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 PN Furbank promptly and courteously gave permission to quote from Turing's unpublished work.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At Turing's death, all of his post 1951 developments remained unpublished.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.
Turing's letter to Coxeter
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 ... You will be inclined to ask how one can move continuously from one integer to another. The reason is this - on any specimen there are different ways in which the parastichy numbers can be reckoned; some are more natural than others. 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. It is natural to count the outermost florets as say 21+34, but the inner ones might be counted as 8+13. Church is hopelessly confused about it all, and I don't know any really satisfactory account, though I hope to get myself one in about a year’s time. (From Coxeter (1972)).This quote comes from a paper of the wonderful Donald Coxeter, who quotes it at the beginning of a paper on (from memory) some of the number-theoretic properties of Fibonacci lattices. He died at 96. (Irritating for me because that was two weeks before I heard of this quote and tired to contact him, but no doubt more irritating for him).
Googling to see if this quote has appeared anywhere else, I am amused but not surprised by the prevalence of google hits for "Church is hopelessly confused".
Brothers Bravais
This image arrived in Cambridge, before the railway age, by horse and boat. It sat unseen there inside a volume in which the pages remained uncut, on a high high shelf in a musty library.
From Bravais and Bravais (1837)
Geometrical phyllotaxis
In this section we put aside the radically new contribution of Turing (a mechanism for dynamic production of lattices) to discuss his (slightly) more conventional treatment of the static properties of lattices, more commonly called geometrical phyllotaxis. His opinion was that the theory had been partially
expounded...by some previous writers but often in a rather unsatisfactory form, and with the emphasis misplaced (p62 of Turing (1992).)
though he was neither first nor last to consider that his theory was the best. This work was (mostly) published for the first time in 1992 in his Collected Works.
This kind of analysis has a long history, at least as far back as the brothers Bravais , but Turing's geometrical theory added several new insights: flow matrices, the `hypothesis of geometrical phyllotaxis' discussed below, and the `inverse lattice', a Fourier representation of the patterns essential to understanding many of the archive pictures though not discussed further here.
Turing's lattice theory
Part I of the Morphogen Theory of Phyllotaxis (see p49 of Turing (1992)) is a fairly coherent and fully worked out manuscript. Two theorems are of particular relevance here. For any lattice, such as the Bravais one, there are not just two rather obvious parastichies (here 2 and 3 are drawn with dashed lines) but a whole series of less obvious ones, which can all be defined relative to lines from the origin (i.e. the point labelled 0) through the other numbered points. The 1-parastichy is the solid line, and the eye can pick out the 4 parastichy by visualising a line through the points numbered 0 and 4 and 8 and so on. What Turing called the `principal parastichies' were the ones in which the nearest points in the parastichy were closest to the origin -- in other words Figure 5 has principal parastichy (2,3) because the points numbered 2 and 3 are the ones closest to the point numbered 0. (The geometrical details are related to but different from, say Jean (1994) : primarily because Jean also needs to ensure that the parastichies wind in opposite directions round the cylinder). A second key theorem is that the third parastichy (in this case 1 since 1 is the next closest point) must be the sum or difference of the first two parastichy numbers, a theorem Turing proves neatly on page 57 of Turing (1992).
On Tour 2004
I will be giving a talk on Turing and morphogenesis at a conference in Manchester on June 5th this summer. Two days later will be the 50th anniversary of his death, and will see a plaque unveiled outside his house in Wilmslow.
Last year I got invited to a meeting in Switzerland, which was very nice, so it was pleasing to get this invitation and feel I'd secured my place on the international Turing gravy train. I was slightly less pleased when I realized it was in Manchester, so all I can claim in expenses from the organisers is a £2 tram ticket.
Dynamic phyllotaxis
The Turing instability by itself, then, can’t provide an explanation for the generation of Fibonacci phyllotaxis. However there are strong indications in Turing’s later manuscripts, particularly the fragmentary Outline of the development of the daisy, that he had conceived an additional mechanism to provide that explanation.
That mechanism is that there is a small arena in which the Turing instability is at work, laying down spots in lattices and then leaving them behind as the arena follows the growth of the plant. Moreover that arena itself changes, growing in diameter, providing a continuously changing lattice to which the the theories of geometrical phyllotaxis could be applied.
The first evidence of this is in Morphogen Theory of Phyllotaxis Part I; in section 13 Turing establishes a formalism of flow matrices for the change in lattice parameters with a parameter called time, adding that
a convenient way of picturing flow matrices is to imagine the change in the lattice as being due to the leaves being carried over the surface of the lattice by a fluid whose velocity is a linear function of position. (Turing (1992) p75)
This was a way of modelling phyllotactic patterns, building on the continuous change models of Richards (1948). But this remains an essentially static picture of spots being passively transported over a changing geometry. What Turing was able to go on and create, with the aid of his new spot-generation model, was a concrete model for dynamic phyllotaxis.
