deodands

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 It may be relevant that quite a large number of the archive sheets (unpublished in Saunders (1992)) are concerned with the dynamics and the stability of lattices.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.

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 Furbank.

This 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 Furbank.

Also 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'.

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.