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

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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

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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

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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

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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.

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Grain boundary (detail) Circles (conformal transf.) quasiperiodic array dis\hex\dis\dis\hex\dis\hex\dis... k (= l 1 ) l (= m 1 ) m (stop) -> k 1 (new) l 1 (= k) m 1 (=l) k = l + m on each grain T1 : imposes 90 0 symmetry (seen in Voronoi cells) Truncated squares : local pattern for crystal growth (crit. point of T1)

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In praise of the T1 local, 90 0 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.)

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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)

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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...

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Pentagonal dipyramid In 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

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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,...

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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,

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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

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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.

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

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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....

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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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Dislocation glides away under shear square lattice of struts + defect (pentagon) under shear (i) odd circuit must be interrupted by one extended edge (AB) (ii) triangulation (extended edges -----) (iii) vertices have degree 6 except o (5) and * (7) (iv) Flipping edge ===== (v) dipole o+ is dislocation, that glides

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