deodands

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.