deodands

Lattice generation

So far, the discussion has been in terms of a one-dimensional pattern wrapped around a ring. What happens if we have the same reaction and diffusion mechanism but now allow it to act in a two-dimensional arena?

In terms of the cannibals and missionaries, we might imagine that the beach of the island is now rather wide. In this case, the Turing instability generically generates not a ring of points but a lattice of points. Might this by itself be enough to explain the occurrence of Fibonacci phyllotaxis? With no constraints, (i.e. an infinite cylindrical arena and ignoring the complexities of the inception and quenching of pattern formation), the instability typically generates hexagonal lattices such as in the figure (See Murray (1993)).

(more)

and this class of lattices certainly include some Fibonacci ones. But it also includes many that are not Fibonacci.

Might it be that there are other constraints acting to select Fibonacci ones? First there are the geometrical constraints arising from the particular arena. Clearly patterns on a cylinder must be different from those on an infinite plane, and different again from those on a cone, but this does not promote any special Fibonacci structure. Then there are the dynamical constraints: the pattern does not suddenly appear, but emerges as a result of nonlinear interactions between morphogens over time. Finally there are growth constraints: during the emergence process, the arena itself may be growing with the plant.