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 It may be relevant that quite a large number of the archive sheets (unpublished in Saunders (1992)) are concerned with the dynamics and the stability of lattices.
Posted by Jonathan at September 25, 2003 02:30 PM