deodands

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.

By Rhodes?

We must find new lands from which we can easily obtain raw materials and at the same time exploit the cheap slave labour that is available from the natives of the colonies. The colonies would also provide a dumping ground for surplus goods produced in our factories

This is allegedly a quotation from Cecil Rhodes. But did he actually say this? While the sentiments seem entirely consistent with his imperialism (appalling even by the standards of the day), the language sounds wrong to me. I first found this quote in The No-Nonsense Guide to Globalization (Wayne Ellwood, Verso, 2001), which cites an article in the Ecologist (Empires Without Armies, by Edward Goldsmith, the Ecologist, 29(2) May/June 1999, 154-157). That article doesn't give any source for the quote beyond 'explicitly stated in the 1890s by [...] Cecil Rhodes.

You can this quote all over the net, but I have not yet found a single pointer to a source. Honourable mentions go to Nicholas Hildyard and Wayne Ellwood for citing the place where they found it, but that's not quite the same.

The earliest dated version I can see is from a
1996 talk
by Nicholas Hildyard of The Corner House.

aha: from
the Edward Goldsmith site:
R. Dumont and N. Cohen, "The Growth of Hunger: A New Politics of Agriculture", Marion Boyars, London 1980. UL: 235.d.98.1. Dumont and Cohen, in a text stuffed with citation, also don't give a source for this quotation. Whilst in the UL I browsed half a dozen biographies of Rhodes, some fairly revisionist, and found no mention of this quotation. But I did get a hint that Rhodes was taken (admiringly) as an examplar of imperialism by one Hobson, around 1900, in work that was in turn used by Lenin. So may be that's a source to look.

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.