Fibonacci modern

At Turing’s death, all of his post 1951 developments remained unpublished.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 Wardlaw is reported to have maintained a long interest in Fibonacci phyllotaxis though it seems to have gone unpublished. (Email from Vidyanand Nanjundiah, 20th March 2003; 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).

Figure 13 All possible parastichy pairs can occur, but only one branch is continuously reachable from the simplest symmetric case. Thin lines: theoretically possible parastichies; triangles: observed parastichies found in numerical simulations from various starting conditions. From Douady and Couder (1996). © Academic Press 1996 COPYRIGHT NOT CLEARED

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 (Figure 13). This local constraint is what Turing would have called the Hypothesis of Geometrical Phyllotaxis.

Fibonacci phyllotaxis: A modern story