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Phyllotaxis: Crystallography under rotation-dilation

Explore the fascinating patterns of phyllotaxis in this crystallographic study. Discover the role of foam, T1 phyllotaxis, grain boundaries, and spiral lattices. Observe the growth, detachment, and mode of truncation in this mathematically dense and visually stunning representation.

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Phyllotaxis: Crystallography under rotation-dilation

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  1. Nick Rivier Jean-François Sadoc Jean Charvolin Newton 2/14 Phyllotaxis: Crystallography under rotation-dilation, mode of growth or detachment A foam ruled by T1

  2. Phyllotaxis • Red : hexagons • Blue: penta • Green: hepta • A foam (z=3) on substrate (plane, sphere, cone, cylinder) with axial symmetry • Fibonacci # pervasive • layers • Grain boundaries: circles z=4, square cells, crit. pt of T1 down (in) complete layers (penta are inclusions) up (out) penta are in next layer • Parastichies (visible spirals) • Core

  3. Spiral lattice • 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 2π/ , where = (1+√5)/2 is the golden ratio. • Requirement of equi-sized florets constraints the radial law of the generative spiral

  4. Generative spiral, spiral lattice • a) the pineapple (not quite correct at polar caps) spherical phyllotaxis (13,8,5) • b) spiral lattice on plane (here, Voronoi cells not equi-sized) • c) spiral lattice on cylinder tangent to sphere (generative spiral (regular) not drawn) - a good representation of a) • d) cylinder flattened on a plane

  5. 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.

  6. Grain boundary (detail) • Circles (conformal transf.) • quasiperiodic array dis\hex\dis\dis\hex\dis\hex\dis... • k (= l1) l (= m1) m (stop) -> k1(new) l1 (= k) m1 (=l) • k = l + m on each grain • T1 : imposes 900 symmetry (seen in Voronoi cells) • Truncated squares : local pattern for crystal growth (crit. point of T1)

  7. In praise of the T1 • local, 900 symmetry • hexagons (chair) into hexagons (zig-zag) • hexagon is a « square » local pattern for crystal growth • perpendicular directions go through • old parastichies perp to new parastichies (inv./conf. trf.)

  8. Grain boundary under T1 • image of grain boundary on a square lattice • Main parastichies 8 and 5 perp. • 13 cells, all truncated squares (5 penta (o), 5 hepta (*), 3 « hexa ») • it is the mode of truncation that flips • bdary (13,8,5)/(8,5,3)

  9. Detachment Remove initial point (s=1) on gener. spiral. Lattice\s=1 invariant. Voronoi cells invariant except s=1 disappears e.g. sphere n≤75 • First layer (5,6,6) • Second layer has 8 cells s = (4,7,10,5,8,11,6,9) cyclic • pentagonal cell s=1 has four neighbours s = (2,3,6,9,4) cyclic, start of parastichies 1,2,5,8,3, all Fibonacci as it should • Now, s=1 detaches. Affects sequence s=1,2,.. thus (o,-,.,+,.,.,.,.,-,.,...), First cell is now s=2. Sequence (5,6,6),[5,5,5,5,5],6,6,6... invariant • Indeed: (5,6,6),[5,5,5,5,5],6,6,6... x (o,-,.,+,.,.,.,.,-,.,...) = (o)(5,6,6),[5,5,5,5,5],6,6,6...

  10. Pentagonal dipyramidIn the foam, detachment or disappearance of pentagonal cell • Essential topological transformation (disconnection of a point in a pentagonal environment on the surface of a convex cluster) • Corresponds to disappearance or detachment of pentagonal cell A. Cell C gains a side, cell D and E remain invariant, the other two lose a side • AB disconnect • The pentagon C. DE . is a (2D) dislocation that can be annealed away

  11. Detachment (ctd) Likewise, sequence (5,6,6),[(6,6,6,6,6),(6,6,6),(5,5,5,5,5)],6,6,... is invariant under detachment of 1 with a T1 on s=4 (.,.,.,-,.,.,.,.,+,.,.,+,.,.,.,.,-,...) that shifts the frst gb [(6,6,6,6,6),(6,6,6),(5,5,5,5,5)]. (13 cells, too small to have 7 hepta but with the topological charge +5 (+1) of an hemisphere) Displace gb by T1 on its first hepta cell ...,6,[7,7,7,7,7,6,6,6,5,5,5,5,5],6,6,... x ...,.[-,.,.,.,.,+,.,.,+,.,,.,.],-,.,.. = ...,6,6,[7,7,7,7,7,6,6,6,5,5,5,5,5],6,...

  12. Spherical phyllotaxis • n cells, genrative spiral symmetrical/mid-equator • n = 16-29 :(5,6,6),[5,5,5,5,5],6,6,6... , invariant/removal of s=1 • n = 43-75 : (5,6,6),[(6,6,6,6,6),(6,6,6),(5,5,5,5,5)],6,6,.., invariant/removal of s=1 and T1 on s=4 • n ≥ 81: (5,6),[(7,6,6,6,6),(5,6,6),(5,5,5,5,5)],6,6,6,6,[(7, 7,7,7,7,7,7,7),(6,6,6,6,6),(5,5,5,5,5,5,5,5)]…, new gb of 21 cells, first layer with 2 cells only, invariant,

  13. Core, planar • Cell, (s=0 at origin) disappears from sequence (5,5,6,7),(7,7,6,5,5,6),[(6,6,6,6,6,7,7,7),(6,6,6,6,6),(5,5,5,5,5,5,5,5)]… • With two T1, one obtains (5,6,6),(6,6,6,6,6,6),[(7,7,6,6,6,6,6,7),(6,6,6,6,6),(5,5,5,5,5,5,5,5)]... NB: innermost gb has 21 cells, the 13-cells gb in spherical phyllo. has been crushed

  14. Natural history of agave • 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.

  15. Agave • 13 8 5 (to 8 5 3) to 5 3 2 • ... to 3 2 1 topological end and death

  16. Agave, details • Spherical phyllotaxis (13,8,5) (5,6,6),[(6,6,6,6,6),(6,6,6),(5,5,5,5,5)],6,...,6,[(5,5,5,5,5)(6,6,6)(.,.,,,.,). • Polar circle [(5,5,5,5,5)(6,6,6)(.,.,,,.,)] • Further growth on cone tangent to sphere at polar circle through complete gb. [(5,5,5,5,5)(6,6,6)(7,7,7,7,7)], then through 2 more gb, to (3,2,1) phyllo, the mast, ie. ...,6,[(5,5,5,5,5),(6,6,6),(7,7,7,7,7)],[(5,5,5),(6,6),(7,7,7],[(5,5),(6),(7,7)],6,6,6....

  17. North polar circle bounding spherical phyllotaxis (13,8,5) • spherical polar cap • [(5,5,5,5,5),(6,6,6),(6,6,6,6,6)](6,6,5) • or continued on cone(s) • [(5,5,5,5,5),(6,6,6),(7,7,7,7,7)],[(5,5,5),(6,6),(7,7,7)],[(5,5),6,(7,7)],6,6,6,... • ending as cylindrical mast (3,2,1)

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