Turing's letter to Coxeter
According to the theory I am working on now there is a continuous advance from one pair of parastichy numbers to another, during the growth of a single plant ... You will be inclined to ask how one can move continuously from one integer to another. The reason is this - on any specimen there are different ways in which the parastichy numbers can be reckoned; some are more natural than others. During the growth of a plant the various parastichy numbers come into prominence at different stages. One can also observe the phenomenon in space (instead of in time) on a sunflower. It is natural to count the outermost florets as say 21+34, but the inner ones might be counted as 8+13. 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. (From Coxeter (1972)).This quote comes from a paper of the wonderful Donald Coxeter, who quotes it at the beginning of a paper on (from memory) some of the number-theoretic properties of Fibonacci lattices. He died at 96. (Irritating for me because that was two weeks before I heard of this quote and tired to contact him, but no doubt more irritating for him).
Googling to see if this quote has appeared anywhere else, I am amused but not surprised by the prevalence of google hits for "Church is hopelessly confused".
Brothers Bravais
This image arrived in Cambridge, before the railway age, by horse and boat. It sat unseen there inside a volume in which the pages remained uncut, on a high high shelf in a musty library.
From Bravais and Bravais (1837)
Geometrical phyllotaxis
In this section we put aside the radically new contribution of Turing (a mechanism for dynamic production of lattices) to discuss his (slightly) more conventional treatment of the static properties of lattices, more commonly called geometrical phyllotaxis. 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).)
though he was neither first nor last to consider that his theory was the best. This work was (mostly) published for the first time in 1992 in his Collected Works.
This kind of analysis has a long history, at least as far back as the brothers Bravais , but Turing's geometrical theory added several new insights: flow matrices, the `hypothesis of geometrical phyllotaxis' discussed below, and the `inverse lattice', a Fourier representation of the patterns essential to understanding many of the archive pictures though not discussed further here.
Turing's lattice theory
Part I of the Morphogen Theory of Phyllotaxis (see p49 of Turing (1992)) is a fairly coherent and fully worked out manuscript. Two theorems are of particular relevance here. For any lattice, such as the Bravais one, there are not just two rather obvious parastichies (here 2 and 3 are drawn with dashed lines) but a whole series of less obvious ones, which can all be defined relative to lines from the origin (i.e. the point labelled 0) through the other numbered points. The 1-parastichy is the solid line, and the eye can pick out the 4 parastichy by visualising a line through the points numbered 0 and 4 and 8 and so on. What Turing called the `principal parastichies' were the ones in which the nearest points in the parastichy were closest to the origin -- in other words Figure 5 has principal parastichy (2,3) because the points numbered 2 and 3 are the ones closest to the point numbered 0. (The geometrical details are related to but different from, say Jean (1994) : primarily because Jean also needs to ensure that the parastichies wind in opposite directions round the cylinder). A second key theorem is that the third parastichy (in this case 1 since 1 is the next closest point) must be the sum or difference of the first two parastichy numbers, a theorem Turing proves neatly on page 57 of Turing (1992).
