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Phyllotaxis: Crystallography under rotationdilation, mode of growth or detachment A foam ruled by T1 Nick Rivier Jean-François Sadoc Jean Charvolin Newton 2/14 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 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 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 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. 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) 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.) 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) 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... 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 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,... 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, 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 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 seedsloaded branches arranged in the (3,2,1) phyllotaxis, the final topological state before physical death. Agave • 13 8 5 (to 8 5 3) to 5 3 2 • ... to 3 2 1 topological end and death 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.... 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)