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At Turing's death, all of his post 1951 developments remained unpublished.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 Wardlaw is reported to have maintained a long interest in Fibonacci phyllotaxis though it seems to have gone unpublished.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.
Routes to phyllotaxis
In addition to direct numerical simulation, there is evidence that Turing explored a more analytical approach to the Fibonacci phyllotaxis problem. The best evidence comes from two sheets in the National Archive for the History of Computing. One (AMT/MAN/4) is a diagram displaying possible parastichy transitions, from the homogenous (Hom) state up to 4+7 parastichies.
A similar sheet (MAN/M/1) includes the comment
Probable paths: Hom ->(0)R -> (0+1) -> (1+1) -> (1+2) -> (2+3) -> (3+5) (MAN/M/1)
The question is what Turing meant by ‘probable’. It might be a simple harking back to the hypothesis of geometrical phyllotaxis, but on another sheet (MAN/M/8) he classifies a number of possible transitions by more empirically geometric observations.
| (0+2)->(1+2) | An unlikely move |
| (0+2)->(2+2) | Quite possible, with [indecipherable] |
| (0+2)->(2+3) | Quite poss. and favoured by a / component (eg some zygomorphy) |
| (1+1)->(2+2) | Almost inevitable |
| (2+2)->(2+4) (2+2)->(2+3) |
In competition. (2+2) -> (2+3) is favoured by 5 < 6, but (2+2) -> (2+4) by 6=2+4. Latter probably favoured by fast [unreadable] of conc. |
| (1+2)->(2+3) | Requires a breakdown process. Can probably only fail by too quick growth, leading to stationary patterns? |
These kinds of parastichy transitions were not entirely new: van Iterson (1904) studied static sphere packings and generated a parameter map of all possible such packings. Turing at one point dismisses the ‘touching circles hypothesis’ – that each new point is introduced as though it was at the centre of a hard disk of a certain radius - although he is referring here to the process on a static cylinder: as discussed elsewhere touching circles is an adequate model provided the cylinder is allowed to grow in diameter.
There is clearly a concern here with the dynamic stability of given phyllotactic patterns and their dependence on the rate of growth of the morphogenetic arena. In identifying the Fibonacci transitions as the probable ones, he is trying to identify reasons why the Hypothesis of Geometrical Phyllotaxis might be true. It may be relevant that quite a large number of the archive sheets (unpublished in Saunders (1992)) are concerned with the dynamics and the stability of lattices.
Parts of the daisy
The petiole is the green bit under the petals, which forms into a number (usually 13, says Turing) of bracts. The yellow flowerhead is called the capitulum.
I once did research in a Plant Sciences department, you know.
I do not know the copyright owner of this picture. Let me know if it's you and
I'll remove it or credit you as you wish.
There really are 13 bracts, look:
(Copyright
Eva Ekeblad, used by
kind permission)
Spin doctors
[Update 2006: the abstracts have disappeared from the website...]
Searching for updates on the microtubule issue, I came across a page of abstracts from a meeting at the University of Arizona last spring.:
The resulting scalar field of self-alienated surplus labor value bounded by utility emerges as self-commodified consciousness, cast in semiotic terms that reflect Foucault’s transformation of the Marxist “mode of production” into a QFT-Weberian “mode of information.”
You (search for Quantum Field Theory and the Critical Semiotics of Digital Mind) are Alan Sokal and I claim my five pounds.
You know that old gag where everyone in the room of criminals turns out to be an undercover police officer? Maybe the whole meeting will out itself. (It would be a great reversal of the tedious academic race for priority: everyone would be trying to publish their own confession of parody writing last, so that all the preceding authors would be seen to have been taken in. Although it would be rather tense for the authors, like a game of Reversi).
This meeting is a serious one, with a talk from the highly respected Roger Penrose and other serious researchers. As I wrote elsewhere, I din't try very hard to understand Roger Penrose's book on quantum mechanics and consciousness after I found it misleading in the one area I knew a little about. Some of the authors on this page are clearly trying to grapple with serious problems about consciousness and mind and reality in a quantum world. When they figure out the answers, they can tell me. But judging by the state of the field they have a way to go (all taken from that page):
A novel theory of consciousness is proposed. We postulate that consciousness is connected to quantum mechanical spin since said spin is embedded in the microscopic structure of spacetime and may be more fundamental than spacetime itself. (Mount Sinai Medical Center)
If consciousness is the foundational substratum of the universe, principles developed in perennial philosophical systems should be even more universally applicable and cut across all levels of the cosmos, "internal" (e.g. individual mental and psychic, etc.) as well as "external" (e.g. collective unconscious, physical, etc). We sketch here a possible new prescription for a unified "science" that will encompass ordinary natural science and extend it to realms where it has not been extended up to now. (George Mason University)
If consciousness is a quantum rather than classical mechanical phenomenon, then these basic parameters of contemporary social scientific discourse and debate will be called into question, and radically new models of human agency, social structure, and the epistemology of social inquiry will need to be developed. (University of Chicage: possibly Sokalian?)