In the later work, Turing typically expressed what we now think of as his reaction diffusion model in forms similar to
(*)
with
and
(cf
equation III.1.2 of MTP, p107 of Turing (1992)).
Here U(x,t)
is the morphogen and V(x,t) is the ‘poison’; the psi function represents the implicit solution of its partial differential equation
in terms of the slower diffusing morphogen. The H terms represents the
effect of the poison on the morphogen, the G term the morphogen’s
autocatalytic nature, and the phi term
its diffusive nature, parameterised by the natural wavenumber k0.
However in Outline of the Development of the Daisy, there is a crucial extra spatiotemporal term I(x,t)U:
(**)
(note that the Saunders edition has a typo for the H and the phi). An even more revealing version of this equation is in AMT/C/27/28:
More © PN
FurbankThis is a version of the morphogenesis equation allowing for growth in the apical meristem region and the possibility of dynamic phyllotaxis. Also, below the first line, there is a list of the numerical parameters which must be specified to allow computational solution. In addition, there is a sketch of an enlarging apical meristem region
Turing not only conceived this idea; he clearly made substantial progress with a numerical implementation of it. At one point of the Daisy draft, he comments on the number of parameters needed:
when actual computations are being carried out the number of quantities to be specified is again increased. (AMT/C/24/12: omitted from the Saunders version)
in a manner suggesting that he had already done so in practice. Moreover one of the subroutines that has survived is labelled KJELL, and AMT/C/27/C25, entitled Kjell theory, works out the algebra in Fourier space of the coefficients of an equation such as the daisy one, including the crucial growth term:
More © PN
FurbankAlso in the archives are a number of solution plots. Based on their form they are probably solutions of the reaction-diffusion equations of forms similar to (*) or (**). Whether they are directly relevant to the Fibonacci problem or as more general illustrations of morphogenesis is hard to say. My speculation is that they are attacks on the Fibonacci problem, not least as there is little evidence of any other computationally active project. One example is from AMT/K/3/8:
More © PN Furbank The H. of G. P.
A key staging post in creating an explanation of Fibonacci phyllotaxis is what Turing called the Hypothesis of Geometrical Phyllotaxis.
After creating a theory of lattices on cylinders, Turing went on to consider lattices of more variable geometry. This raises the question of what kinds of transformations of parastichy numbers are possible when a phyllotactic lattice is deformed. As it is deformed, the principal parastichies will in general remain unchanged. They will only change when a new lattice point from a different parastichy moves so as to become closer to the origin, but generically the point that does so must have previously been the third parastichy. Thus one of the two principal parastichy numbers, together with the third parastichy number, will become the new principal parastichy number. Turing showed (p72 of Turing (1992)) that if that third parastichy number never lies between first and second parastichy number then a Fibonacci property, once begun, would persist. This constraint he named the Hypothesis of Geometrical Phyllotaxis.
The figure shows an initial lattice on the left-hand side with principal parastichy numbers 2,3 and 1. The lattice is then squeezed and stretched, and as this happens the principal parastichy numbers change. The 'hypothesis of geometrical phyllotaxis' (HGP), is satisfied in this example.
It was this result which was surely the cause of Hoskins' view, reported in Max Newman's 1955 Royal Society memoir, that Turing had shown that
if a phyllotactic system is Fibonacci in character then it must change, if at all, to another Fibonacci system (AMT A/7; section contributed by Hoskin to MHA Newman's obituary in Biographical Memoirs of the Royal Society AMT A/6)
But the truth, as Turing recognised, is that this relies on the HGP being true and the theory so far provides no reason why it should be. Indeed he wrote (p72 of Turing (1992))
the phyllotactic systems of botany do not arise in this way’ More
However this idea is still worthwhile: the key insight it embodies is that of continuous change. Phyllotactic lattices are not laid down all at once on an infinite cylinder: they are produced locally, node by node, and the resulting pattern is also deformed by growth.
We have evidence of Turing working out these ideas in 'Outline of the development of the daisy'.
The H. of G. P.
A key staging post in creating an explanation of Fibonacci phyllotaxis is what Turing called the Hypothesis of Geometrical Phyllotaxis.
After creating a theory of lattices on cylinders, Turing went on to consider lattices of more variable geometry. This raises the question of what kinds of transformations of parastichy numbers are possible when a phyllotactic lattice is deformed. As it is deformed, the principal parastichies will in general remain unchanged. They will only change when a new lattice point from a different parastichy moves so as to become closer to the origin, but generically the point that does so must have previously been the third parastichy. Thus one of the two principal parastichy numbers, together with the third parastichy number, will become the new principal parastichy number. Turing showed (p72 of Turing (1992)) that if that third parastichy number never lies between first and second parastichy number then a Fibonacci property, once begun, would persist. This constraint he named the Hypothesis of Geometrical Phyllotaxis.
