a gold mine.” He said, “I’ll give you six months. If you haven’t written a thesis on this topic in six months, I’m going to give it to one of my students because it’s too good not be done by somebody.” I didn’t need six months.

MP: Did Ky Fan who was then at Santa Barbara ever come up to Berkeley?

Banchoff: I was always terrified by the possibility that Ky Fan would end up having some conversation in Mandarin Chinese with Shiing-shen Chern in which he mentioned that he had a student named Banchoff, and Ky Fan would yell, “Banchoff will never be a mathematician, never, never!” Chern would then call me up and say, “I’m sorry, I just heard. . . .”

As it did happen, I ran into Ky Fan at a meeting in New Orleans. I went up to him and said, “Dr. Fan, I was in one of your courses. I’m Tom Banchoff.” He said, “Weren’t you in my freshman course as a . . . ?” I said, “No, you’re thinking of Jim Livingston.” “Didn’t you take the course with. . . .” I said, “No, that was Jim Wirth.” He said, “Banchoff, oh yes. Oh, yes. I’m happy to see that you have developed into a mature mathematician.” And I smiled.

MP: How did you end up going from Berkeley to Harvard?

Banchoff: I’m embarrassed to tell you how simple it was. Chern asked me, “Have you thought about what you want to do next year?” I said, “I think I’d like to go to the Institute for Advanced Study.” He said, “That’s a good idea. What about teaching? Would you like to teach?” I said, “Oh, yes, I really do want to teach.” “Well,” he said, “What about Harvard? Would you like to teach at Harvard?” I said, “Yes, I think so.” He said, “I’ll recommend you for a Benjamin Peirce Instructorship.” A week later I got a letter offering me a Benjamin Peirce Instructorship. I went to Harvard, and, in fact, I had a wonderful teaching job the first year. I got to teach freshman calculus, which I really wanted to teach and totally enjoyed. I remember George Mackey on the phone, trying to figure out what else I would teach, and he said, “There’s this one course—mathematics for nonmathematicians—would you be interested in that?” I said “What is it?” He said, “It’s a general distribution course for juniors and seniors in the social sciences. You try to find topics that are interesting to these students.” I said, “I’d love to teach a course like that. Is it hard to get to teach a course like that?” He said, “Would you like to teach it next year?” I said, “That would be great.” There was an audible sigh on the phone, and he said to me, “The funny thing is, every year, there’s actually somebody who wants to teach that course.”

The Two-Piece Property

In part of that course, I developed for the first time a notion that turned out to be really my favorite theorem about the spherical two-piece property. An object has the spherical two-piece property if it falls into at most two pieces when you bite it. For example, a baguette does not have the two-piece property, nor does a long pretzel or anything long because you bite it so that it falls into three pieces. Biting of course, means intersecting it with a sphere. And so the question is if you intersect something with a sphere, does it fall into more than two pieces. Now a nonspherical ellipsoid can fall into three pieces, so it doesn’t have the spherical two-piece property. But the sphere does, as does the hemisphere. So does a doughnut, that is, a torus of revolution. In any case, I developed this for my students in that class as I was going along trying to teach them what mathematics was as a creative process. I told them what I was working on. By the end of the second year, the second time I presented it, I really had a good idea of what the classification theorem was. I gradually understood it more and more and had to develop the methods. That investigation started the whole subject of what’s called taut submanifolds. It all began with the spherical two-piece property.

MP: How did you go from Harvard to Brown?

Banchoff: That’s also embarrassingly simple. I gave up my third year as a Benjamin Peirce Instructor to go to Amsterdam as a postdoctoral research assistant to Nico Kuiper. While I was there I got an unsolicited letter from Brown offering me an assistant professorship. The Chair, Wendell Fleming, said he had been up to Harvard, and Shlomo Sternberg had recommended me as a geometer and a good teacher. That was that.

Dean Banchoff Returns to the Classroom

MP: You started at Brown in 1967 and became dean of students in 1971. That’s quite a departure from the academic life. How did your mathematics colleagues react to your becoming dean of student affairs?

