This is a bridge chapter, not the full treatment promised by this book's index — the complete version is reserved for Vol VII. What follows states the real, citable biology and physics this chapter will build on: Turing's reaction-diffusion instability, Wolpert's positional-information (French flag) model, the auxin/PIN1 transport mechanism actually shown to generate plant phyllotaxis, and Douady & Couder's 1992 physical demonstration that the golden angle is a dynamical attractor, not a biological program.
Alan Turing's 1952 "The Chemical Basis of Morphogenesis" showed that two diffusing, mutually reacting chemical species (an activator and an inhibitor with different diffusion rates) can turn a spatially uniform state unstable and self-organize into a periodic pattern — spots, stripes, or spacing — without any external template. Wolpert's positional-information model (the "French flag" model, 1969) is a related but distinct idea: a single morphogen gradient, read against fixed concentration thresholds, tells cells which of several discrete fates to adopt depending on where they sit in the gradient.
For plant phyllotaxis specifically, the mechanism experimentally established is neither classic two-species Turing reaction-diffusion nor a simple gradient-threshold French flag read-out: Reinhardt et al. (2003, Nature) showed that the plant hormone auxin, actively transported by the polar PIN1 efflux carrier, accumulates into local concentration maxima at the growing shoot tip, and each maximum instructs the initiation of a new leaf or floral primordium at that position — a transport-based patterning mechanism, related in spirit to Turing-type instabilities but not identical to the original two-morphogen reaction-diffusion system.
Separately, Douady & Couder (1992, Phys. Rev. Lett. 68, 2098) showed with a physical experiment — magnetized ferrofluid droplets added at a disc's center and pushed outward by a radial field, repelling each other — that the divergence angle between successive droplets converges to the golden angle as the outward advection speed is slowed, because that angle minimizes the repulsive energy between successive elements. The same Fibonacci-indexed spiral order seen in plants was reproduced in a physics lab with no biology involved at all, and confirmed numerically. This is one of the cleanest published demonstrations that a phyllotactic pattern is a physical self-organization attractor rather than something requiring biological machinery to specify it directly.
This is a checked arithmetic identity connecting phi to the golden angle; it says nothing on its own about auxin, PIN1, or any specific plant's leaf spacing, which is a separate, independently measured biological fact (also generally close to 137.5°, and cited as such across the phyllotaxis literature).
The operator-chain framing this book uses elsewhere (K as threshold, F as fold) applies naturally to morphogen-threshold read-out in the French-flag sense, and the auxin/PIN1 mechanism gives a real, specific, transport-based instability to build a K-operator account on, rather than an idealized two-species Turing system that plant phyllotaxis does not literally use. The full chapter, reserved for Vol VII, is where that operator mapping will be built out in the same C→K→F→U table format used elsewhere in this book. This bridge chapter's job is only to get the citations and the arithmetic right in advance, so the later chapter isn't built on an uncited or approximate version of either.
Turing, A.M. (1952). The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B 237, 37–72.
Wolpert, L. (1969). Positional information and the spatial pattern of cellular differentiation. J. Theor. Biol. 25, 1–47.
Reinhardt, D., Pesce, E.R., Stieger, P., et al. (2003). Regulation of phyllotaxis by polar auxin transport. Nature 426, 255–260.
Douady, S., Couder, Y. (1992). Phyllotaxis as a physical self-organized growth process. Phys. Rev. Lett. 68, 2098–2101.
See also: Book VI Index · Vol VII, Ch A (Ada Lovelace) for the other "Turing."