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

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 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

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 seeds-loaded 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)