Prof

by Dermot Turing

  BILL OF RIGHTS – 1 W&M c.2

  An Act declareing the Rights and Liberties of the Subject and Setleing the Succession of the Crowne.

  The Lords Spirituall and Temporall and Commons pursuant to their respective Letters and Elections being now assembled in a full and free Representative of this Nation takeing into their most serious Consideration the best meanes for attaining the Ends aforesaid Doe in the first place (as their Auncestors in like Case have usually done) for the Vindicating and Asserting their auntient Rights and Liberties, Declare That […] excessive Baile ought not to be required nor excessive Fines imposed nor cruell and unusuall Punishments inflicted.

  In 1689 the question of what amounted to ‘cruell and unusuall Punishments’ was first considered in the English courts. One Titus Oates had been complicit in the Popish Plot and convicted in 1685. His sentence was a fine of 2,000 marks, pillorying four times annually, life imprisonment, defrocking, being whipped from Aldgate to Newgate, and from Newgate to Tyburn two days later. In 1689 he petitioned for release from this sentence, and was pardoned and pensioned off. Unfortunately the case gave little guidance to judges in future centuries. To my mind, allowing a subject liberty only on condition of taking hormone injections and receiving psychotherapy might count as ‘excessive Baile’ and almost certainly as ‘cruell and unusuall Punishment’.

  In 1791 the Eighth Amendment to the United States Constitution was adopted. Exactly tracking the wording of the United Kingdom Bill of Rights 1688, this too prohibits cruel and unusual punishments. And, as you might expect in the United States, there is a lot of jurisprudence and commentary on what constitutes a cruel and unusual punishment. Any sentence which is new or ‘contrary to usage’ may be regarded as ‘unusual’, and the Amendment has been used to challenge a wide range of sentencing practices and prison conditions, and to improve the lot of offenders. The same principles ought also to apply by virtue of the European Convention on Human Rights. I do not think that it was legal to force Alan Turing to undergo hormone treatment.

  The more justifiable part of the ‘treatment’ was the psychotherapy. A study by Dr Mary Woodward was published in the gloriously named British Journal of Delinquency in 1958, which has a blazon across its front cover bearing the words ‘Special Number on Homosexuality’. Maybe that is enough to tell you all you need to know. Dr Woodward’s patients were the homosexuals, mostly referred by the courts and probation officers, whom she treated in 1952 and 1953. Dr Woodward says that psychotherapy alone was the treatment given in all but ‘five cases, which were also treated with the hormone method’. Her results are noteworthy, because, according to her conclusions, psychiatric treatment actually worked.

  Treatment appears to be most successful (resulting in a loss of the impulse) with bisexuals who are under 30 years in age, who have not started overt homosexuality until their late ‘teens and have not a very long habit of activity. This result can, however, be achieved with older patients, provided they are co-operative and can be given treatment over a long period. Although the strength of the homosexual impulse is little diminished, behaviour can be changed in the direction of greater control or discretion among older homosexuals with a long history of persistent activity.

  For every psychiatrist like Dr Woodward, eager to find that their therapy was successful, there were plenty who knew that a finding of ‘success’ said more about the state of mind of the psychiatrist than that of the patient. Lytton Strachey had complained in a letter written in 1923 that ‘psychoanalysis is a ludicrous fraud’. The author of the Oxford University study referred to previously wrote: ‘psychiatry to-day suffers from the exaggerated expectations of the general public. Mental treatment, like prison, cannot be expected to turn a recidivist thief into a reliable cashier or a confirmed homosexual into a happy husband and father.’

  So, Alan’s sentencing was procedurally flawed, partly illegal, and ineffective as to the rest. Not that much of this would have registered with any court in 1952 given the prevailing social attitudes. There was not much point in applying for the sentence to be quashed, as this would probably have resulted in Alan being sent to prison, ghastly publicity, and him losing his job. Furthermore, by being put on probation, Alan’s conviction did not count as a ‘conviction’ by virtue of Section 12 of the Criminal Justice Act. Being put on probation, and being ‘treated’ because he had a ‘condition’, was illogical and wrong in principle, but it was the easiest way for him to keep his job and forget the whole nasty business. M.H.A. Newman was more than just a character witness for Alan at the end of March 1952. The University of Manchester has not retained personnel files from this era, but it seems that Newman was sufficiently autonomous in his department to square things with the university authorities.

