Phyllotaxis: Crystallography under rotation-dilation, mode of growth or detachment. A foam ruled by T1
Seminar Room 1, Newton Institute
AbstractCo-authors: Jean-François Sadoc (LPS Orsay), Jean Charvolin (LPS Orsay)
Phyllotaxis describes the arrangement of florets, scales or leaves in composite flowers or plants (daisy, aster, sunflower, pinecone, pineapple). Mathematically, it is a foam, the most homogeneous and densest covering of a large disk by Voronoi cells (the florets). Points placed regularly on a generative spiral constitute a spiral lattice, and phyllotaxis is the tiling by the Voronoi cells of the spiral lattice. The azimuthal angle between two successive points on the spiral is 2p/ t, where t = (1+v5)/2 is the golden ratio.
If the generative spiral is equiangular (Bernoulli), the phyllotaxis is a conformal (single) crystal, with only hexagonal florets (outside a central core) and zero shear strain. Florets of equal size but not all hexagonal are generated by points on a Fermat spiral. There are annular crystalline grains of hexagonal florets (traversed by three visible reticular lines in the form of spirals, called parastichies) separated by grain boundaries.
Grain boundaries are circles of dislocations (d: dipole pentagon/heptagon) and square-shaped topological hexagons (t: squares with two truncated adjacent vertices). The sequence d t d d t d t is quasiperiodic, and Fibonacci numbers are pervasive. The two main parastichies cross at right angle through the grain boundaries and the vertices of the foam have degree 4 (critical point of a T1) . A shear strain develops between two successive grain boundaries. It is actually a Poisson shear, associated with radial compression between two circles of fixed, but different length. Thus, elastic and plastic shear can be readily absorbed by a polycrystalline phyllotactic structure described by several successive Fibonacci numbers. The packing efficiency problem is thereby solved: One grain boundary constitutes a perfect circular boundary for the disk into which objects are to be packed.
An application of phyllotaxis to growth can be seen in Agave Parryi. Structurally, it spends almost its entire life (25 years, approx.) as a single grain (13,8,5) spherical phyllotaxis, a conventional cactus of radius 0.3 m. During the last six month of its life, it sprouts (through three grain boundaries) a huge (2.5 m) mast terminating as seeds-loaded branches arranged in the (3,2,1) phyllotaxis, the final topological state before physical death.
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