The figure shows an initial lattice on the left-hand side with principal parastichy numbers 2,3 and 1. The lattice is then squeezed and stretched, and as this happens the principal parastichy numbers change. The 'hypothesis of geometrical phyllotaxis' (HGP), is satisfied in this example.
It was this result which was surely the cause of Hoskins' view, reported in Max Newman's 1955 Royal Society memoir, that Turing had shown that
if a phyllotactic system is Fibonacci in character then it must change, if at all, to another Fibonacci system (AMT A/7; section contributed by Hoskin to MHA Newman's obituary in Biographical Memoirs of the Royal Society AMT A/6)
But the truth, as Turing recognised, is that this relies on the HGP being true and the theory so far provides no reason why it should be. Indeed he wrote (p72 of Turing (1992))
the phyllotactic systems of botany do not arise in this way’ More
However this idea is still worthwhile: the key insight it embodies is that of continuous change. Phyllotactic lattices are not laid down all at once on an infinite cylinder: they are produced locally, node by node, and the resulting pattern is also deformed by growth.
We have evidence of Turing working out these ideas in 'Outline of the development of the daisy'.
Parastichy number
An explanation of what a parastichy number means.Here are four representations of the same idealised growing tip (green) on which primordia (brown) are forming, later to become stem branching points, or florets, say. The first image shows the physical arrangement of the primordia on the tip. The image below gives a view from above, while that to the right stretches the surface of the tip so that it has a constant diameter. The image below that unrolls the resulting cylinder into a plane. In each representation the same primordia are linked by one example parastichy (red). A parastichy which is obvious to the eye in one representation is not necessarily so obvious in another. In what follows we concentrate on the last, planar representation.
Given any of these lattice plots, the eye picks out obvious straight lines, which correspond to spirals on the original plant. These are called parastichies.
To understand where the parastichy number comes from, pick one point and number the rest in sequence up the cylinder:
What Turing defined as the principal or first parastichy was, loosely, the number of the closest point to 0. Here the first parastichy number is 3, because the nearest point to 0 is 3:
The second parastichy number is 2. Note how the parastichy lines go the opposite way around the cylinder:
The third parastichy number is 1. Note that 3-2=1: in general the third parastichy is always the sum or difference of the first two. Turing proves this theorem on p57 of Turing (1992).
Turing is one of several who have constructed theories of geometrical phyllotaxis: the static properties of lattices. His opinion was that the theory had been partially
expounded...by some previous writers but often in a rather unsatisfactory form, and with the emphasis misplaced (p62 of Turing (1992).)
but he wasn't the last to consider that his theory was the best.
The details of Turing's definition are related to but different from those of Jean (1994) primarily because Jean also ensures that the parastichies wind in opposite directions around the cylinder.
Lattice generation
So far, the discussion has been in terms of a one-dimensional pattern wrapped around a ring. What happens if we have the same reaction and diffusion mechanism but now allow it to act in a two-dimensional arena?
In terms of the cannibals and missionaries, we might imagine that the beach of the island is now rather wide. In this case, the Turing instability generically generates not a ring of points but a lattice of points. Might this by itself be enough to explain the occurrence of Fibonacci phyllotaxis? With no constraints, (i.e. an infinite cylindrical arena and ignoring the complexities of the inception and quenching of pattern formation), the instability typically generates hexagonal lattices such as in the figure (See Murray (1993)).
(more)
and this class of lattices certainly include some Fibonacci ones. But it also includes many that are not Fibonacci.
Might it be that there are other constraints acting to select Fibonacci ones? First there are the geometrical constraints arising from the particular arena. Clearly patterns on a cylinder must be different from those on an infinite plane, and different again from those on a cone, but this does not promote any special Fibonacci structure. Then there are the dynamical constraints: the pattern does not suddenly appear, but emerges as a result of nonlinear interactions between morphogens over time. Finally there are growth constraints: during the emergence process, the arena itself may be growing with the plant.
Fourier modes and dispersion relations
(It's not you: this entry makes little sense. One day it might.)
The natural mathematical framework for the analysis of Turing's spot-formation theory comes from the idea of Fourier modes. In order to make sense of the pictures in the archive it's necessary to understand a little of this theory, at least graphically.
Fourier modes and dispersion relations
Figure 6 A dispersion relation.
Consider a pattern like that of Figure 5: this can be thought of as a wave rising and falling around the ring four times. This number of times that the wave pattern fits around the ring is called the wavenumber, and we call it k. We can imagine other patterns with wavenumbers of none, one, two, three, and so on. Now suppose we set up an island with an initial pattern of each of these wavenumbers, and allow the reaction-diffusion dynamics to act. Over time, each of these patterns will either grow or decline, and we can measure this rate of growth and plot it against the wavenumber (Figure 6). This graph is called the dispersion relation. Clearly, it is those patterns which grow fastest in time which will be most visible. Thus the highest region of the curve (or more precisely the highest point corresponding to an integer wavenumber) gives the wavenumber, and thus the characteristic length scale, of the pattern which the reaction-diffusion mechanism will generate.