  The remaining question now was how Alan Turing would cope. To Philip Hall, who had congratulated him on his Royal Society fellowship a few months before, he wrote:

  Dear Philip,

  Your letter very welcome. I am afraid I didn’t make my communication very clear. I am both bound over for a year & obliged to take this organo-therapy for the same period. It is supposed to reduce sexual urge whilst it goes on, but one is supposed to return to normal when it is over. I hope they’re right. The psychiatrists seemed to think it useless to try and do any psycho-therapy.

  The day of the trial was by no means disagreeable. Whilst in custody with the other criminals I had a very agreeable sense of irresponsibility, rather like being back at school. The warders rather like prefects. I was also quite glad to see my accomplice again, though I don’t trust him an inch.

  Yours ever

  Alan

  So, Alan was going to take it with fortitude and good humour, if with scepticism when it came to the psychotherapy. Donald Bayley, his old colleague from Hanslope Park, said that Alan regarded the whole business as a ‘joke’, but this jollity may have been old-fashioned grinning and bearing it. At least the newspapers had been kind to Alan: apart from the intrepid Alderley and Wilmslow Advertiser there was little coverage, for there was a bigger crime story to fill the pages of the popular press that day. At the Old Bailey a male nurse, an Indian married to a German woman, was on trial for poisoning his wife’s food, with the assistance of the housekeeper with whom he was having an affair. That was much more fun to read about than another routine ‘gross indecency’ sentencing, even of an FRS.

  Orthogonal trajectories

  The FRS had significant work still to do on his theory of morphogenesis. Having dealt with animal spots in his 1952 paper, it was time to turn to the problems of organism development in plants. Mapping the behaviour of chemicals diffusing across a two-dimensional surface had provided a possible explanation for the appearance of blotches on animal skins. But what about three-dimensional things like fir cones and the position of leaves on plant stems? This was the problem of phyllotaxis. Confusingly enough, important work on phyllotaxis had been done by a man called Church. Not, in this case, Alonzo Church, but Arthur Henry Church, who had written a book called On the Relation of Phyllotaxis to Mechanical Laws in 1904.

  The first part of Church’s book is entitled ‘Construction by Orthogonal Trajectories’ – in other words, looking at things sideways. Church explained that you could map the nodes from which leaves emerge from a plant stem onto a two-dimensional sheet, by imagining that you had made a vertical cut along the stem and unpeeled the skin. The nodes tend to spiral up the plant in opposite directions, but when laid out on the unpeeled sheet they sit in straight parallel lines, with the leaves at uniform intervals. These are called ‘parastichies’. The number of parallel spirals – parastichy numbers – seem to belong to the Fibonacci series, as Alan had noted in his letter to J.Z. Young. Similarly, the arrangement of seeds in a sunflower head is typically one of spirals in opposite directions, and the numbers of spirals are also typically from the Fibonacci series. But what determines where a leaf node sprouts, or where a sunflower seed is positioned?

  As Bernard Richards, Alan’s las
t research student, observed, Alan’s own thought process was to look at problems ‘orthogonally’. Alan’s reaction-diffusion theory might be able to explain all this.

  According to the theory I am working on now there is a continuous advance from one pair of parastichy numbers to another, during the growth of a single plant. During the growth of a plant the various parastichy numbers come into prominence at different stages. One can also observe the phenomenon in space (instead of in time) on a sunflower. Church is hopelessly confused about it all, and I don’t know any really satisfactory account, though I hope to get one myself in about a year’s time.

  To get to a satisfactory account, Alan would have to solve the partial differential equations on which his theory was based. By choosing a variety of inputs, the equations could be solved numerically using the Manchester Mark 1 computer. The output data were written onto graph paper by hand, and the graph paper was shaded to show points of concentration. Alan was using the idea of the contour map to model the growth and development of plants.