Banchoff: I don’t think anybody objected. I still taught the third-semester honors calculus course. First semester I also taught a graduate course. I also co-taught a freshman seminar modes of thought course on “growth and form in mathematics, biology, and art,” which I didn’t tell anyone about. When the provost asked me to be dean of students, he told me that I wouldn’t have to teach at all. I said, “I want to teach.” He said, “I assumed that you would want to teach. You get paid the same whether you teach or not.” I taught three courses while I was a dean. I think some of the other administrators thought it a little unusual, but nobody said anything to me about it.

Figure 4.9 Young assistant professor Banchoff in 1970 before he was selected dean of students in 1971.

MP: But how did you actually become dean of student affairs?

Banchoff: At the end of my first year at Brown, I was asked to be the membership chairman of the faculty club. I realized that I met the criteria. I was representing youth, and I was representing science.

MP: Was this the “all-around guy” still at work, again?

Banchoff: No. I think it was probably the case of somebody who was very interested in different aspects of Brown University. When I came to the faculty club for lunch every day, I’d sit at the round table where a variety of professors gathered each day telling stories. I drank in all of Brown’s lore, and I’d learn about what was happening. I’d sit in on other people’s courses. I sat in on a course in ethics. I was in the American Association of University Professors. At the same time, I got involved in activities throughout the university rather early. Then I was asked to be on the university housing committee. It was a heck of a lot of work about all aspects of the university, and I did it pretty well. I also played pool at the faculty club, although I wasn’t very good. But I did run the table once.

MP: You were an assistant professor at this point. It’s risky business to take on a nonacademic job and not pay lots of attention to research. After all, research is what leads to promotion.

Banchoff: When I was appointed to the special faculty committee charged with creating a kind of faculty senate, one of my friends took me aside and said, “This may be serious business here. We might be coming up with strong recommendations.” Well, the president resigned half way through that semester, so it was serious business. I wasn’t worried about getting tenure because, frankly, at that time I thought that if I didn’t like it at Brown and if Brown didn’t like me for some reason or other, I could go somewhere else. I was never even the slightest bit worried about political prejudice. I wasn’t a rabble-rouser by any means. I’ve always been a very moderate, middle-of-the-road-type person. I was perceived that way, and I had no trouble talking to authority figures.

MP: How did the deanship offer actually come to you?

Banchoff: The administration wanted an inside faculty person to be the dean of student affairs, and the provost asked, “Will you do it?” It was unusual enough that I felt I should say yes. So I went home and told my wife that I had been asked to be dean. She said, “Did you say yes?” I said, “Yes. I think it’s a really great opportunity. I’ll see what it’s like.” So after a year I realized what you’d have to do in order to do it very well. I felt I’d be better as a teacher than as an administrator, and so did the students. The Brown student newspaper reported, “Dean Banchoff is returning to the classroom—a job for which he is much better suited.” It was the only time that year that I totally agreed with the Brown Daily Herald.

MP: Your passion is in mathematics, especially the fourth dimension.

Banchoff: Correction. My passion is in teaching, and I teach mathematics. Most mathematicians do think of the fourth dimension as being rather commonplace. It is curious that it has become such a big part of my life. But it is. It’s an entree for me to talk to any number of people who probably wouldn’t bother to listen to me otherwise. And since I like to teach and communicate, having an entree is half the battle.

Figure 4.10 Banchoff just back from another encounter with the fourth dimension.

Banchoff Meets Captain Marvel

MP: You said your interests in the fourth dimension started when you were very young.

Banchoff: Way back in fifth or sixth grade. I first read about the fourth dimension in the comic book, “Captain Marvel Visits the World of Your Tomorrow.” One of the panels shows a boy reporter going into a laboratory where a host guide says, “This is where our scientists are working on the seventh, eighth, and ninth dimensions.” And a thought balloon goes up from the boy reporter, “I wonder what ever happened to the fourth, fifth, and sixth dimensions.”