'Cooking' the Turing instability
Figure 7 Cooking the Turing instability
This simple description of the Turing instability explains where a characteristic length scale emerges, but things are a little more complicated. For example, given that all waves of the maximal wavenumber will be growing at an equal rate, and that perturbations are equally likely to occur at all places round the ring, the model can't predict the phase of the pattern: that is it might say there will be four troughs around the ring but it can't say where they will begin. Another complexity is that the simple model predicts that patterns will go on growing forever: in order to prevent this we have to change the conditions after a while. These and other questions made it necessary for Turing to introduce the notion of 'cooking'. As shown in Figure 7, this is simply increasing the dispersion relation, so that the system moves from having no unstable mode, to having one and then possibly more.
In two dimensions, what happens to the dispersion relation? For a one dimensional lattice, we saw that patterns on rings could be described by wavenumbers. What happens when we widen the beach: that is when we consider two dimensional lattices on cylinders? In particular, Turing considered very wide beaches, that is infinitely tall cylinders. To describe a pattern on a cylinder, we still have a wavenumber describing how many times the pattern repeats around the cylinder; but we must also have a number describing how frequently it repeats vertically: unlike the around-wavenumber, this isn't constrained to be an integer, although if the pattern is a lattice it will be. Thus the analogue in two-dimensions of the dispersion-relation is the dispersion plot.
There are examples of these two-dimensional Fourier pictures in the archive:
Where do spots come from? Turing's answer
Turing's 1952 paper, The Chemical Basis of Morphogenesis has become celebrated in its own right for introducing what is now known as the Turing instability, and provides a framework for understanding Turing's later, unfinished work. This section provides a brief non-technical discussion of the Turing instability.
Turing provided a hypothesis to explain the generation of pattern when smooth sheet of cells develop pattern during development in a wide variety of settings including the formation of leaf buds, florets, skin markings, and limbs. According to this hypothesis, chemicals called morphogens generate organs when present in sufficient density, and the pattern is created through mechanisms of reaction and diffusion. The corresponding reaction-diffusion models are by now well known to mathematical biologists, and for the mathematically inclined the books by Meinhardt and Murray (1993) can give much more detail.
Reaction...
More.
One way of understanding the reaction-diffusion process is to borrow an analogy which Turing himself used at least once, although in a slightly different model: cannibals and missionaries. An island is supposed to be populated by a population of cannibals and missionaries. The missionaries are all celibate and thus depend on recruitment from the external world to maintain the population as its members gradually die. Cannibals also die, but can also reproduce, so that the population naturally increases. However when two missionaries meet a cannibal, the cannibal is converted to missionary status. (If this seems a rather imperialist island it might be worth pointing out that under a commoner interpretation the cannibals are the growth promoters and the missionaries are the poison). This tension between production and transformation means that a balance is reached when both populations are mixed together. If this balance is disturbed by a small amount of noise, the tension will act to restore the balance: the system is stable.
...and diffusion
Pattern emerges on a ring.
Now we imagine that the two populations, instead of mixing completely together, are spread out in a thin ring around the rather narrow beach of the island. Now individuals react (that is, reproduce or convert) only with their immediate neighbours, but they also move around at random in a diffusive way. Moreover the members of the two populations move at different speeds: the missionaries have bicycles and move faster. This is enough to destabilize the system. For if there is at any point a small excess of cannibals, say, then this will be followed by excess 'production' of more cannibals, and then of more missionaries (since they have more targets for conversion). Without the spatial dimension the extra production of missionaries would in turn reduce the cannibal excess and the system would return to balance. But because the missionary excess is transported away more quickly, a pattern develops in which there is a near excess of cannibals and a far excess of missionaries. Moreover the distance between these zones of relative excess is determined by the interaction between the reaction and the diffusion: a length scale, which is what is required for the emergence of pattern from non-pattern, has emerged from the dynamics.
Where did the idea come from?
The key to making this idea work is the missionaries' bicycles: more technically that the inhibitor morphogen has a higher coefficient of diffusivity. Once the reaction-diffusion system is set up, a simple linear analysis makes this an obvious requirement for heterogeneity, but that was an analysis that no one, to Turing's knowledge, had done at that time. We have no record of Turing's thought process in developing the model idea and whether the diffusivity constraint came before or after the reaction-diffusion model itself. His analysis in Turing (1952) and Turing (1992)in terms of Fourier modes would have been second nature to him: for example his pre-war project to compute the zeroes of the Riemann zeta function using an analogue computer used a similar basic analysis. The formal theory in The Morphogen Theory of Phyllotaxis shares some structure with the then new quantum mechanics he had learned as a student in Cambridge, but presumably many of the techniques were in the armoury of any applied mathematician at the time. As Allaerts (2003) points out, Jeans’ 1927 book on Electricity and Magnetism, which Turing cited, is a source for many of the techniques, particularly spherical harmonics. But the source of the key scientific innovation is harder to pin down. It's not even obvious that Turing himself appreciated it that it was key: it is hardly emphasised in either Turing paper.
