deodands

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'.