The rest of the story isn’t so good. But the lead-in was really wonderful. I kept thinking about it, and I really did decide that the fourth dimension had something to do with arithmetic. I also remember a comic book adaptation of a famous story, “The Captured Cross Section,” which was written in the 1920s. It has appeared in a number of guises. In one of them, folks are terrorized by this mysterious blob that comes in and changes shape, and someone says, “It must come from the fourth dimension. See, as you put your finger through this napkin you see this?” And so they spear it, and then a scientist gets in this machine that takes him to the fourth dimension, and he comes back with his glasses all askew and his eyes wild, and he’s dead because he’s been in the fourth dimension. I remember that. I’d love to find that book. Once you start thinking about an idea like that, you can carry it in all sorts of different ways. So I did. I used to think about it a lot. But I really got into it when I was a freshman in high school, and by the time I was a sophomore, I really had a full-fledged theory of the Trinity.

MP: Theory of the Trinity, as in theological terms?

Banchoff: A sympathetic biology teacher in my school was standing outside the administration building after class, and I told him that I wanted to tell him about my theory of the Trinity. I started talking to him about this theory about how God really is a manifestation of when the fourth dimension comes into the third dimension, but all we see is a slice. So we think that it is a person like ourselves, and that’s Christ. But there are two other parts we see, so that’s where Trinity comes in, and so forth. I was doing this very intently, and Father Jeffrey was smiling at my seriousness about this, and said, “Why is it so important that you have your theory validated today?” I said, “Because tomorrow I’m going to be sixteen years old.” I was interested in it in high school, and I still remember boring my friends crazy in pizza parlors with my napkin, doodling. I talked to my pastor about my ideas. He’s retired now. He’s seventy-five or so. He said, “Well, Tom, I’d have to say I thought you were probably going to end up in a heresy, but you seem to have done all right.” As a sophomore at Notre Dame, I did my theology term paper on the fourth dimension and the Trinity.

MP: Let’s talk about the reactions that you get from your listeners and your readers when you discuss the fourth dimension. It’s clearly an exotic topic.

Banchoff: It’s kind of mysterious and jazzy, and people think of it as kind of offbeat. They come to find out about it out of curiosity. Most of the people I talk to are kind of bemused by the experience. They like the visuals a lot, especially the computer graphics films we started making in 1968. It’s a topic that makes people feel challenged. It’s something they hadn’t thought about before, and it stretches their minds in some way. Many people say, “I didn’t quite understand it all, but if I think about it a little bit more, I think I’ll get it.” I say, “That’s okay. I don’t understand it all myself.” It would be terrible if they understood it. Well, it’s funny, because in a sense you do want to leave people with the idea that there are many different kinds of fourth dimension. That’s the biggest message. As I say in my talk, “I started out this talk by saying what’s a one-word definition of the fourth dimension? What would it be?” Many people say “time.” Why would that be the wrong answer? Because it’s the wrong question or a wrong question. Time isn’t the fourth dimension; it’s a fourth dimension. And it’s something as prosaic as making an appointment with somebody on Third Avenue, Fourth Street, the fifth floor at 6 p.m. You could write 3–4–5–6 in your appointment book. But if you talk about time as a fourth dimension, not the fourth dimension, then that puts people off the defensive a little bit. Then they may be willing to think about something else as being many-dimensional. And then you go from there. The idea of slicing a hypercube seems like a reasonable thing to do even though nobody has ever really had to do it up to that time. In our hypercube movie, we stayed with the really basic, fundamental, and elementary stuff. It’s hard enough. The things that are raised there are central—what we’re actually experiencing when we see the rotations of the four-dimensional cube, what it means to see something rotated, and to understand what rotate means, and what perspective means. These are questions that we should be asking about our ordinary experiences. These things make people look at their own experience in a different way. We don’t have a lot of converts running out to do four-dimensional stuff, but at the end of my courses and even at the end of my lectures, I have people doing three-dimensional things differently than they did before. That’s sort of being conscious of the dimensionality of experience. It’s a kind of a paradigm of experience as a way of organizing what we do and feel and having to come to terms with it. Visualization is very useful if you can separate things into different parameters and translate them into dimensions and translate that into some geometric form that you can look at, think about, and manipulate. Then you have a better chance of making sense out of whatever complicated thing you’re trying to deal with. That’s, I guess, the basic message.