There are some rather more patently crackpot targets in the page too, but I chose these quotes from abstracts where the authors give academic affiliations.
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.
Article references
I've made a more up to date bibliography on Turing and morphogenesis.
Morphogenesis
The process by which living things develop organised structures.
For a mathematician, one canonical question in morphogenesis might be how a uniform sheet of embryonic skin breaks symmetry and develops zebra stripes. A biologist investigating the same problem might ask what genes are promoting and suppressing each other, and how the proteins they encode interact.
Microtubules
I got all excited when I came across a reference to Fibonacci numbers in microtubules in Roger Penrose's Shadows of the Mind. But now I'm not excited, or rather only excited about microtubules.
Microtubules are hollow tubes, around 25nm in diameter, which help form the cytoskeleton. Microtubules are a polymer of tubulin dimers, which are connected head to tail to form protofilaments which, when assembled side by side, form the cylinder, although that's probably not how they are actually made in vivo. Now the exciting thing was that one common arrangement is to make the cylinder from 13 parallel lines of protofilaments rolled together, and 13 is a Fibonacci number, right? Exciting because we understand enough about Fibonacci structure in plants to see that under the right circumstances you should be able to see it elsewhere (like Douady and Couder's oil drops), but in truth there are no other biological examples. There are some chemical ones I think.
So I read some papers: Li et al, Structure 10: 1317 (2002) and Inclan et al J Cell Science 114:413 (2000). These are why modern molcular biology is so exciting - sequence analysis, structure data, molecular modelling, functional studies, astonishing imaging coming together to offer the beginnings of a real explanation of cellular processes. It turns out that the 13 protofilament tubule has a rise of one and a half tubulin dimers per turn, so that there is a seam running along the tube where the two different monomers become neighbours. First problem for a phyllotaxis like explanation - with aperiodic boundary conditions you're lost. It also turns out that, while most microtubules have 13 protofilaments, they can have between 9 and 16. Second problem. Final problem: it looks as if the assembly mechanism is template driven in some way, which makes it hard to envisage a continuous change along a parastichy tree as an explanation.
So, lose enthusiasm for that idea. Let's go back to Penrose and have another look. He does to his credit say
However, one should not get carried away with such considerations; for example,the "9" that occurs in the bundles of microtubules in cilia and centrioles is not a Fibonacci number. Penrose, Shadows of the Mind
You might wonder what all the fuss was about 13, then. Also, Penrose gives a picture, Fig 7.8, imagining a microtubule split open and displaying a 5+8 parastichy (but not the 1 1/2 unit rise). Where did this picture come from? The only reference in the text is that 'it is apparently found (at least normally) that this pattern is made up of 5 right-handed and 8 left-handed helical arrangements' but there's no reference given for this claim.
Why I am going on about this? Well I already had a caricature view of Penroses's thesis, something like "we perform Turing uncomputable operations when we think, and we couldn't do that with deterministic, digital computer-like brains, and the way it can happen is because quantum mechanics, and quantum coherence in particular, are not averaged out up at the biophysical level of the brain". Always thought it seemed like nonsense, and now I have the book on my desk it's a chance to find out what this manifestly bright man knows that I don't that makes him say such apparently silly things.
Unfortunately I flunked my chance. Philosophy makes my eyes glaze, no matter how I try, and a benefit of my long mathematical education was to learn skip all formulae, so that's most of the book unread. It's towards the end of the book where Penrose tries to figure out where the QM is getting to act at this scale and points the finger at those microtubules. But that's an unconvincing discussion when it talks about things I know about, which is always pretty discouraging. So maybe Penrose will remain unread, and the mystery of the apparently silly will remain unsolved, at least until I'm in my philopause.
Another original view
From Golden Section in Enigmatic Martian Monuments and Terrestrial Flora and Fauna, by Dr Demetrio G Lahoz in Tetet-95, whatever that is. Nice pictures, though:
As if ETI [extra-terrestrial intelligence], knowing that phi was going to be so good for sunflowers alleviated to them the billion-year long process of naturally learning to encode it in their genes AND the creator ETI Himself artificially implanted phi in their quasi-soul just a few megayears ago.
What is Fibonacci phyllotaxis?
Try my page.
