Phyllotaxis: Crystallography under rotationdilation, mode of growth or detachment
A foam ruled by T1
Nick Rivier
Jean-François Sadoc
Jean Charvolin
Newton 2/14
Red : hexagons
Blue: penta
Green: hepta
• A foam (z=3) on substrate (plane,
sphere, cone, cylinder) with axial
• Fibonacci # pervasive
• layers
• Grain boundaries: circles
z=4, square cells, crit. pt of T1
down (in) complete layers (penta are
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
• 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)
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,-,.,+,.,.,.,.,-,.,...) =
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 ...,.[-,.,.,.,.,+,.,.,+,.,,.,.],-,.,.. =
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
• With two T1, one obtains
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.
• 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)
• 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.
North polar circle bounding spherical
phyllotaxis (13,8,5)
• spherical polar cap
• or continued on
• ending as cylindrical
mast (3,2,1)