Figure 4.11 Banchoff studying a hypercube. He first became interested in the fourth dimension as a fifth grade student.

Figure 4.12 As A Sphere passes through Flatland, the two-dimensional section changes, starting as a point, reaching maximal extent as a circular section, and then reducing to a point as it leaves Flatland.

The Only Thing I’m Good At

MP: Maybe you’re luckier than a lot of mathematicians. You’ve had the good fortune of choosing an area that is imaginable for a wide group of people.

Banchoff: Well, I didn’t have much choice. It’s the only thing I’m good at. I simply followed my own lights.

I feel lucky because I do have some phenomena to work with. The phenomena I study are things that I can present to people, and a number of people can appreciate them quite independently of understanding why I care about them, how I created them, or what they’re for. They are beautiful, after all. They are intriguing. All my life I’ve worked with polyhedra, I’ve worked with drawings, I’ve worked with models, and now, in the last twenty years, with computer graphics. There’s no question about the fact that the images that come out are fascinating. It’s sort of a different kind of fascination than fractals. Fractals are fascinating because you keep going and going and going—you never run out. But I work with things like hypercubes, which are so simple that you can describe them in a line; and yet you can watch the film “The Hypercube” three hundred or four hundred times and always see something new.

The inexhaustibility is not because the object itself is infinitely complicated; it’s just that the set of relationships that are available there is infinite. There are infinitely many views that you can take of an object, and there just seems to be no limit to the ways of combining these insights. You show it to people, and people react to these things at many different levels. You’re sharing, and you’re telling them some of the things that it means to you. You’re not telling them, “This is insight number 397 or number 398.” You’re just saying some words and letting them organize their own insights. And that’s rather nice. It’s a very pleasant time to go up there and parade some of your mathematical children in front of people, and they applaud, or at least they smile.

Most mathematicians probably aren’t so lucky. I think I mentioned this before: I’m one of the few people who can actually tell people what his doctoral thesis was about. Most mathematicians have no chance to tell. Of course, after I tell what my thesis is about, I have to explain to them why it’s not trivial.

MP: Your work certainly gets a lot of coverage in the popular press.

Banchoff: I remember the very first time there was an article about our work, a Washington Post article on the fourth dimension. One of my mathematician friends, who was in Washington for the mathematics meetings, came up to me and said, “It was a nice article, but I don’t see why people are interested in that. Our work on norms on Banach spaces is much more interesting.” I didn’t say anything, except that there’s no accounting for taste. What can I say? Some people are fascinated by this business, while others ask “What does this have to do with the price of tomatoes?”

Figure 4.13 Banchoff signing copies of his book Beyond the Third Dimension.

MP: Who asked?

Banchoff: Tom Zito was the Washington Post reporter who interviewed me. And so I started talking about using dimensional representation for brain research and for various economic models. Real mathematicians know you don’t have to answer that question about what math research is “good for.” You can say, “I don’t have to answer that question.” It’s a right answer because, as a matter of fact, I am excited. I got a call just a month ago from Tom Webb who’s a geologist-paleoclimatologist in a building next to mine at Brown. We’d been looking for years for a chance to collaborate on something. His data were four and more dimensional. His data have to be understood visually, and I think they can be understood from a four-dimensional point of view much better than any other point of view, and he thinks so too. He borrowed the Hypercube film to show to a bunch of geophysicists out in Colorado to try to convince them that they have to forget about the two-dimensional representations and go to families of three-dimensional representations sliced various ways—honest to God, four-dimensional stuff. His data sets are all four-dimensional in the sense that you have latitude and longitude to locate a position of a core sample from a dried-up lake bed. A core sample is a very nice way of translating time into space because the further down you go, the further back you go in time. And then, when you’re down there, you can look at something like the density of spruce pollen and use that information to tell you what was happening in the neighborhood of the core at that time. There’s enough experience spread over a two-dimensional plane that you can make a whole family of time series. And you really develop a four-dimensional continuum of two-space, one time and one density. Seeing how these things interact, sliced in long transects, river beds and so forth, you get three-dimensional slices in different ways, not just trivial ways but ways that respect the geometry and the geology of the situation.