Turing was not alone in arriving at these ideas. Jean states that similar ideas were also introduced by Kolmogorov, Petrovski and Piskunov (1937), and Rashevsky (1940). Nanjundiah (2003) discusses the (lack of) influences in more detail. And the splendid Ralph Abraham's website, in an article on the mathematical formalism of Rupert Sheldrake's morphic resonance fields, mentions (but gives no reference for) as relevant the 1924 work of Roland Fischer, about which I know nothing. (Can't find that refeence now, but it is mentioned here
The 1952 paper actually dealt with a number of important and more complex issues usually glossed over in the standard undergraduate accounts. Turing also discussed tricky issues of mode selection and the effect of noise. He also extended the model to two dimensions to produce an example of dappling. It's hard to understand from his description exactly how the example was drawn, but it's possible to understand enough to see why Turing thought this was this least interesting of the cases. This paragraph will make more sense one day.
By the time of the drafting of The Morphogen Theory of Phyllotaxis, the theory had been developed yet further, particularly by a representation in terms of spherical harmonics, and by an application to the particular case of a sphere, done as an MSc project by Bernard Richards (1998). This more general theory, which has been recently reviewed by Allaerts (2003), is, though relatively technical, conceptually a fairly straightforward development of the original idea.
Turing's interest
When and where did Turing's interest in Fibonacci phyllotaxis come from? We know that at school Turing was well acquainted with D'Arcy Thomson's classic On Growth and Form that discusses it; decades later, Turing is recorded as discussing daisies and fir-cones during off-duty periods at Bletchley Park (Hodges p207). We have little concrete idea of his thinking on the subject until 1951. When Turing returned to Cambridge for a year in 1947-1948 he attended the undergraduate physiology lectures of Lord Adrian, and Hodges has plausibly speculated that his prime interest by now was the possibility of a logical description of the nervous system. Indeed it was in a correspondence with a leading physiologist on the needs of a physiological theory of the brain that he continued:
'...my mathematical theory of embryology...is yielding to treatment, and it will so far as I can see, give satisfactory explanations of
(i) gastrulation
(ii) polygonally symmetrical structures, e.g. starfish, flowers
(iii) leaf arrangements, in particular the way the Fibonacci series (0,1,1,2,3,5,8,13,...) comes to be involved
(iv) colour patterns on some animals, e.g. stripes, spots and dappling
(v) pattern on nearly spherical structures such as some Radiolara...' more
Whatever the original trigger, these were strong claims and it is the aim of this website to examine why Turing felt able to make them and claim (iii) in particular. In the same month he also wrote in a letter that
'Our new machine is to start arriving on Monday. I am hoping to do something about 'chemical embryology'. In particular I think I can account for the appearance of Fibonacci numbers in connection with fir-cones' more
He certainly could do something about chemical embryology. By November of that year he had submitted a paper to Philosophical Transactions. This paper, The Chemical Basis of Morphogenesis, has become celebrated in its own right for introducing what is now known as the Turing instability, and provides a framework for understanding Turing's later, unfinished work. In 1952 he wrote that he had
`Had quite a jolly time lecturing on fir cones' (more)
in Cambridge, and in 1953 wrote to HSM Coxeter:
...During the growth of a plant the various parastichy numbers come into prominence at different stages ... Church is hopelessly confused about it all, and I don’t know any really satisfactory account, though I hope to get myself one in about a year’s time. (Coxeter 1972).
Between 1952 and 1954 he drafted parts of a paper on the Morphogen Theory of Phyllotaxis. This work was left incomplete, and indeed Robin Gandy wrote, after Turing's death, that
'When I was staying with Alan the weekend before Whitsun he also told me more or less where the computations had got to; but since his methods were so individual, he was unmethodical, I imagine it will be almost impossible for anyone to go on with the programme where he left off.' more
In fact Nick Hoskin did manage to make some progress with preparing the work for publication, and Bernard Richards provided a third section based on the MSc thesis he started under Turing. But the resulting typescript was not published until 1992.
Fibonacci Phyllotaxis
This entry describes the phenomenon of Fibonacci phyllotaxis. Many other authors have done this: a couple which are on line are Knott, who concentrates on the elementary mathematics of Fibonacci numbers and the informative and good looking interactive website at Smith. Also see the references for printed discussions.
Phyllotaxis means here the arrangement of structures, such as leaves or florets, in plants.
