The Oxford Road Show
I gave my Turing and Fibonacci phyllotaxis talk at the Turing 2004 conference yesterday. It was at Manchester University, on Oxford Road. 10 minutes into my talk, the fire alarm went off, and I ended up bellowing most of the rest of it on Oxford Road, above the roar of the buses echoing across the pavement (it's a quality bus corridor you know).
But anyway, the talk (6Mb Powerpoint) itself is online. The work it is based on is here.
Acknowledgements
Twenty years ago, I saw pictures of spots and stripes in the Turing Archive. I have always wanted to know what the pictures were of, both narrowly (which equations, which parameter values?), and more broadly: which questions were being asked and why? I still don't know the answers, though we can begin to make some guesses. The archive is too fragmentary for much more to be possible, but I hope that making my own notes available will help others to have a go. I don't think that it's very likely that any answer will be very useful scientifically: in fact I disapprove of Turing's approach to biology. (Though the record does tend to suggest I am wrong there.)
Many people deserve thanks for their help with this project. Andrew Hodges started it all for me when he gave a talk in King's, and much more recently has been both helpful and encouraging. Michael Halls, then the Modern Archivist at King's, invited me to see the original Turing material: an experience I have never forgotten. I was doubly amazed first to find myself back near King's and secondly to find that I could contribute to understanding the pictures I saw then. His successors as Archivist, Jacqueline Cox and Ros Moad, have been efficient, friendly, and helpful. Christof Teuscher invited me to a meeting in Lausanne which made much of this material take shape. Rebecca Hoyle has been encouraging, and explained some useful material about lattices, and Nick Hoskin shared his insight. Jon Agar, at the Manchester National Archive for the History of Computing, helpfully provided photocopies. PN Furbank promptly and courteously gave permission to quote from Turing's unpublished work. I'm grateful to Lesley Robertson and DJ Mabberley for help with copyright issues.
Turing and modern Fibonacci phyllotaxis
This entry attempts to put Turing's achievements in the 1951-1954 period in context with subsequent work in the twentieth century.
At Turing's death, all of his post 1951 developments remained unpublished. Hoskin, Newman and Gandy tried to prepare what could be prepared for publication, but none of them had any particular expertise in the problem. Bernard Richards might have developed his MSc with Turing (on reaction diffusion systems on a sphere) into this broader question, but moved on to other areas (Richards 1998). Unsurprisingly the work remained almost unknown. The only citation I've found before 1992 came at one of Waddington's select meetings on theoretical biology held at Lake Como in the late 1960s, where Scriven described his
treatment, developed from Turing's paper on morphogenesis, based on transport processes to move things from place to another. (Robin Grands [sic] has a Turing manuscript for the nonlinear case treatment) (p321 of Waddington (1970))
Turing had discussed the morphogenesis work with Wardlaw, who subsequently published several papers explaining and discussing the reaction-diffusion hypothesis (Wardlaw 1953, 1954). Wardlaw is reported to have maintained a long interest in Fibonacci phyllotaxis though it seems to have gone unpublished. (I'm grateful to an email from Vidyanand Nanjundiah on 20th March 2003 on this point; Professor Nanjundiah believes Wardlaw talked on this topic at a 1974 Mosbach Colloquium).
The subsequent literature of phyllotaxis is substantial, and I have been primarily guided by the various surveys in Jean and Barabe (1998) for this section. Some of the subsequent studies of phyllotaxis concentrated on, and gave more rigorous mathematical theories of, the 'static'; phyllotactic problem of the classification of lattices, and , for example, the relationship between the divergence angle and the visible opposed parastichies (Adler et al (1997), Jean (1994) ). A second strand used numerical approaches based on dynamic models in which the appearance of a new point was governed by a rule which was some variant of 'far away from previous points'. Some even used reaction-diffusion equation to do so: Veen and Lindenmayer (1977) were the first to do this).