The lucky thing about collaborations like this is that I get to meet all these great people outside mathematics. People come to me with problems. In one way or another, I interact with lots of people. New problems get suggested. And I follow them up. Not all of them. You get a dozen nibbles a year and maybe one or two get beyond the second phone call.

MP: Give me a little sampling of these people who are calling you up.

Banchoff: I worked with an anthropologist who has two projects that involve mathematics. One has to do with the ranging behavior of aboriginal tribes, hunter-gatherers, who operate in places where there are no large predators. So wide circles represent their range from the camp, and then after a while they move to another place—not too far away, but not with too much overlap from the original camp. So the overlapping behavior of circle patterns over time is something that we happen to be discussing when we are also talking about the geometry of circles in a plane.

The same fellow, Richard Gould, was doing underwater archaeology. They were studying a site in the Caribbean where there had been a number of ships that had sunk and then fill was brought in on top of them. They’re trying to excavate them, but it’s very difficult to excavate things under water because, if you take them out of water, they rot. So you really have to be able to work at the underwater level. So what you have is a slice of a large hull. And by looking at that slice, they’d like to predict exactly what the aspect of the rest of the hull is. So it’s very geometric in the sense that you’re trying to do a reconstruction from a small amount of data. That’s very exciting.

MP: How about artists?

Banchoff: Artists, writers, dancers.

MP: Dancers?

Banchoff: I cooperated with Julie Strand-berg, a choreographer who did a wonderful piece a couple of years ago: DIMENSIONS. It was performed in New York and twice at Brown. It was absolutely great. It was so thrilling. I cry every time I see it. It’s a sad story (chuckling). There’s this square, you see. It starts out with all these poor creatures who are limited to a two-dimensional space. They dance with their backs up against the wall. They can’t even pass each other unless they go over the top. They can do cartwheels, but they can’t do somersaults. Then if they go into Egyptian mode, they can sidle back and fourth, but they can’t really pass one another. But after we see a lot of these contortions and so forth, there’s a visitation from a being from a higher dimension on the screen. And everybody runs off except one square. She stays around to find out what’s going on and to have a chance to see people operating in the third dimension. There is a pas de deux that gives me the chills just to see it. It’s a marvelous thing watching professional dancers do this with the square watching; the square is wearing gray leotards and these people are the rainbow. And then finally she very timidly comes up and is welcomed and is pulled in between the two of them. Finally they take her out and liberate her into the third dimension. But in the end she has to return to the plane. It’s very sad.

The Joy of Teaching . . . Undergraduates

MP: Brown focuses on undergraduates. Was that part of the original attraction of Brown for you?

Banchoff: I like teaching the undergraduate courses more than I like teaching the graduate courses.

MP: Why?

Banchoff: I always feel a great sense of duty when doing the graduate courses that I have to cover lots of material whether I particularly find it interesting or not. I had about fifteen students this year, and not all of them were very geometric. I thought I had to make it possible for them to get something out of it even if they weren’t geometric. The geometric ones I had no problems with, and they had no problems. They got involved in it, and I’ll have one or two thesis students out of it, I’m sure. I felt I just couldn’t produce stuff that’s formal enough even though I’m fairly formal myself. Differential geometry is even harder because there’s a lot of technical stuff you have to do in the first-year course.

MP: Do you really think that’s true that you must get through all that technical stuff? Are you worried that in the process you may be damaging attitudes? Can the joy of doing the subject be lost by pushing people through all the techniques?