Here's an sunflower, where some florets have been removed to make it easier to see some spirals. The shallower spiral, or parastichy, at the bottom repeats all the way around. Counting the removed florets, there are 34 such spirals. The steeper spiral repeats 55 times. The astonishing thing is that in most of the sunflower head you examine, you will also find 34 and 55 spirals. Or if not that then 21 and 34 together, or maybe even 55 and 89. And of course these numbers are adjacent members of the Fibonacci series: 1,1,2,3,5,8,13,21,34,55,89,... Perhaps the more astonishing thing biologically is that you don't find near approximations: if the spirals are clear you won't find 34 spirals one way and 57 the other.
This phenomenon is not restricted to sunflowers. The example below shows the more common case where spirals appear to wind around the cylindrical stem of a plant, in this case a Euphorbia:
These spirals visually joining nearby points are called parastichies and the parastichy number for a spiral counts how many such spirals fit on the cylinder, or equivalently how many points around the plant but not on the spiral have to be skipped in the vertical direction between two points of the spiral.
Thus one prominent parastichy on the left hand slice of the specimen is a 5 parastichy because there are four other branching points spread around the cylinder between any two consecutive points on the spiral. The parastichy in the other direction is a 3 parastichy and the pair is called a (3,5) parastichy pair. Like the sunflower, this specimen exhibits a number of different parastichy pairs, each of which consists of two adjacent Fibonacci numbers. Yet more remarkable is that this property can be found in very many examples in many different species of plants.
Explaining this ubiquity is the problem of Fibonacci phyllotaxis. (A substantial recent book on Fibonacci phyllotaxis is edited by Jean and Barabé (1998)). The appearance of these numbers, or variants on them, turns out to be intimately related to the divergence angle, the difference in angle between successive points on the stem. If that angle in a cylindrical lattice is close to a simple function of the Golden Ratio, then Fibonacci numbers naturally appear. Moreover the Golden arrangement typically has the property of optimal packing. All of these relationships have been closely studied in the mathematical phyllotaxis and number theoretic literature (reviewed in Adler et al (1997) and Jean and Barabé (1998)), and each of them has been adduced at one time or another as the explanation for Fibonacci phyllotaxis, often with varying degrees of mysticism or arguments from evolutionary optimality attached. According to Adler et al (1997) the first to explicitly recognise that Fibonacci numbers were involved in phyllotaxis were Schimper (1831) and Braun (1831).
Another excellent example of Fibonacci phyllotaxis, very well known to Turing, is the fir cone.
Turing's Last, Lost Work
Alan Turing has a wide and still growing reputation as one of the most creative thinkers of the 20th century. His interests, from computing and the mind to information and biology span many of the emerging themes of the 21st century.
Turing (born 1912) was an undergraduate and then Fellow at King's College Cambridge. In his twenties he published a seminal paper outlining what we now call the Turing Machine, establishing many of the key ideas in computing theory and mathematical logic. During the 1939-1945 war he served as a code breaker at Bletchley Park and played a central role in the breaking of the German Enigma code. After the war he largely returned to academia. In 1950 he published a celebrated Mind paper describing the Turing Test for machine intelligence, and in a 1951 paper he proposed the Turing Instability, now a central concept of mathematical biology. Despite this astounding set of eponyms he remained a relatively neglected figure in his lifetime and for several decades after his suicide in 1954.
However a superb biography by Andrew Hodges (and perhaps the need of the newly mature discipline of computer science to find and venerate its own icons) have contributed to a reevaluation. Alan Turing is now well known as a pioneer in the logical and technical development of the computer and of cognitive science, and also widely recognised in mathematical biology.
Less well known is that he spent the last few years of his life further developing his morphogenetic theory and using the new computer to generate solutions to reaction-diffusion systems. Some of this work was published in his lifetime; some, thanks to the editors of his Collected Works, was eventually published posthumously, and some has been preserved unpublished, mainly in the archives of King's College Cambridge. The paper published in his lifetime has turned out to be seminal and very widely cited in the mathematical theory of biological pattern formation, but the rest of his researches have remained obscure and ill-understood. It is the purpose of this website to interpret this last work of Turing's.
Article references
I've made a more up to date bibliography on Turing and morphogenesis.
Guide to online Turing archives
A brief guide to the major web resources of unpublished material by Turing (ie scans or text of key works) in general, not just morphogenesis.
As usual, Andrew Hodges has the authoritative list of sources, but there are two large sites with similar names but different purposes, and enough other sites to make it all rather confusing.
Primary material by Turing
The Turing Digital Archive
This site, www.turingarchive.org.uk, (with which I have some association) only has the pilot project live at the moment. However the project has scanned essentially all of the material in the King's College Archive, and a full site with most of this available should, I fervently hope, be online during Summer 2003. By construction, it has many of the personal letters that Sara Turing and Andrew Hodges drew on for their biographies, many pages of working notes on morphogenesis (which my site is mainly about) and a miscellanea of other things. It is less strong on documents from the history of computing, though there are some gems.
