deodands

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.

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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).