Banchoff: I think it is important to develop a common joy of the subject. I think it’s important for each mathematician to let her or his own joy show through. My strength is not in teaching a basic, technical first-year graduate course. I can do it. But as it happens, I very much like doing it the way we did it in algebraic topology. I taught the first semester in which I presented quite a bit of the motivational stuff, geometric stuff, if you will, with a lot of good problems. In the second semester a very talented assistant professor who likes the formal stuff very much continued with the same class. I think it’s very important for the students to see both. I think it’s terrible if they see only the formal thing, but I think it would be equally terrible or even more terrible if they saw only the intuitive stuff, because in this day and age you have to be able to do the formal stuff.

MP: But the formal technique generally will come more easily if the intuition is well developed.

Banchoff: I think you’re right. I don’t know of any first-rate researchers who don’t have a very well-developed intuition. And I know that it’s easy to be judgmental, but I’ve watched a number of situations in which people sort of tried to follow the coattails of famous mathematicians and never made it anywhere near the stature of the person they were trying to emulate. They were trying to talk the way the person talked, but what they were missing was the fact that that person had a couple dozen years of experience that they never articulated, but it was behind all the right insights that they had. Just talking like somebody doesn’t mean you share [that person’s] insights.

Figure 4.14 Student Deborah Feinberg learning Stokes’s theorem from Banchoff

One of the academic sins is coming up with good results and not doing that last bit of really hard work. It’s too easy to get up and tell your contemporaries about them. Most of the people who are working in a field know my definition of the normal Euler class and exactly what the properties are. But the only people who know that are the people who knew me. The people who are interested in this particular topic generally do know me, so that’s not the problem. The problem really is that I’m not even writing for a larger community outside those individual people whom I run into, and I’m certainly not writing outside of time and space. Writing articles is not just for people of this generation. There are people who will possibly be interested in what I write a hundred years from now. Unless I’m scrupulous enough to write it down seriously, nobody a hundred years from now is going to be able to read it whether they want to or not. So, part of the responsibility is there.

MP: How important is the geometric approach to you?

Banchoff: For me, a lot of times, when I hear a result that sounds very geometric, frequently I can’t understand the proof at all. If it’s really geometric, I want to try to find my own proof. I want to see what it really is. That’s happened to me a lot.

MP: Wouldn’t it be easier, to maybe learn some of this other stuff?

Banchoff: No, no.

MP: Why not? Wouldn’t that enrich you? Wouldn’t that give you even more power?

Banchoff: The fact that I’ve always tried to find the geometric content of theorems, say, about characteristic classes has enabled me to discover some things that other people passed by because they knew too much. They didn’t care about these problems. I wanted to find out how it worked, for example; how it worked for something else. And I would stay with examples until I understood something, and then I would prove something, and then occasionally something new would come out of it. It’s almost a quirk to try to find what I call the geometric content of theorems. Occasionally, it leads to a new insight as it did several times. Again, the sense of creativity comes with a feeling of uniqueness, not that you were the first person to reach the finish line in a race where there were a lot of other people running together and somebody would have gotten there ultimately, but maybe you find another path through the woods that nobody else would have taken.

MP: You’ve nailed some nice problems. What can you compare it to in the experience of others that they might relate to? Is it like winning a race? Is it like pumping a lot of iron—more iron than you ever imagined you could pump?

Banchoff: I think it’s more like taking a walk with a bunch of friends, up on a mountain trail. Maybe other people have gone off in another direction, either ahead of you or coming up behind you. You see something that looks kind of interesting and for some reason or other you come upon a particular rock and look out and see an extraordinarily beautiful landscape, a beautiful expanse, a beautiful sunset, and you call your friends, and they come up and they agree. That’s really beautiful. It might mean that everybody will decide to go off in that direction.

MP: But how long does that last? Do these experiences, these special experiences, come back from time to time? Do you think about them? Can you remember the feelings?

Banchoff: Yes. I can remember some of the feelings from the first time I discovered a polyhedral surface in five-dimensional space that had the two-piece property.

MP: You still can remember that?