It is being constructed thanks to a grant from Elsevier to an informal consortium administered by King's College and using the skills of the King's archivist and a group at Southampton University. In the longer term I'd like to see this site expand a little, particularly to include the primary material currently held at Manchester.
The Turing Archive for the History of Computing
This site, www.alanturing.net, is an archive of more general ambition, covering a range of documents from the early history of computing. The site consists mainly of documents scanned (looks like from photocopies) from the UK Public Record Office. From a Turing perspective it is useful on the NPL years, and has a copy of the report on the Colossus. The site was created by Jack Copeland and his team in the Turing Project, which
exists to develop and apply the ideas of the mathematical logician Alan Turing ... Project members are currently engaged in research on Turing's Neural Network Architectures, his Theory of Morphogenesis, and Super Turing-Machines. Work on the history of computing and the analysis of Turing's philosophical views is ongoing. The Project is also home to the Turing Archive, the most comprehensive single collection in the world of unpublished documents relating to Turing's work in computing.
I'm not quite sure who the Project members are (not Craig Webster any more, I think), but it will be great to read their work.
The National Archive for the History of Computing
This is a proper paper archive held by the University of Manchester. They have scanned a couple of images that you can see, and there are a few more scattered through my website.
Approaching The Chemical Basis of Morphogenesis
From: R ...
To: jonathan@swintons.net
Subject: RE: morphogenesis
Hi Jonathan,
I'm a PhD student in ...
... my supervisor has asked me to understand the morphogenesis paper completely so that I know how these people came up with these equations and especially understand how and why Turing came up with the idea.
... I did pure mathematics many years ago but no applied maths so I know I can understand it with some help
Dear R,
If you want to approach Turing's 1952 morphogenesis paper from a technical point of view, you do need some applied mathematics. I would suggest you start with JD Murray, Mathematical Biology. It's written from a discipline very much in the shadow of the Turing paper. It has split from one to two volumes since I read it so I may be a bit out of date, but it certainly in the old edition the reaction-diffusion sections (half the book) cover most of the technical material in the Turing paper at an undergraduate level. It will be hard work if you have done no applied mathematics at all, but actually it's pretty straightforward stuff.
From a non-technical point of view, there are probably ten billion descriptions of the Turing instability around, so I will only recommend my own, which forms a section in an upcoming book chapter. That also points to a good few more references.
For a relentlessly optimistic view of the value of the Turing instability in understanding biological form, almost any exposition of mathematical biology will do, but Ian Stewart's Life's Other Secret is non-technical, well written and relentlessly optimistic. After reading one, though, I would strongly recommend anyone enthused by (great) books like Stewart's also to read Evelyn Fox Keller's Making Sense of Life for an insight into why biologists are much more indifferent to these theories than mathematicians think they ought to be (it's not technical ignorance: it's about what counts as an explanation).
In terms of 'how and why Turing came up with the idea', the 'why' is relatively well documented. There are several sections of Andrew Hodges' superb and readable biography devoted to Turing's early interests in pattern in nature, his growing interest in the neurophysiological construction of the brain, and the general problem of morphogenesis. As to the 'how' we have little concrete idea. I do have some mild speculations about this in the chapter mentioned above
Alan Turing: the Enigma
Andrew Hodges, Alan Turing: the Enigma.
A great read; don't forget the website.
Mathematical Biology
JD Murray, Mathematical Biology. (My local bookshelf has an apparently new edition of this in two volumes, but Amazon still thinks there is a single edition.)
See Jim with the cheetah. See the spots of the cheetah. Do you like spots? Jim likes spots. The cheetah likes spots. Jim likes the cheetah. See Jim with the model. See the spots of the model. The model has spots. Jim likes the model.
See Jim with the zebra. See the stripes of the zebra. Do you like stripes? Jim likes stripes. The zebra likes stripes. Jim likes the zebra...
See Jim with the butterfly...
Or, this book taught me all I know about the mathematics of morphogenesis. Oddly, no mention of Hox in the index.
Famous for a day
Today's fleeting fame brought an email alerting me to Bob Johnson's Fibonacci resources page for his Durham mathematics undergraduates, which includes a couple of interesting articles such as one by George Markowsky neatly demolishing some of the sillier myths about the Golden Ratio and a 1995 Scientific American article by Ian Stewart. Stewart, as ever, explains things beautifully; it builds up on the Douady and Couder experiments, and a version of the same material, without the hokey dialogue device, made its way into Stewart's book Life's Other Secret. It's a shame Stewart was apparently unaware of Graeme Mitchison's much earlier Science paper which captured most of the key ideas: I think he could have made characteristically good popular use of it.
As featured in the Guardian...really
Whoops. I now seem to be in the paper paper as well as the, um, nonpaper paper. It wasn't really meant to happen. I had sent an email to the journalist involved, mainly out of sheer gratitude at being linked to, but it got routed to the letters section instead and they told me they were going to publish it at quarter to five (when I do real work) and I didn't get a chance to think about whether it was wise or not.