But the earliest, clearest and most undercited explanation for Fibonacci phyllotaxis was developed by Mitchison (1977). Writing in Science, Mitchison deftly used the simple touching circles hypothesis for new points appearing in the cylindrical region formed by the apical meristem, and identified the key parastichies as what Jean would later call the visible opposed parastichies, those winding in opposite directions. He then showed that as the diameter of that region slowly changed, the bifurcations of parastichy number would, as Turing saw, replace one of the pair (m,n) with m<n by m+n, and that as Turing hypothesised but failed to demonstrate, that the new visible opposed pair would have to be (n,m+n) effectively because the pair (m,m+n) would both wind in the same direction. This general hypothesis about which of two possible choices will be made at each stage, combined with the necessary geometric clarity to see that there are only two choices, and a dynamical system which can generate movement through the bifurcation diagram, is what is needed to explain Fibonacci phyllotaxis.
Through the 1990s other workers exhibited lattice Fibonacci structures experimentally (eg Douady and Couder 1996I) computationally (eg Douady and Couder 1996II) or analytically (eg Kunz and Rothen 1992; Levitov 1991; Atela et al 2002). This new generation used a variety of models, but the common feature is that each exhibited a bifurcation tree corresponding to all possible parastichy pairs, and showed, by local analysis at each bifurcation point, that the single branch traversable by continuous variation of a bifurcation parameter was the Fibonacci branch. This local constraint is what Turing would have called the Hypothesis of Geometrical Phyllotaxis.
Despite confident words in 1951, Turing probably did not have a full explanation for Fibonacci phyllotaxis either then or later. Such patterns, we now know, can arise naturally as the product of iteratied creation processes with simple rules. In his reaction-diffusion system he had the first and one of the most compelling models mathematical biology has devised for the creation process. In his formulation of the Hypothesis of Geometrical Phyllotaxis he expressed simple rules adequate for the appearance of Fibonacci pattern. In his last, unfinished work he was searching for plausible reasons why those rules might hold, and it seems only in this that he did not succeed. It would take many decades before others, unaware of his full progress, would retrace his steps and finally pass them in pursuit of a rather beautiful theory.
Turing's progress post 1951
As Turing's theory progresses from reaction-diffusion to lattices and then to parastichy transitions, the surviving documents becomes sparser and less coherent, so assessments of his progress between 1951 and his death on June 7th 1954 become correspondingly more speculative. But speculation is what this entry attempts.
There is no concrete archival support for that claim in 1951 to explain fir cone patterns. A possible explanation is that Turing saw clearly that he had a spot generation mechanism and assumed, incorrectly, that this would be sufficient to generate Fibonacci lattices. There is a quote from a Ferranti engineer, dated before the summer of 1953, that
...with a random starting disturbance the final configuration was displayed on the MkI's monitors. It was always of interest to those of us watching to see what Fibonacci configuration would result. (p65 of Bennett (1996))
Turing was certainly producing spotty patterns by 1953. But it seems more plausible that what the engineer saw was similar to those than explicitly Fibonacci patterns. Support for this comes from a letter of Turing's of May 1953.
None of the fragmentary material can be reliably dated; some of the probably relevant computer printouts are dated May 24th, but give no year. In addition several years of computing would have generated rather a lot of output, so the fact that all we have is a few sheets, and those not obviously archival records, hints that what we do have is the end of a series of ephemeral documents. So a speculation would date the latest analysis to within weeks of Turing's death. It is then likely that this was what Gandy was referring to when he wrote of hearing of Turing's individual and unmethodical computatiions.
In considering Turing's state of mind at his suicide, Hodges wrote that
Possibly the morphogenetic work had turned out plodding and laborious. It was three years since he had claimed he could account for the fir cone pattern and he had still not achieved it when he died. (p492 of Hodges)
The morphogenetic work was not, I think, plodding: the bifurcation tree of parastichy numbers was new and, as discussed below, on the right lines. The computer simulations, even for the author of Computable Numbers (or more relevantly of the first programming manual), must though have been laborious and frustratingly slow to get right. Although it was apparently producing at least some meaningful output, Turing might have become the first to appreciate the sheer craft needed by computational biologists. Probably Turing had not, indeed, accounted for Fibonacci phyllotaxis when he died, but he had got much further, and in the right direction, than he was in 1951.