Banchoff: Oh yes, that was back in that laundromat on the east side of Berkeley. I was folding laundry, and there it was.

The essence of the creative act is putting ideas together in a way that nobody ever has before and maybe nobody is likely ever to do again. You come up with something truly original, not just before other people do, but quite independently of what anybody else ever might do. Maybe I’m wrong. You can’t prove that somebody else wouldn’t have come along and done it exactly that way sometime later on. But it doesn’t have very much meaning to say that. I have a few things I’m proud of that I can show people, much in the same way that my friends who are artists would come in and show their favorite sketch or their favorite oil painting or their favorite ceramic sculpture. I really envy people who can actually hang something up on the wall. It’s kind of hard to hang up a theorem.

Dalí Meets “The Mathematician”

MP: Who is the most unusual person you’ve ever met?

Banchoff: Salvador Dalí. He’s certainly the most outrageous person I’ve ever met.

MP: What was the occasion?

Banchoff: He called us. The Washington Post had written this article about our work back in 1975. They had included a picture of me holding my own folded hypercube, hair sticking out, Daliesque. And as an insert in the back of the picture, they had a picture of Dalí’s Corpus Hypercubicus, body on the hypercube, the famous surrealist painting. When I came back to Rhode Island, my computer science collaborator, Charles Strauss, said to me that he was very surprised to see that they had included that picture in the paper because, although Dalí is well known for his love of publicity, he loves it on his own terms, and he was surprised that the Washington Post hadn’t gotten Dalí’s permission to use this. A couple of weeks later, when I came back from class, there was a note saying: “Call so-and-so in New York representing Salvador Dalí.” I said to Charlie, “What do you think?” And he said, “Well, it’s either a hoax or a lawsuit.” I called up, and a woman answered the phone and said, “Oh, yes, Señor Dalí would like you to come to New York and meet him.” So, I said to Charlie, “What do you think?” And he said, “The worst we can get out of it is a good story.” So we went down to New York City, and we got not just one but a whole set of good stories. We met Dalí at the St. Regis Hotel in the cocktail lounge. He was holding court. All sorts of strange people would come up and bow to him in a sense and be asked to sit in some large circle around him.

Figure 4.15 The mathematician Banchoff with the artist Salvador Dalí.

MP: How were you introduced?

Banchoff: As the “Mathematician.” Actually, not then. But he had called us, and he wanted to know about what we were doing. He was very curious about stereoscopic oil painting and Fresnel lenses and various other ways of displaying it. We hit it off very well. Since he was picking my brain, I thought I would pick his too, so I asked him about his painting. I showed him my model of the hypercube. He looked at it and said, “I may have this.” It wasn’t a question; it was sort of a regal statement. I said, “Well, yes, you may have it.”

MP: What did he mean by that?

Banchoff: He wanted it for his new museum in Catalonia. Later on, I saw the model now exists; there’s a model of it in his museum in Spain. No attribution, mind you. I asked him where he got the idea. He said, “Ramon Llull,” or something like that. I don’t want to try to repeat his accent because in public he would speak with kind of a mixture of French and Spanish and English and Catalan. Anyway, I finally realized who he meant. I said, “Ramon Llull, do you mean the famous twelfth/thirteenth century poly-math?” He said, “Yes, yes. You know Ramon Llull?” I said, “Well, yes, strangely enough I do.” I sat in on a course taught by a friend of a friend on the history of the Church in Spain and Italy in the twelfth and thirteenth centuries. I took it because it was something different from math and something I could do instead of math. It was a break, like reading detective novels. Well, the teacher mentioned Llull, so I read some of his work, kind of a fatuous work in which he proved all the major theorems of antiquity, and various other things. Dalí was very impressed that I knew who Llull was. I was very impressed that Dalí knew who Llull was. So, he invited us down. My wife and I came down the following week because he wanted to see our films, and we showed our films to him and his wife and his managers. Every spring he would invite us to come back down and bring our videotapes and films and slides, and he’d show us what he was doing. That went on for about ten years.