The letter 1) it doesn't make any sense if you don't know about the way people use jargon like 'emergent', 2) it doesn't make much sense if you haven't spent hours thinking about Turing and phyllotaxis (and I know no one else in the world who gives a toss)
3) on reflection I think it's maybe also wrong, so its good that it's incomprehensible. One function of the original letter was self-advertisement, so there was a result.
The spark was that someone said that the study of 'emergent' phenomena goes back to Turing's work in the 50s on the development of plant structure. It's true that he developed a model of how things like stripes could apparently 'emerge' from homogenous tissue. But the way people use words like 'emergent', particularly the Santa Fe complexity theory types is more to do with the notion of a self-organising complexity that arises as a feedback between the state of the system and the response of individual agents to that state (traffic jams, termite mounds, tipping points, blah), and that you can't analyse in terms of that individual response. The point of the letter was that Turing was doing 'good old fashioned applied mathematics': ie exactly doing that kind of analysis you're not meant to be able to do for emergent systems. And if that counts as analysing emergent phenomena then like that man in Moliere, applied mathematicians have been doing complexity theory all along.
Why I think I might be wrong is that there might, after all be a qualitative difference between, say, using Maxwell's equations to predict radio waves, which is a very domain specific, experimentally-bound kind of applied mathematics, and the Turing approach which is much more an approach asking how in principle certain kinds of structure can come from certain kinds of inputs. Turing's is perhaps much closer to the complexity theory viewpoint. I'm just in the middle of reading an interesting book by Evelyn Fox Keller on why biologists loathe this latter view.
But I am now the sort of person that writes letters to the Guardian.
Self-organisation is not reductionist
Why Turing stripes are not a self-organising system....
"Can things self-organize ? Yes, any system that takes a form that is not imposed from outside (by walls, machines or forces) can be said to self-organize. The term is usually employed however in a more restricted sense by excluding physical laws (reductionist explanations), and suggesting that the properties that emerge are not explicable from a purely reductionist viewpoint" Self-Organizing Systems FAQ for Usenet newsgroup comp.theory.self-org-sys
In other words, if you have a proper theory, it's not a self-organised, emergent system
Manchester details updated
I have updated the section of my Turing archive bibliography dealing with the Manchester NAHC archive.
As featured in the Guardian...
Sort of. Linked to in the online edition but despite excitedly digging through the recycling box to find last weeks paper (see, Lomborg, there is a point to recycling) not in the paper copy.
It's not quite clear from the article, but it sounds to me like an Eric Bonabeau talk at ETCon claimed that the concept of emergence goes back to Turing's morphogenesis work. (Hence a link to my site). Lovely to have my site referenced, but whoever said it, it's a weak claim. If 'emergent' means anything, it means properties that you can only see in a large enough interacting system, whereas Turing's reaction diffusion theory is good old fashioned applied mathematics. It only predicts emergent pattern in the sense that radio waves are an emergent property of Maxwell's equations. You might claim that his theory of Fibonacci phyllotaxis comes a bit closer: it has at least an idea of complexity increasing through time via a simple iterated dynamics, but the complexity is that the parastichy numbers are 89 and 55, say, not 3 and 5. These are differences of degree, not of kind: there's not anything to 'emerge'. Andrew Hodges does say somewhere that Turing was in a position where he could have gone towards chaos theory, but that remains a might have been.
I think the silliness most likely resides in the mushiness inherent whenever people start using words like 'emergent'. As the great and very cute John Maynard Smith says of complexity theory: 'Absolute fucking crap. Crap with good PR but crap' (via Andrew Brown's Darwin Wars).
Turing after the instability
Most of my site is about Turing's approach to the specific morphogenetic problem of Fibonacci phyllotaxis, so it's good to see a paper (Allaerts W: Fifty years after Alan M. Turing. An extraordinary theory of morphogenesis. Belgian Journal of Zoology 133(1): 3-14 2003) on Turing's more general reaction-diffusion theory based not on the well known 1952 paper but on his later works finally published in 1992.
On the other hand, the Belgian Journal of Zoology don't seem to have mastered online publishing ("free full text of special issues"). Personally I have pretty much stopped reading any new academic papers that I can't get online, so I hope Dr Allaerts' paper doesn't have the same fate as a chemistry paper published in German about the time all the chemists started publishing in English instead (1914?) or a Domesday book published for a special edition of the BBC Micro.
Changes in my Turing site
For the last few years, I've maintained a website on Alan Turing's work in morphogenesis. It's a pretty austere if not incomprehensible site, but I think I'm going to be able to improve the expository side over the next few months.
I'll try and record changes and less formal notes here.
I also hope that all the TrackPingBackRSSComment widgets built into the wonderful Movable Type will allow a bit more interaction.
