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SUMMARY:Phyllotaxis: Crystallography under rotation-dilation\, mode of gro
 wth or detachment. A foam ruled by T1 - Rivier\, NY (University of Strasbo
 urg)
DTSTART:20140226T145500Z
DTEND:20140226T151500Z
UID:TALK51102@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:Co-authors: Jean-François Sadoc (LPS Orsay)\, Jean Charvolin 
 (LPS Orsay) \n\nPhyllotaxis describes the arrangement of florets\, scales 
 or leaves in composite flowers or plants (daisy\, aster\, sunflower\, pine
 cone\, pineapple). Mathematically\, it is a foam\, the most homogeneous an
 d 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\, whe
 re t = (1+v5)/2 is the golden ratio.\n\nIf the generative spiral is equian
 gular (Bernoulli)\, the phyllotaxis is a conformal (single) crystal\, with
  only hexagonal florets (outside a central core) and zero shear strain. Fl
 orets of equal size but not all hexagonal are generated by points on a Fer
 mat spiral. There are annular crystalline grains of hexagonal florets (tra
 versed by three visible reticular lines in the form of spirals\, called pa
 rastichies) separated by grain boundaries.\n\nGrain boundaries are circles
  of dislocations (d: dipole pentagon/heptagon) and square-shaped topologic
 al hexagons (t: squares with two truncated adjacent vertices). The sequenc
 e d t d d t d t is quasiperiodic\, and Fibonacci numbers are pervasive. Th
 e 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 sh
 ear strain develops between two successive grain boundaries. It is actuall
 y 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 th
 ereby solved: One grain boundary constitutes a perfect circular boundary f
 or the disk into which objects are to be packed.\n\nAn application of phyl
 lotaxis to growth can be seen in Agave Parryi. Structurally\, it spends al
 most its entire life (25 years\, approx.) as a single grain (13\,8\,5) sph
 erical phyllotaxis\, a conventional cactus of radius 0.3 m. During the las
 t six month of its life\, it sprouts (through three grain boundaries) a hu
 ge (2.5 m) mast terminating as seeds-loaded branches arranged in the (3\,2
 \,1) phyllotaxis\, the final topological state before physical death.
LOCATION:Seminar Room 1\, Newton Institute
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