Maker of Patterns

by Freeman Dyson

  WEDNESDAY

  Tuesday being the great day when I was to meet Miss Hill, I repaired to Mill Lane in good time to see her at nine and found a large class of people, and when Miss Hill came in, she singled me out and said she was glad to see me. She must have been told about me by Besicovitch. The class was fairly elementary, but it is definitely worth going to for vocabulary and points of idiom if nothing else. She began with half an hour’s communal translation from English to Russian, and then half an hour back again. She is a good teacher, and I picked up quite a few new ideas during that hour. I found Henry Padfield, a man whom I knew vaguely at Winchester. They expect me to take part in their performance of the Kossovo Ballads at the end of the term. It is being recorded for a broadcast on Kossovo Day sometime in June. She says that I shall probably have to appear in a crowd of warriors or peasants or something, but I have not heard any more as yet.

  FRIDAY

  I am still going to Mill Lane at nine in the morning on Tuesdays and Thursdays and like it very much; I am also going definitely to appear as one of a crowd in the Slavonic Society’s Yugoslavian binge. That was not what I intended, but it will be amusing. The performance is next Wednesday so that they do not require very much from me in the way of going to rehearsals; as far as I know there is one on Sunday evening and probably only one after that. So it will not be at all burdensome. I went to Besicovitch last Wednesday. I found there a young man to whom Besicovitch had given similar instruction a year or two ago; it appears that Besicovitch is in the habit of improving the Russian pronunciation of anyone he can get hold of. At any rate this young man had come back to see Besicovitch, and he recited the same verses that I had begun reciting to him. So we all three sat and conversed in Russian, and I did very well; it is so much easier with three people, as you can let the other two carry on while you get your next remark straight. He said that he was in the army in an officers’ educational course and was hoping to get to Russia. Although he speaks the language quite fluently, he said that it was very unlikely that they would have him in that capacity. They are, however, offering special scholarships to any students who are prepared to learn Japanese or Persian.

  Last Wednesday, a week later, Besicovitch asked me to come again and discuss one or two mathematical problems. I arrived at eight-thirty in my usual clothes and found a regular social gathering with subfusc suits and long dresses. I was just preparing to run away and hope nobody had seen me, when Besicovitch welcomed me, and we sat down on a sofa and talked mathematics while the others played bridge. Besicovitch has the most perfect manners of anybody I have ever met, and made me feel more or less comfortable. He always introduces the guests, and escorts them to the door with a low bow, and generally behaves like a landed aristocrat. Mrs. Besicovitch was there, also a Miss Thompson and four more people who played bridge with each other, and whom I did not see much of except for handing round toast and tea at ten o’clock. Besicovitch was as amusing as usual in spite of the uproar. He tried to introduce Miss Thompson into the circle, but she is not a good linguist; she has been learning Russian from Mrs. Besicovitch, she said, for years, but she could not understand anything Besicovitch, still less I, said; so we finally relapsed into English. I shall be very interested to know more about Besicovitch, as I hope I shall in time. When he asked Miss Thompson whether she, having toured Lapland, could speak Lappish, I asked him whether he, having lived at Perm, could speak Tartar. He said he was surprised that I should have thought of that, as his native language was actually Crim Tartar. His family lived in Crim Tartary and spoke Crim Tartar among themselves, including his elder brothers and sisters, but when he was born, they decided only to speak Russian to him so as not to spoil his chances of speaking Russian like a native. I never thought when I read The Rose and the Ring that such a place as Crim Tartary existed, or that I should ever meet a native from it. But what sort of people his family were he did not say. As far as I remember, in the book the sympathy of the reader was not on the side of Crim Tartary.

  Thackeray’s The Rose and the Ring: A Fire-side Pantomime for Great and Small Children (1854), was one of the favorite books of my childhood. Written under the pseudonym Mr. M. A. Titmarsh, it is a spoof of a romantic fairy story, involving the warring states Paphlagonia and Crim Tartary.

  TUESDAY

  The Kossovo play was a success, beyond all my expectations. I was told to go to a rehearsal last Sunday week at eight-fifteen p.m. and after waiting in the dark for half an hour discovered that that rehearsal was really at two-thirty in the afternoon. So I came for the dress rehearsal on Monday without any previous experience; and that dress rehearsal lasted from four till eight, during which time three out of ten scenes were rehearsed; it was about the most incompetently managed rehearsal I have ever attended. However, on Wednesday night when the King and Queen of Yugoslavia and the entire staff of the Yugoslav forces were situated in the front row, and the BBC had microphones and cameras all over the place, things were fortunately very different. The main difference was caused by a handsome and clever young man who took the part of the Bard, who was not performing on Monday. The play being a dramatised version of ballads, there is a great deal of monologue by the Bard while the other people sit in silence. Thus it was a great asset to find that this young man had a marvellous voice, which made the rather indifferent versification sound highly poetic. This man was called Dimitri by everybody, and it was not till afterwards that I discovered he was the long-lost Prince Obolensky. Apart from him the chief part was Tsar Lazar, played by Henry Padfield whom you may remember from Winchester. Padfield was considerably brighter, since he went with the Home Guard on their twenty-eight-mile night march on Saturday night, and had recovered by Wednesday. The performance on Wednesday was very effective and was broadcast to America and to Yugoslavia during the week, and the queen of Yugoslavia was in tears all the time. The BBC sent Miss Hill a letter of congratulation which she read out proudly this morning.

  On Thursday there were no microphones and cameras, but otherwise it was the same, and again was successful; at the end the actors were invited by the king to a party in the Arts Theatre, to go on till one o’clock. The ballads of Kossovo are very interesting and well worth listening to; I do not imagine anything is known about their authorship, but it is a beautiful story. Before the battle St. Mary sent a falcon from Jerusalem to the king, with the message that he must choose either to win a great victory or to reign in a heavenly kingdom, in which case he was to draw up his whole army to a solemn Mass on the field of Kossovo, and then they would know certainly that the whole army would perish.

  King Peter was in exile in London, still recognized by the British government as the lawful sovereign of Yugoslavia. His agent General Mihailovich, in Yugoslavia, was supposed to be organizing resistance to the German occupation. Meanwhile the Communists led by Marshal Tito were organizing a more effective resistance. The British government later transferred recognition from the king to Tito.

  I am now comparatively idle; this morning Dirac gave his last lecture, so that will be the last time I shall have four lectures in a morning. I have found that listening to all these lectures is just about as much as I can do; Dirac reached a climax of difficulty in his last lecture. Littlewood has also finished, so I have only one more of Miss Hill, Hardy, Newman, and two of Besicovitch.

  MONDAY

  Besicovitch is without doubt a complete dear. On Wednesday he invited me to come with him after lunch for a walk to speak Russian, and we talked about some problems he had given me a few weeks ago. I told him a few incipient ideas I had had on the subject, and he was so encouraging that I spent the next two days in a fervour of activity, and actually proved two quite interesting results of which one may be new. This is the way I have always wanted to spend my days, but it is seldom that it really happens. On Friday the weather was lovely and all Besicovitch’s supervising was over, and he took me for another and longer walk round the country, during which he discoursed largely on Russia; and Friday, Saturda
y, and Sunday I spent an idyllic existence, doing research in the mornings and evenings and talking Russian in the afternoons. Indeed I have been talking so little English this week that I shall soon be forgetting it.

  It appears that I am not the only person on whom Besicovitch lavishes his goodwill, for there was a most unexpected incident on Saturday when we came in to Neville’s Court on our return; a small figure at the other end of the court outside Besicovitch’s rooms rushed down the court straight at him, and he shouting “My darling” embraced her right off her feet. She was about twelve years old, but as they went to his rooms then I did not discover who she was. On Sunday, however, as we were returning through the backs we met another young girl, rather smaller this time, plus a father. The three of them talked a very peculiar conversation, the girl to her father in Czech, the girl to Besicovitch in English, and Besicovitch to the father in Russian. Besicovitch was if possible even more delightful with this young friend than with the other, and they chased each other round the lawns to the great delight of the many passersby. It appeared that both of the girls were Czech refugees, but they were not related; the second one was called Ladochka, lived at Brno, and her father escaped sometime after Hitler arrived by leading a trade mission to Rumania and taking his family with him. They both spoke perfect English, though the father did not. It is certainly no wonder that people are so fond of Besicovitch.

  I have had a Russian game of billiards with Besicovitch. In the billiards I did very well on the whole, making 109 while he made 163; he is fairly good. Hardy the other day made a new record break of 119 and was enormously pleased with himself.

  MAY 19, 1942

  In my second year as an undergraduate at Cambridge, I registered for national service. I was in the Home Guard, which was supposed to help the army defeat the Germans in case of an invasion. In 1940 the threat of an invasion had been real, but in 1942 the Germans were heavily engaged in Russia, and nobody took the threat of an invasion seriously. It was lucky that we never had to do any real fighting. The Home Guard required me to take part in occasional night exercises. The invasion became more and more unlikely as the years went by, but the exercises continued until the end of the war. They helped to sustain the wartime spirit that made England a friendlier country during the war years.

  I went on the exercise on Saturday after all. We sat in the parade ground till ten p.m., when my section set out for Grantchester on bicycles. The idea was to stop the Welsh Fusileers from capturing Cambridge, so we were stationed near a bridge at Grantchester; two men to demolish the bridge and the rest to give protection to them. We took up a very halfhearted defensive position on the roadside, lying along the hedge, and waited for orders to demolish the bridge, or the arrival of the enemy. We had a beautiful notice saying Bridge Demolished and a small piece of explosive to make a bang. Of course, as always when I am on an exercise, the enemy never came anywhere near Grantchester, though they overran about a third of Cambridge. We stayed under that hedge from eleven p.m. till eleven a.m. It was impossible to take enough clothes to be able to keep warm, though the weather was quite fine. Last night I spent at the hut fire-watching. I was out from twelve till two-thirty but slept pretty well the rest of the time. Fortunately the weather was again good. But this military activity does take up a great deal of energy when they have night operations.

  FRIDAY

  The other evening I played a game of billiards with Besicovitch and had the honour to be directed in the strokes I should play by Hardy, who is a first-class player. He sat in an armchair and criticised both of us. In the end he arose and said that he would enter the game on the seventh break from now and would play one break; if the score he made was of the form 4n+1, it would go to me, if of the form 4n+3, it would go to Besicovitch, and if it was even, it would be null and void. He urged us to play so as to leave either a good or bad position as suited our scores at the moment. When the moment came, he scored zero, and so no ill-feeling was caused. In the end I won the game.

  NOVEMBER 10, 1942

  Since Thursday I have spent most of my time working on the theory of partitions. I discovered, by experiment, a striking result in the theory; and I have been trying to prove it, without success, and when I got tired of trying to prove it, I started verifying the result for various special cases, a process which can go on ad infinitum; so I have been rather busy. The partitions of 4 are five in number, namely 4, 3+1, 2+2, 2+1+1, 1+1+1+1. Ramanujan discovered that the number of partitions of any of the numbers 4, 9, 14, 19, 24, 29, is always divisible by 5. If in each of the partitions of 4 you take the number of parts and subtract it from the largest part, you get in the five cases 4–1=3, 3–2=1, 2–2=0, 2–3=–1, 1–4=–3. You have the five numbers 3, 1, 0, –1, –3, and one of them (0) differs from 0 by a multiple of 5, one of them (1) differs from 1 by a multiple of 5, one of them (-3) differs from 2 by a multiple of 5, one (3) differs from 3 by a multiple of five, and one (-1) differs from 4 by a multiple of 5. What I have discovered is the following: If N is any one of the numbers 4, 9, 14, 19,—and you write down all the partitions of N, you can treat each of them as I did before, i.e., you can subtract the number of parts from the largest part. You get a set of numbers, which you can arrange in five groups according as they differ from 0, 1, 2, 3, or 4 by a multiple of 5. In the case N=4 there is just one number in each group. I find that there is always an equal number in each of the five groups; for instance, when N=9 there are just six in each group. But I cannot prove it in general. You will see that this result explains why the total number of partitions is always divisible by 5, because the total number of partitions is got by adding the membership of all the five groups.

  The partitions of 5 are seven in number, namely 5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1. Ramanujan proved that the number of partitions of any of the numbers 5, 12, 19, 26, 31, 38 is always divisible by 7. My result then carries over word for word to this new case; you get seven equal groups instead of five, when N is one of the numbers 5, 12, 19, 26, 31, I have verified this for N=19 which comes under both cases. You get 490 partitions, and from these you get 490 numbers by subtracting the number of parts in each from the greatest part. Then if you divide them by the scheme 0, 1, 2, 3, 4 you get five equal groups of 98 numbers. If you divide them by the scheme 0, 1, 2, 3, 4, 5, 6 you get seven equal groups of 70.

  I published my discovery about partitions in the student magazine Eureka which was the journal of the Archimedean Society (1944). The conjecture which I had verified numerically was finally proved analytically in 1953 by my friends Oliver Atkin and Peter Swinnerton-Dyer.

  NOVEMBER 17, 1942

  I made a closer acquaintance with an undergraduate called [Fritz] Ursell whom I like very much. We had a session on Sunday night, he providing coffee and I cake. He is a refugee from Düsseldorf, and talked a lot about the refugees. He confirmed me in the opinion that the policy of the English government and trade unions toward refugees is both uncharitable and foolish. In particular I was impressed by the fate of the majority of his family who stayed in Germany; they are now all in a ghetto in Lublin. There was only the choice between clearing out altogether and complete defeatism. It was not the weaker elements that became refugees, but all the people who had sufficient energy and moral courage to wish to preserve their standards, and to carry on their lives, at the cost of heavy and immediate sacrifices. However, I will not preach any further. Ursell’s father is a quite eminent doctor, and he is not allowed to practise; though in fact there is and always will be enough work for any quantity of doctors.

  It is difficult to remember how much we knew or cared about the deliberate killing of Jews by the German government. We certainly knew that Jews were persecuted and dismissed from their jobs. Fritz Ursell was one of several Jewish friends who had escaped to England and told me how badly their relatives were treated both by the Germans and by the British. The full extent of the horrors in the concentration camps became known only as the camps were overrun in the last year of the war. H
itler’s decision to implement the Final Solution and massacre all the Jews was kept secret in Germany and was not known to us in England until the end of the war. Both in Britain and in the United States, the governments had a deliberate policy of suppressing information about the killing of Jews, knowing that in both countries the quickest way to lose public support for the war would be to give the impression that we were fighting the war to save the Jews. Neither Churchill nor Roosevelt was fighting the war to save the Jews.

  NOVEMBER 30, 1942

  On Saturday I went to tea with Besicovitch and enjoyed it more than any tea party I have attended for a long time. There were present two young Diracs, by name Gabriel and Judith. Gabriel is seventeen and a first-year undergraduate from St. John’s, Judith fifteen and at a school in Cambridge. They are Hungarians by upbringing and knew a lot about Central Europe; also they used to row on the Danube with von Neumann the great topologist, an almost legendary figure now in America. They came to England about four years ago, but speak the language very well. Gabriel is an ardent member of the Communist Party and kept the ball rolling from the start. Besicovitch himself is not a Communist but is the most persuasive arguer for it that I have ever struck, as he never makes wild statements or resorts to political clichés. The young Dirac is reading mathematics but is at present more interested in politics. I sometimes think I ought to become more political; it is a very fruitless activity but it does keep one from getting bored. Ursell the refugee was telling me that the real crisis of reaction in Germany was in 1921–22 when all the competent Communist leaders were murdered. He is however of the opinion that Russia will keep them from doing it again this time. I hope so, as I am sure England would not raise any objection. On Saturday I played a game of billiards with Hardy. He is very good. He plays incredibly fast; made a break of 53 in about three or four minutes. The score was about 240–140, and I had also a heavy handicap.