  PHYLLOTAXIS AND FIBONACCI – THE TURING DRAWINGS

  Alan Turing’s beautiful drawings in the King’s College Archive in Cambridge were never written up by him, and considerable posthumous effort has been put in by scholars – notably Professor Jonathan Swinton – to interpret them. They connect directly to the work done by Church, but Turing’s theory went far beyond Church’s work, which was largely descriptive and analytical. Church was trying to fit biology and mechanics together. He saw the problem of shape as one of packing: how and where would a growing plant fit in the next floret or leaf node. Alan was going further: could his chemical theory of morphogenesis explain the emergence of florets and nodes in the places where they appear, and thus reveal the reasons behind Fibonacci series being found in fir cones and sunflowers?

  The diagrams shown here illustrate some part of what Alan was doing with his mathematical models:

  Image (1) is a split-and-unpeeled stem of a hypothetical plant. Spirals of leaf nodes can be followed upwards going left to right and (rather steeper in gradient) right to left. The Fibonacci number 21 can be counted on the left-to-right spiral, being the total number of leaf nodes which you have to bypass on the stem before you reach another node on the same spiral, which is directly above your starting point. The drawing also shows a lattice structure. Lattices emerge from the reaction-diffusion theory the same way that growth-points appear on a ring (as Alan had shown for the tentacles of Hydra in his paper on the Chemical Basis of Morphogenesis).

  Image (2) is another unpeeled stem. Professor Swinton interprets the spots in the left-hand part of the drawing as growth-points whose positions on the stem are constrained like beads moving along a wire. Where they stabilise will depend on a small number of factors, including mutual proximity, leading to the pattern in the other part of the drawing.

  Image (3) is a photograph of a sunflower head in the King’s College Archive and (4) is Alan’s schematic redrawing, with the seed positions numbered. Spirals are clearly visible in both clockwise and anticlockwise senses. The number of spirals in either sense typically falls somewhere on the Fibonacci series.

  Image (5) is a theoretical representation of the growing tip of a stem. The doughnut shape is the ‘apical meristem’ beneath or (in a multi-floret flower like a sunflower or a daisy) around the growing tip, which is the area in which cell differentiation occurs leading to the sprouting of leaves or florets. This schematic assumes that the florets appear in positions determined by the Fibonacci angle (137º – a figure arrived at from the ratio of successive numbers in the Fibonacci series). But why do they do so?

  Reaction and diffusion are not sufficient to explain Fibonacci numbers, as Alan’s draft paper on applying morphogen theory to phyllotaxis acknowledges. Alan recognised that growth – the change in shape over time – would also change the geometry. The Kjell computer routine introduced an additional term into the equations to bring this into account; Professor Swinton thinks this may have been the key to the Fibonacci problem.

  Alan Turing’s phyllotaxis drawings were much neater and more attractive than his usual scruffy work done for his own benefit. It seems quite likely that the plan was for these to be used to illustrate a new paper, one that, with the rest of his work on Fibonacci numbers, was never completed.

  A short story

  Mrs Dixon was the cleaning lady at the Royal Society-funded Computer Laboratory in Manchester. It was all in a day’s work dealing with the untidy professors who left their notes about and got excitable if the blackboard was cleaned. She also expected science fiction to become reality in a place where they were operating an electronic brain. But to find an empty pair of trousers in the middle of the floor, like a time-travel scene with a vanished body, was a rather close encounter with the surreal.

  Prof, the owner of the trousers, had simply gone for a run. He was supposed to change in his office – the Royal Society grant hadn’t extended to changing rooms – and how the trousers had come to be where they were would remain a mystery. One person the runner encountered in the Cheshire countryside was Alan Garner, who later found fame as a writer of fiction for children. He was interviewed in 2012:

  GARNER: It must have been about 1950, and when I was that age, which would have been 16–17, I was a serious athlete. And in those days people didn’t go round clogging up the roads with jogging, so when I was out training, for me to see somebody else running was a very strange experience. And so I fell into talking with the strange man, who was quite a different shape from me: I was tall and thin, and he was stocky with a great barrel chest. It was strange to see him running because he didn’t run, he was hammering the road: he was running into the ground, not over it. And he had a very extraordinary voice. It was an aristocratic English voice, high pitched, but he had the most remarkable sense of humour. I realised immediately once we got talking that we had quite a lot in common, which was a sense of the absurd. And that’s how it grew. So we’d make loose agreements, you know, ‘Will you be out running on Tuesday? Okay, I’ll see you then.’ And this went on for nearly three years until I went to do my military service.

  INTERVIEWER: Was it mainly running, when you would get together and talk with him?

  GARNER: It was only running. And I realised that why he was doing it. He was thinking. He was using running to think.

  Monopoly, Newman-style. Alan was trounced by M.H.A. Newman’s young son William playing on this board of William’s design.

  Lyn Newman, Max’s wife, was also part of Alan’s inner circle after his move to Manchester. She too had a literary streak and was accepted on the fringes of the Bloomsbury group. Before the war she had produced a literary journal read by the likes of Virginia Woolf, Graham Greene, John Maynard Keynes and others. When her children were older she went on to write some more substantial books, as well as the foreword to Sara Turing’s biography of her famous son. Alan was good with the children, as Newman’s son William described:

  Alan now became a close family friend of ours. Despite his shyness he seemed to me more approachable than the other mathematicians who visited us. His choice of presents on my birthdays was especially thoughful and generous: a splendid steam engine one year, a little hobbyist’s toolkit another. He played games with me and Edward1, and lost ignominiously to us at Monopoly. He came with us on a brief spring holiday in Criccieth, North Wales, where we rented a house. I remember Alan running on the beach, disappearing into the far distance and coming racing back. When he later bought a house in Wilmslow he would sometimes run the dozen or so miles from there to our house in Bowdon. Once I heard a noise in the early hours of the morning and went to the front door to find Alan dressed in running gear. He wanted to invite us to dinner and, thinking us all asleep but having nothing on which to write, was posting through our letter box an invitation scratched on a rhododendron leaf with a stick.

  There was more to the game of Monopoly than this simple account might suggest.

  I remember a lat
er phone call from Alan, asking me if Edward and I had a Monopoly set. I replied that we did, and was delighted when he suggested we three might play a game when he next visited. At the time it didn’t occur to me to mention that our ‘set’ was home-made, with a ‘board’ drawn by me on a sheet of paper. I could also have told him that its layout was somewhat unorthodox – I’d added an extra row of properties, diagonally connecting the ‘GO’ square to the square on the opposite corner. I cannot recall why I’d done this, but I may have wanted to provide a choice of routes. In the end Alan lost – neither Edward nor I showed him any mercy.

  And there was a bit more to the jog along the beach, too:

  One afternoon of overcast skies and threatened rain, Alan changed into blue shorts and disappeared for a short time. When we asked him where he had been he pointed out a promontory of Cardigan Bay seven or eight miles north-west, inaccessible by road. We might have entertained the idea of walking there, but not without carrying a meal and macintoshes with us, scarcely without resting an hour or so on the way. For us it would have been a day’s outing, but Alan did it between lunch and tea. From that day – although his normal walking gait was uninspired and almost shambling – we all felt awed, as if Mercury had joined our circle of acquaintances.

  Lyn Newman persuaded Mercury to read War and Peace and Anna Karenina – something to add to Alan’s diet of reading which had, apart from the likes of John von Neumann, been limited to Jane Austen. There was another literary figure in his life, too: Alan had taken a holiday with Robin Gandy in 1948, a short let on a house in, predictably enough, Wales. A non-mathematician was in the party – P.N. Furbank, who was a scholar of English from Emmanuel College, Cambridge. Nick Furbank fitted in surprisingly well, maybe because he knew E.M. Forster, who was living at King’s; he would go on to become Forster’s biographer. Nick became a frequent participant in holiday expeditions and one of Alan’s close friends. Maybe as a result of rubbing shoulders with authors, Alan ventured briefly into the uncertain world of fiction himself. There survives a six-page manuscript, which is not dated but seems likely to have been written in mid-1952. Here is the start: