Structure of Bénard convection cells, phyllotaxis and
crystallography in cylindrical symmetry
N. Rivier, R. Occelli, J. Pantaloni, A. Lissowski
To cite this version:
N. Rivier, R. Occelli, J. Pantaloni, A. Lissowski. Structure of Bénard convection cells, phyllotaxis and crystallography in cylindrical symmetry. Journal de Physique, 1984, 45 (1), pp.4963. <10.1051/jphys:0198400450104900>. <jpa-00209739>
HAL Id: jpa-00209739
https://hal.archives-ouvertes.fr/jpa-00209739
Submitted on 1 Jan 1984
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J.
Physique 45 (1984) 49-63
Classification
Physics Abstracts
47.25Q - 61.00
-
JANVIER
1984,
49
87.45
Structure of Bénard convection cells,
phyllotaxis and crystallography in cylindrical symmetry
N. Rivier
(*+),
R. Occelli
(+), J.
Pantaloni
(+) and A.
Lissowski
(* ~)
(+) Dynamique des Fluides, Université de Provence, 13000 Marseille, France
(*) Blackett Laboratory, Imperial College, London SW7 2BZ, U.K.
(~) Dept. of Psychology, Polish Academy of Sciences, Warsaw, Poland
(Reçu le 20 mai 1983, révisé le 21 septembre, accepté le 26 septembre 1983)
Ceci concerne la cristallographie à deux dimensions en symétrie cylindrique. Des défauts (cellules
hexagonales) sont introduits nécessairement par la symétrie, et, dans les structures de Bénard, des cercles
de glissement de dislocations servent à dissiper un faible cisaillement du à la rotation terrestre.
Des cercles de glissement apparaissent naturellement en phyllotaxie (arrangement des florets dans les fleurs
composées) et la structure des marguerites, ananas, etc., constitue la première étape de la construction de structures
de Bénard. Toutes ces structures sont engendrées par un algorithme élémentaire de Théorie des Nombres. Elles
sont auto-similaires et localement homogènes, engendrées par un seul nombre irrationnel 03BB. L’homogénéité
demande que 03BB soit un nombre Noble, et explique la prolifération des nombres de Fibonacci en phyllotaxie.
Une fois la structure construite, il est élémentaire de simuler et d’analyser sa fonte.
Résumé.
2014
non
This paper is concerned with crystallography in two spatial dimensions, in the presence of cylindrical
Defects
symmetry.
(non-hexagonal cells) are imposed by the symmetry and glide circles are necessary to dissipate
a weak, steady shear associated with the earth’s rotation.
Glide circles occur naturally in phyllotaxis (leaf or floret arrangement), and the structure of daisies represents
the first stage of the construction of Bénard patterns. Both types of structures can be generated by an elementary
algorithm, which constitutes a physical application of number theory. The structures are self-similar and locally
homogeneous. They are generated by a single, irrational number 03BB. Homogeneity imposes 03BB to be a Noble number
and explains the pervasiveness of Fibonacci numbers in phyllotaxis.
Once the ideal structure is constructed, melting can be simulated and analysed.
Abstract.
2014
1. Introduction.
Classical crystallography consists of an enumeration
of the infinite, space-filling patterns made by repetition of identical cells. One then introduces defects
as local faults in the pattern, almost as an afterthought, despite their universal occurrence in real
crystals, and their overriding effect on the physical
properties of crystalline matter. The perfect crystal
corresponds to a strict energy minimum, however
inaccessible experimentally.
The situation is very different if the system is finite.
The few, infinite pattern (space groups) of classical
crystallography are incompatible with most boundary
conditions, and defects are a necessary ingredient of
the resulting pattern which minimizes the energy.
The classification and description of these finite
patterns is still an open problem [1], which we shall
solve here for a particular class of systems.
This paper is concerned with crystallography in
spatial dimensions, in the presence of cylindrical
(or axial) symmetry. The symmetry is imposed by
boundary conditions, by conditions of growth (as
in plant phyllotaxis), or by external agents like the
earth’s rotation. In the absence of cylindrical symmetry and for identical cells, the problem has been
completely solved by bees and 19th century crystallographs, and nothing more need to be added to the
classic enumeration of space groups. (Nonperiodic,
or random patterns still await classification, but
they require at least two different kinds of cells [1-3]).
We shall search here for crystalline structures filling
the space available, with as much homogeneity,
and made of as similar, isotropic cells, as is compatible
with the boundary conditions. The whole structure
should be described by a single algorithm which
must be as simple as possible.
There are two direct fields of application. Cellular
two
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198400450104900
50
occur in Benard convection, whereby a
fluid heated from below exhibits convective motion
above a certain temperature threshold. The centre
of a cell corresponds to the hot, rising fluid, and the
vertices of the pattern to the cold, descending fluid.
A perfect infinite pattern would form a hexagonal
(honeycomb) lattice. Cylindrical symmetry is imposed
by the shape of the container, and, as we shall see,
by the earth’s rotation [4]. It is also believed that
the solar granulation is also a result of convective
motion [5]. There, too, some non-hexagonal cells
must appear for topological reasons (Euler’s theorem). A different manifestation of cylindrical crystallography can be found in the structure of daisies,
sunflowers, pine cones, etc., called phyllotaxis (or
leaf-arrangement), which constitutes a space-filling
problem solved by some algorithm or code governing
the successive generation of cells or florets from the
stem, and whose manifestation is through a sequence
of numbers of opposite spirals (the Fibonacci
sequence) [6, 7]. Here, cylindrical symmetry, and
the resulting structure, are imposed by successive
generation from a central axis, whereas in Benard
convection, the pattern appears more or less at
once in the whole fluid.
As a consequence, this paper can be read at three
different, self-contained levels. It is first and foremost
a description of the structure of B6nard cells in cylindrical symmetry, that is, the solution of a problem
of fluid dynamics (sections 2, 4, 5 and Figs. 1, 2, 7).
The second level is that of phyllotaxis, where the
emphasis lies on successive generation of cells, for
which coding replaces packing as the problem solved
by nature (sections 3, 4 and Fig. 6). Finally, this paper
is an essay on crystallography where defects are
necessary ingredients imposed by symmetry or boundary conditions.
patterns
B6nard-Marangoni convection patterns, defects and
cylindrical symmetry.
Recent experiments [4, 8] on the Benard-Marangoni
convection in cylindrical containers with large aspect
ratio, yield the following clues to the structure of
2.
convective cells.
1) The cellular pattern is rotating. The trajectory
of an individual cell is a spiral, with angular period
that of the Foucault pendulum [4].
2) The cell rotates about itself with the same
period, but undergoes large, apparently random
fluctuation of magnitude 2 7r/5.
3) The pattern (Figs. 1, 2) always contains a finite
number of penta- and heptagonal cells. It can never
be made perfectly hexagonal. This remains so for
rectangular containers, but the number and position
of non-hexagonal cells varies with the geometry of
the container. It also varies with the Marangoni
number : if the temperature of the bottom of the container is increased further above To, the convective
Fig.
1.
-
structure
Photograph of a Benard-Marangoni convective
[4] in a silicone oil layer (thickness about 1 mm).
threshold temperature, the pattern begins to melt [8].
Knowledge of the crystalline structure (with its
necessary number of defects, see below), is a prerequisite for understanding the dynamics of melting.
Similar patterns (Fig. 2) are obtained in nematic
liquid crystals, with hydrodynamic instability induced
by circular shear [8, 9]. The shear rate plays the part
of the Marangoni number in the melting of the structure.
Observations (1) and (2) point at the Coriolis
force of the earth’s rotation as the main agent in the
evolution of the cellular structure (1). Benard cells
are convection patterns, the liquid flowing in the
free (upper) surface from the centre of the cell (hot
point) to the vertices (cold points). Under Coriolis
force, both individual cells and the network as a
whole are subjected to a steady, weak rate of shear.
Crystalline patterns respond to shear by motion
(glide) of dislocations which represents the most
efficient means of dissipating shear energy.
To a first approximation, the cellular pattern is
clearly the solution of the purely geometrical problem
of filling an area with roughly isotropic and equalsized cells. The cells are deformable, and repel each
(1) Whether the Coriolis force due to the earth’s rotation,
is entirely or partially responsible for the rotation of the
cellular pattern, is still a matter for debate. It is not a prerequisite for the theory presented in this paper. The manifest
experimental facts on which the following discussion is
based, are that the pattern is rotating, and that there is
some dissipative mechanism. The advantage of assuming
Coriolis forces is that it suggests directly the presence of
circles of dislocations (Coriolis --&#x3E; shear dislocation
glide). But cylindrical symmetry, and the necessity to
screeri the strain due to the 6 pentagonal cells necessary
from topology, leads to the same result. Of course, an alternative solution is a random, polycrystalline mosaic.
51
packed with convex objects of equal size
triangular pattern with hexagonal cells.
A cell can be defined geometrically as the region in
space closest to the centre of one particular object.
In this case the space is partitioned by Voronoi or
Wigner-Seitz cells. The most obvious defects are
penta- or heptagonal cells (Figs. 1 and 2). There are
An
area
produces
a
also a few vertices of coordination, 4 instead of 3,
but these are not topologically stable (a small deformation splits them into two normal vertices of coordination 3) and have a short lifetime in Benard
patterns [11].
Penta- or heptagonal cells are positive or negative
disclinations (rotation dislocations), sources of positive or negative curvature. They are topologically
defined objects which are structurally stable, that is,
they keep their identity under small deformations.
Opposite disclinations attract each other. A dipole
of disclinations, a pair pentagon-heptagon, is a
(translation) dislocation, since two successive rotations of opposite sign about parallel but distinct
axes make up a translation. Such dislocations are
easily observed in figure 2. Exactly as isolated charges
are screened by polarizing the surrounding medium,
the strain energy of isolated disclinations is screened
by dislocations [12]. This is why a radiolaria has
more non-hexagonal cells than the 12 it needs topologically [13]. Dislocations, as the element of polarization of the cellular network, is the mechanism
whereby strain energy is locally screened.
Topologically, a finite cellular pattern must contain
6 isolated positive disclinations (pentagonal cells).
This results from Euler’s theorem, relating the number
of cells F, edges E and vertices V
Fig.
z.
-
Keconsiruaion 01 ine cenuiar
structurew igner-
Seitz-Voronoi construction). a) In a nematic liquid crystal
with an instability induced by a circular shear [8, 9, 26].
b) In a standard Benard-Marangoni experiment [4, 8, 26]
0
heptagon, X octagon, 0 + dispentagon, +
location.
=
=
=
=
other to maintain their correct size, which is given
thickness [4]).
to the magnetic
ball model than to the bubble-raft model [10] popular
to visualize dislocations and grain boundaries. The
Coriolis force and boundary conditions are small
perturbations on the main requirement of areafilling by isotropic cells of equal size. The specific
nature of the interaction between cells, which should
come out of the differential equations of hydrodynamics, is assumed to have an even smaller effect,
and will be neglected.
by hydrodynamics (a = 1.9 e, e
The repulsion is soft, more akin
=
for a finite, simply connected, two-dimensional network. (The cell at infinity is not counted.) Let Fn
be the number of n-sided cells, and Ep, Yp, the edges
and vertices on the perimeter of the network. We
have the valence relations
because every
separates
two
edge, except
those
on
the
perimeter,
cells, and
edge joins two vertices, and every
(tetravalent vertices are not topologically stable : they can be split into 2 trivalent
vertices by a small deformation), except those on the
perimeter of the network, which are all divalent
(this defines perimeter vertices topologically). The
valence relations included in Euler’s formula yield
because every
vertex is trivalent
A natural method of closing peripheral cells is by
every rim segment by two perimeter edges
replacing
52
defect-free grains, which can glide on each other,
are seen in large daisies and sunflowers, whose structure can also be regarded as the solution of a geometrical problem in cylindrical symmetry [15], which
we shall discuss in the next two sections.
3.
3.
Natural closure of the cells-on the boundary
of the container. This convention eliminates the distinction
between boundary and interior cells in Euler’s formula 2.4.
Fig.
-
Ep and one perimeter vertex Vp (Fig. 3), thus Ep
=
2
Vp
and, by (2.4) the network contains 6 positive discli-
nations. The natural method of closure does not
apply to containers with sharp corners, for which
a full analysis of the right-hand side of equation 2.4
is necessary. A more intuitive and geometrical derivation is due to Dormer [14]. Consider a cigar-shaped
surface, half of which contains our finite pattern.
The cigar, being homeomorphic to a sphere, must
contain 12 positive disclinations. This is easily obtained
from equation 2.1, with 2 instead of 1 on the righthand side (sphere), and equations 2.2, 2.3 with
Ep Vp 0 (no boundary). The cylindrical section
contains no disclinations, and the two hemispheres
6 each by symmetry.
In the experiments of Pantaloni et al. [4, 8], the
edges of the peripheral cells are perpendicular to
the rim of the container. However, precise knowledge
of the boundary conditions (whose justification
remains an open problem), is not necessary for the
argument of this paper, where only cellular patterns
and an overall cylindrical symmetry need be assumed.
(The structure and stability of some convective
structures (rolls) under different conditions, have
been discussed recently in reference 30).
In conclusion, Euler’s theorem imposes the presence of 6 positive disclinations or pentagonal cells
in the midst of the network. Their curvature, or strain
energy, must in turn be screened by dislocations,
which are pentagon-heptagon dipoles. Moreover,
the shear induced by the earth’s rotation is also dissipated by dislocation glide. However, cylindrical
symmetry requires the glide lines to be concentric
circles, instead of the straight glide lines or planes of
conventional crystals under shear. Disclinations and
dislocations are an essential part of the structure,
imposed by boundary conditions and by topology.
They must not be regarded as defects which can
be annealed out by some heat treatment.
The key problem is to produce a structure with
concentric glide circles. It turns out that concentric
circles of dislocations, forming boundaries between
=
=
Phyllotaxis (leaf
or
floret arrangement).
The inner florets of an aster, a daisy or a sunflower,
the scales of a pine cone or a pineapple, form an
area-filling structure in which neighbouring « cells »
are arranged on spirals or parastichies. A hexagonal,
cell belongs to three parastichies, but most botanical
cells have the shape of a lozenge, and belong to only
two manifest parastichies of opposite chirality. In an
overwhelming number of cases, the number of lefthanded and right-handed parastichies are two consecutive Fibonacci numbers. In the largest sunflowers
(see, e.g. [13]), the structure stretches from one pair
of consecutive Fibonacci numbers to the next higher,
when one goes from the centre to the periphery. It is
also said that the number of parastichies can be
increased by intensive cultivation [16].
It seems, therefore, that a family of structures
can be generated by a simple code or algorithm,
capable of dealing with affine (scaling) transformations or with breeding, and that the Fibonacci series
is the external manifestation of this code. Coding
operates upon generation of new florets from the
stem or the centre. It fixes the angle of successive
florets. The resulting structure is obtained simply
by younger florets pushing the older ones from the
centre. The code is itself the practical translation
of a biological variational principle, which may well
be the most efficient sharing of horizontal space
between florets or leaves in order to share the sun,
rain or air [6]. As far as the physicist is concerned,
we have a genuine crystallography, with a few, areafilling structures, characterized by an algorithm or
code. Moreover, the structures obtained are selfsimilar radially, and, like the Benard convection
pattern, dominated by the geometrical requirement
of filling space with roughly isotropic cells of equal
size (scaling due to growth notwithstanding). The
symmetry is obviously axial and, as we have argued
in the preceding section, this automatically introduces defects in the structure. Finally, one would
expect the code to be as simple as possible, in order
to generate the most probable structure consistent
with the geometrical constraints.
[The only structural information which is encoded
(cf. algorithm 4.1) is the angle A between florets,
and the fact that successive florets appear at regular
time intervals. The shape of each floret (whether it
is a pentagon, hexagon, etc.) is not coded. It is simply
a consequence of filling an area (Voronoi construction).-The fluidity of the structure must be emphasized. Indeed, nothing prevents the shape of a particular floret to vary with time.]
53
In the following section we shall construct such
structures from first principles. The agreement with
botanical structures is manifest (cf. Fig. 6). Moreover,
shall associate the omnipresence of the Fibonacci series or the Golden section r in phyllotaxis,
with self-similarity and homogeneity of the structure.
The structure contains defects, is finite, and homogeneity can no longer directly be associated with
global translational invariance. This justification of
the Golden section is new, although Coxeter [15],
following Tait, has given a different reason (absence
of intermediate convergent) which is mathematically
sufficient, but much less overriding structurally.
As a bonus, we shall find that the Golden section,
and Fibonacci series, are given by the simplest code.
Finally, because they have circular glide lines (Fig. 6),
the botanical structures must be very similar to
their hydrodynamic counterpart.
we
daisy : crystallography in axial symmetry.
The construction proceeds by steps. We shall end
up with a genuine crystallography in cylindrical
geometry, i.e. with only a few possible structures,
identified by an elementary algorithm, or code, which
are structurally stable and sufficiently flexible to
accomodate elementary defects or excitations. Whether
the physical system will take up such structures, or
select a polycrystalline structure with polygonized
grain boundaries, will depend on the relative strength
of the cylindrical perturbations (size of the circular
boundary relative to that of the liquid, strength of
shear stress, etc.). The same dilemma has been investigated by Farges in atomic clusters of n atoms.
Small clusters (n
200) take up polyicosahedral
which
minimize (locally) the energy,
configurations
but cannot fill space, and larger ones (n &#x3E; 200)
have the closed-packed, space-filling configurations
associated with crystals.
4.1 Symmetry suggests polar coordinates for the
cell centres, r(l), 0(l), where 1 = 1, ... labels the individual cells, proceeding outwards from the centre
of symmetry. Once the centres are specified, the
cells are constructed by Voronoi partition of space
around each centre (random Wigner-Seitz cells).
The construction is unique. Uniformity requires
that 0 is proportional to d, and homogeneity in cell
densities, that r increases as 11/2, on average. Curvature of the cellular « substrate » (as occurs in some
plants), or growth of the florets, can be accommodated by any monotonic function r =.f(/). Thus, the
algorithm reads
4. The
where a is the typical cell’s linear dimension (its
area is na2), and A is the only parameter characterizing the structure. For definition, 0 A 1. The
algorithm 4.1 introduces cells regularly on a gene-
or genetic spiral in the reverse order of successive leaves budding from the stem of a plant.
The continuous relation r - (J1/2 associated with
(4.1) is obviously radially self-similar. The discrete
version (4.1) which corresponds to discrete cells
will also be self-similar, in a remarkable connection
with theory of numbers, as we shall see below.
The area can be regarded as partitioned into concentric, quasi-circular shells containing an integer
number of cells, m say, such that 0(l + m) L-- 0(l) [17].
It follows immediately that A cannot be rational.
If it were, Â.
A/B say (A, B integers), all cellsI &#x3E; B
would have their centres precisely on the same B
azimuths (lying on top of each other), and the resulting
cellular structure would resemble a spider’s web
(Fig. 4), which is hardly desirable as a solution to the
problem of filling area with isotropic cells. The shells
correspond, therefore, to successive rational approximations of A, or a division of 2 n by successive integers,
the number of cells pdr shell.
rative,
=
Fig.
4.
with A
-
=
The
spider-web,
constructed from
algorithm
4.1
13/21 rational.
4.2
Theory of numbers yields these successive rational
approximations of A elegantly and directly [18, 19].
Any number can be represented as a continued fraction,
1, with qi integers &#x3E; 0. This decomposition is
unique. Thus, A { qi }, is given by the set of integers
qi’ which constitute a code. If A is rational, the continuif A
=
ed fraction is finite. It is infinite for irrational A.
54
Successive truncations of the (infinite) continued
fraction yield convergents, which are rational approximants to A
where
Am’ B. are (relatively prime) positive integers,
monotonically increasing with m. These convergents
yield the successive shells : B. is the number of cells
in the shell described by the mth convergent, and Am
the number of turns of the generative spiral necessary
2 nÂ.(l + Bm) =
to fill the shell, since 0(l + Bm)
2 n(A"JBm) I, mod. 2 n rr 0(1),
2 n(A"JBm)(1 + Bm)
mod. 2 n. One expects celll + B. to be neighbouring
cell 1, and B. is the number of secondary spirals, or
parastichies, corresponding to the mth convergent
to A. In fact, the cells are hexagonal, and through each
cell pass not one, but three lines of neighbours (i.e.
three parastichies). Thus, for any cell correspond not
one but three numbers
Bj, Bm &#x3E; Bm - 1 &#x3E; Bm - 2’ hence
the necessity of having not one, but three convergents
to A simultaneously. Continued fraction representation
provides an infinity of convergents to A. We shall
define the circular grain m as those shells with B.,
Bm - 1, Bm-2 numbers of parastichies. These grains are
immediately conspicuous in sunflowers, and in figure 6,
generated by algorithm (1) with A 1/T, the inverse
of the Golden section. In plants, the B’s are numbers
=
=
=
of the Fibonacci series.
One can show that the successive convergents
Am/Bm of A given by (4.3) are alternatively smaller
and larger than A [20], improve as m increases, and that
they are the best possible approximation at any order
(any better rational approximation will have larger
denominator B). The numerators and denominators
of the convergents satisfy recursion relations
cell) be generative. If it is not, one has a glass instead
of a crystal [3].
A theorem of Lagrange states that A is represented
by a periodic continued fraction (after a few stages, i.e.
qj arbitrary forl m, periodic thereafter) if, and only
if, A is a quadratic irrational, i.e. a solution of the
quadratic equation aÂ.2 + M + c 0, a, b, c, integers.
=
4.4 Finally, homogeneity of the structure (the condition that the frame of reference { Bm } is independent
of the cell considered) requires that, after a few arbitrary qi, the continued fraction must merge into the
inverse of the Golden section I/T
1)
=
1(,]5 -
Such A have been called Noble numbers
by
=
Percival
[24].
Thus 1/i p qi
1, the simplest possible code !
the
is
of
a
solution
quadratic equation t2 - t - 1 = 0,
(i
and the numerator and denominator of its convergents
{ Am }, { B. }, constitute the Fibonacci series
{ Bm } 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, familiar
to rabbit-breeders. On the other hand, the geometric
properties of -r have inspired many Pythagorician metaphysicists, art critics and architects [6, 17, 21].)
The proof of the theorem (4.5) goes as follows.
Consider an arbitrary cell 1. Its neighbouring cells are
labelled 1 ± Bm, I ± Bm - 1 and 1 + Bm - 2 (Fig. 9).
Homogeneity requires that the {Bm} constitute a
suitable frame of reference, independent of the value
of 1. Thus, one can go from 1 toI + B. either directly,
along the B. parastichy), or through cell I + Bm-2
(along Bm - 2 and Bm - 1 parastichies). To be consistent,
=
=
...,
1 ’dm (4.4), and the inverse Golden
which yields qm
section 1/T.
There are other reasons for ending up with lit.
=
that the numerology of parastichies { B. I (e.g. the
Fibdnacci series) is given directly by the code { qm }.
Finally, one can rewrite (4.4) as (Bm - Bm-2)
qm Bm-1, which implies that the density of B’s (and
A’s) is largest for smallest q’s. It also suggests an
elegant geometrical representation of A and its convergents, due to F. Klein ([18], p. 111).
so
grain corresponds, as we have seen,
parastichies m : (Bm, Bm _ 1, Bm - 2). A grain
boundary is a circle of dm dislocations, corresponding
to pentagon-heptagon pairs, with sm solitary hexagon
in between (Fig. 6) linking grain m - 1 : (Bm-1,
Bm- 2, Bm - 3 ) to grain m : (Bm, Bm - 1, Bm - 2 ). It corresponds to switching from one set of convergents of
4. 3 The choice of A, and of the few possible structures, to the next better one [22]. Two sets of parastichies out
is now narrowed by the following arguments. We have of three continue through the grain boundary to the
seen that A must not be rational. It should not be
next grain, namely Bm - 1 and Bm - 2. The set of flattest
transcendent, either, because, in that case, the conti- spirals, Bm-3’ ends on 8m B. - 3 solitary hexagons
nued fraction (i.e. the sequence { qi }) would not be of the boundary, and is replaced in the next grain m
periodic, and the code { qi } would be infinite. Alter- by Bm steepest spirals, stemming out of all 2 d. + sm
natively, radial homogeneity, or self-similarity, requires cells (penta, hexa and heptagons) in the grain bouna periodic continued fraction. This can be regarded in
dary. Bm and Bm-3 have opposite chirality. Clearly,
physics (unlike biology where coding is essential) as a
weak assumption of simplicity, but it is the familiar
requirement of crystallography that the pattern (unit
i)
=
to
a
A circular
set of 3
=
55
Fig.
5.
-
False
phyllotaxis, constructed from algorithm 4 .11
}, b) Ànot merge
with
{ 2 }, a)c) AA=4/(Il
3) {{ 3,3 },3, 4which
§l/13 +- 15-)
do not merge
in to the inverse of the Golden section. See discussion in
appendix A.
=
=
Structures { qm } = 2 or 3, which do not have the same
desirable properties of self-similarity as {qm}
1
more
bits
to
encode
(Fig. 5) require
[23].
Homogeneity only requires qi 1 after. a few
stages, i.e. with the first few qi arbitrary. We shall take
advantage of this arbitrariness in section 5 to select the
structure of convective cells which is the most similar
to a closed-packed crystal. On the other hand, in
botany, coding is governing the successive generation
of florets or leaves, and this would imply overwhelmingly qi 1 for all i and A 1/i. However, minor
deviations have been observed in about 10 % of
Norwegian spruce cones (cf. [6], p. 923 or [15], p.174) :
6 % exhibit 4-7 parastichies, and 4 % the unlikely
4-6 parastichies, whereas the Fibonacci series requires
5-8. 4-7 is sometimes called the Lucas sequence
([7], p. 83), and is achieved with q1 3, qi &#x3E; 1 1, i.e.
A
1/(3 + lit). It merges into I/T after a few stages,
and does not correspond to an essential change of
coding. We are unable to generate the 4-6 phyllotaxis
in that fashion (unless one uses ql
4, q2 = 0,
2, qi&#x3E; 3 1, i.e. Â. 1/(6 + I/T), but the 4 spiq3
rals should not be apparent [23]), and would welcome
less indirect evidence than a second-hand quotation
=
=
=
=
=
=
is consistent with equation 4.6, or with the recursion
1. The procedure is repeated at
relation 4.4 for qm
every grain boundary, with dm+ 1 = dm + 8m’ 8m+ 1 = dm.
Such normal phyllotaxis (ending in 1/i) is summarized
in table I. The structure is self-similar in that every
grain or grain boundary is similar to every other.
=
=
=
=
=
=
ii) The highest density of successive convergents of
(i.e. the smoothest sequence of convergents) corres- ([6], p. 923).
ponds to { qm} 1, i.e. Â. 1/(... llr), as was pointed
On the other hand, serious coding errors (the
out in connection to recursion relations 4.4. The
« false phyllotaxis » of Coxeter [15]), with A not
smoothness is a quantitative formulation of the homoterminating as 1/T, give strongly spiralling structures,
geneity in covering space by isotropic cells.
shown in figure 5, very different in appearance from
iii) Last, but not least, the simplest possible code the normal phyllotaxis of figure 6. In these structures,
{ qm } 1 is required to generate the entire structure. the spiralling is conspicuous from the centre outwards,
=
=
=
56
Table I.
-
Normal
phyllotaxis (the Daisy)
Grain characterized by 3 parastichies numbers m : (Bm, Bm-1, Bm - 2). Grain boundary characterized by dm dislocations
and sm isolated hexagonal cells. A parastichy extends through 3 grains, starting from one of the 2 d + s cells of the inner’
boundary and ending on one of the s isolated hexagons of the outer boundary :
Illustrated in
figure
6. Radial
self-similarity
is manifest and faultless.
and the cells are no longer hexagonal, but nearly
rectangular. The structure is no longer radially selfsimilar (compare the different grain boundaries of
Fig. 5). This is because the requirement of homogeneity
within each grain governs the choice of rational
approximations of A, which will not only consist of the
(principal) convergents {Am} and {Bm} given by
the recursion relations 4.4 (which violate homogeneity
for qm &#x3E; 1), but also of intermediate convergents Bm(a)
defined by :
a is an integer. The convergents A (’)IB (’) are
longer alternatively smaller and larger than A, and
they are no longer the best possible approximation of
a given order. It turns out that most grains consist
then of only one principal convergent and two intermediate convergents, (B::), B (cx - 1), Bm-l)’ except for
groups of two which have two principal and one
where
no
intermediate convergents (Bm, B("--’), Bm-1) and
(B (1)m+ 1, Bm, Bm-l)). The parastichies labelled by intermediate convergents are carried through 2 grains only
and are not visually manifest, whereas the parastichies
labelled by the principal convergents (Bm, say) are
carried through qm+ 1 + 2 grains. It is this continuation
of parastichies through more than 3 grains which gives
the structure its spiralling appearance. The structure of
non-Fibonacci phyllotaxis is discussed in appendix A
and summarized in table II. It is also entirely determined by the requirement of homogeneity within.each
gain (4.6), but the articulation between successive
gains is different from normal phyllotaxis, and grains
are no longer self-similar.
In this section, we have shown that homogeneity
and self-similarity of the structure imply that A must
merge into the Golden section, A = 1/(ql + 1/(... + l/r)),
after a few qi ¥- 1.
The structures are few and generated by a very
simple code { ql’ ..., 1 } which is the equivalent of the
unit cell in crystallography. They reproduce perfectly
(Fig. 6) the botanical arrangements, but their conspicuous spirals are absent in B6nard-Marangoni convective structures. Nevertheless, radial self-similarity, and
homogeneity are overriding requirements, and as they
are due to the azimuthal distribution of the cells (l)
with A terminating as 1/T, this part of the algorithm
should not be tampered with. The radial distribution
of the cells, on the other hand, can be adjusted as long
as r(l) remains a monotonic function of 1 increasing as
11/2
on average.
We must stress that the selection of appropriate A is
based on the theory of numbers, so that the general
appearance of the structure (cell shapes, etc.) is not a
continuous function of Â..
Compare, indeed, figures 4 and 6. The sequence of
successive approximations to Â., and therefore to the
structure, is given by its continuous fraction representation (4.2), and it is through their continuous fractions that different A and their structures can be
compared.
generated in this section have
cylindrical symmetry and glide circles (circular grain
boundaries) capable of dissipating shear strain. These
are the features anticipated in § 2 for B6nard-Marangoni patterns.
The structures
5. Construction of the
Benard-Marangoni convective
structure.
The Daisy structure (Fig. 6) shows conspicuous spirals,
which are absent in the B6nard convective structures
of figures 1 and 2. The spiralling in plants originates
from their growth. Florets or leaves are generated one
after the other from the centre of the flower or the stem
of the plant, at regular angles. In hydrodynamics,
on the other hand, the whole cellular structure is a
collective phenomenon generated almost instantaneously (even though the peripheral cells are formed
last when the temperature difference is increased above
threshold). However, the daisy structure with its
57
within
cylindrical symmetry, one should replace the
generative spiral by concentric, circular shells.
The first 4 shells should fit, as much as possible, a
perfect triangular structure (hexagonal cells) in the
neighbourhood of the geometrical centre. This selects
the first few qi of the possible A which were hitherto
arbitrary. The triangular structure has its cells on
circular layers containing n cells at distance p(n)
from the centre,
with b, the lattice spacing. The histogram follows
r
a JT (4.1), as befits a uniform density of cells of
area na2
(J3/2) b2, thus bla (2 nlV3)1/2 1.9.
The first few layers in (1) can be grouped together in
=
=
=
concentric shells with non-decreasing
to three alternatives
numbers nm
of
cells, according
We shall require that the first few shells generated by A
(4.1) contain approximately the same number of cells
as those of the perfect triangular lattice, thus
This selects the first few qm which were arbitrary in
section 4, and the following A
1/(ql + 1/(...1/i)) :
=
Â.o = { 1 } = lit (5. 2c), Â.6 = { 6, 1 }, Â.s = { 5, 1 } (5. 2b)
Â62 = { 6, 2, 1 } and A52 f-5, 2, 1 } (5.2a). Conventional notation, underlying the periodic qi in
or
=
has been used.
The first 4 shells will then have the same radius as the
triangular lattice (1), thus, instead of r(l)
one has r( 1 )
0, r(2-7) 1.9 a, ... Thereafter, the
generative spiral is replaced by circular shells :
shell i contains Bi cells, labelled by li I
li + B,
and, because hexagonal cells are staggered, the shell is
divided into two halves, thus
=
daisy structures, parametrized by : a) Ao
1/i { 1}, b) À,6 2/(11 + 1-5) = {6, 1}. Notice the
manifestself-similarity, and the glides circles. 0 penta-
Fig.
6.
-
Two
=
a, ,/1(4.1),
=
=
=
=
=
gon, +
=
pentagon, 0
+
=
dislocation.
circular grain boundaries, is so apt to dissipate shear
strain that this feature should be conserved in hydrodynamic patterns subjected to Coriolis force. The
azimuthal part of the algorithm 4.1, with ,the proper
selection of A terminating as 1/r to guarantee homogeneity and self-similarity, must therefore be kept.
On the other hand, the radial law r(1 ) should be modified to eliminate spiralling.
It is expected that the generative spiral is responsible
for the spiralling of the parastichies, and, to get rid of it
,
If Bi is odd, we choose Bi ± 1 arbitrarily for the first
shell of the grain, and alternating for successive shells
of the same grain (if there are any). This is the only
arbitrariness in the structures which are parametrized
by one of the Noble A selected above by equation 5.3.
The structure Â.6
{ 6, 11 is illus2/(11
trated in figure 7, and the full radial law is given in
appendix B. The spiralling has disappeared, as was
anticipated. The structure is not dissimilar to the
experimental Benard patterns illustrated in figures 1
=
+ J5) =
58
Benard structures of
figure 7. We expect them to be
topologically stable,
hexagonal cells becoming
more and more regular, as they are experimentally
(compare Fig. 2 to Fig. 7). This relaxation algorithm
is a local constraint forcing hexagons to be regular.
(We are grateful to P. Manneville for his specific
coments on this point.)
with
6.
Melting of the Benard convective structure.
Once the « crystalline )) structure has been constructed,
it is easy to simulate its melting by random fluctua-
Benard structure, constructed by the method
Fig. 7.
described in Section 5 and appendix B using A
{ 6, 1 }.
Change of the radial algorithm (4. 1) has eliminated the
spiralling of the daisy (fig. 6) while retaining self-similarity
and glides circles. 0 pentagon, +
heptagon, 0 + dislocations.
-
=
=
=
=
argued that some experimental
polycrystalline » than our theorepatterns
tical structure which is, by construction, a collective
cellular structure extending over the whole (cylindrical) container. In the absence of a variational
principle sufficiently general and elementary to yield
the cellular pattern, it is not possible to determine
under which conditions one should obtain one single,
collective, cellular structure like that of figure 7, or,
conversely, a random aggregate of « microcrystals ».
Moreover, the geometry of random polycrystalline
mosaics is still in its infancy [3, 25].
There is also the problem of defining a cell. Our
patterns are generated by specifying the centres, or
seeds of the cells, which are then determined by Voronoi construction. This topological snap-shot can be
relaxed rapidly by the following iteration method :
replace the original position of the centre of each cell
by the centre of mass of its Voronoi polygon, repeat
and 2. It may be
are more «
the Voronoi construction with the
new
centres, and
so on :
Centres
(4.1)
or
-
Voronoi -
Centres of mass
(5.4)
The daisy structure of figure 6 is topologically stable
under this relaxation procedure (only the details of
the sequence of dislocations and solitary cells on grain
boundaries vary at a few stages of the iteration procedure. The numbers of dislocations (d) and solitary
cells (s) are invariant). We have not yet relaxed our
Fig. 8.
Melting of the Benard structure of figure 7.
a) Small departure from threshold (low T excitation)
Random azimuthal fluctuations : 15 x 10-2, b) Large
departure from threshold (melting). Random azimuthal
fluctuations : 0.8. The number of non-hexagonal cells has
increased from the crystalline structure (fig. 7).
-
59
tions of the positions of the cells centres. A useful test
of our contention that the structure constructed in
section 5 is the true, crystalline structure
despite
its non-hexagonal cells introduced inevitably by the
boundary conditions is that the number of nonhexagonal cells does not decrease under increasing
fluctuations.
Experimentally, Benard convective structures
become more disordered as the temperature difference
AT between lower and upper surfaces of the fluid is
increased about the threshold for convective instability.
They melt eventually, at a temperature, and according
to a mechanism, which are not yet known with precision [26]. Accurate description of the crystalline
structure is a prerequisite to any investigation of its
melting. In this Section, we simulate disorder and
melting at two temperatures AT above the convective
instability threshold (Fig. 8). The first example corresponds to a low temperature excitation of the crystalline
structure, whereas the second structure is clearly
liquid. A complete analysis will be the subject of a
future publication.
Because the essence of the crystalline structure
(self-similarity, homogeneity) lies in the azimuthal
part of algorithm 4 .1, we chose to restrict random
fluctuations to the azimuth 0(l) of every cell. Let the
root mean square deviation in the position of each cell
be measured by x. Fluctuations are homogeneous, so
that x is independent of the cell’s position. The azimuth
is
-
-
where e = ( - 1)n (m/10) with the integers n and m,
0
10 produced by a random number genen, m
rator. Results are exhibited in figure 8 for : a) x = 10-2,
a low temperature excitation, and b) x
0.8, which is
sufficient to melt the structure. The low T excitation
hardly changes the number of non-hexagonal cells,
but the grain boundaries have been modified. Grains
are no longer perfect annuli, and polycrystallinity
=
begins
to
develop.
The cells should become more regular after the
relaxation procedure, discussed in § 5, is applied.
7. Conclusion.
We have shown 2-dimensional cellular structures with
cylindrical symmetry constitute a genuine crystallography. The requirements of homogeneity and selfsimilarity are overriding, and sufficient to reduce
drastically the number of possible structures. This is
exactly as in conventional crystallography of infinite
structures, where the geometrical requirements of
translational and rotational invariance
the space
cut the number of 3-dimensional structures
groups
to 230. But in the present situation, the requirements
are topological, not metric. We have related these
requirements of homogeneity and self-similarity with
-
-
the numerical concept of code, which is obviously
relevant in biology, but also in physics via the entropy.
The simplest possible code, qm
{... 1 }, yields the
whole 2D crystallography in cylindrical or axial
symmetry; which is parametrized by Noble numbers.
It is therefore no accident that the Fibonacci (or
Lucas) integers are so pervasive in phyllotaxis. (The
open question is how does the plant transfer digital
( { qi }) into geometrical (A) information, i.e. compute
the continued fraction.)
As for any’crystallography, we have been concerned
solely with global structures, while recognizing that
polycrystallinity may occur for containers of very
large aspect ratio. This implies that the centre of
symmetry plays a privileged part, whether biologically
(through growth) or physically (because of the constant
rate of shear due to the Coriolis force). Indeed, it was
the search for a structure with glide circles, suitable
for the dissipation of shear energy, which motivated
our study of botanical structures, where circles of
dislocations are the boundaries between self-similar
grains of closed packed florets. Homegeneity is sufficient to explain the preponderance of Fibonacci
numbers in phyllotaxis, and imply that Noble numbers
parametrize the structure. As a corollary, the structure
is radially self-similar (Fig. 6). False phyllotaxis
(non-Noble numbers, appendix A) is not self-similar.
However, the homogeneity requirement (4.6) is so
strong that intermediate convergents, satisfying (4.6),
impose themselves in the description of the structure
(Fig. 5). These intermediate convergents were first
introduced in this context by Coxeter [15], but on
purely geometrical grounds. The physical requirement
of homogeneity is new. It is also the first time, as far as
we are aware, that self-similarity has been emphasised
in phyllotaxis and in Benard patterns. (In a different
context, recent investigations of the stability of conservative or dissipative dynamical systems have used
Golden or Noble numbers to parametrize perturbations in return (Poincare) maps, and their successive
convergents to carry renormalization [27, 28, 29]. The
pattern of orbits is then self-similar [29]. These references have been brought to our attention during
redaction of this manuscript).
The daisy (Fig. 6) is a direct representation of the
Noble number A, through the elementary algorithm
(4.1). It exhibits in the number of its parastitichies the
series of successive convergents to A which yield a dense
series of self-similar grains.
To be homogeneous, a cellular structure in cylindrical symmetry must be parametrized by an irrational
number A. At a given radius, there is an integer number
of cells, and A is approximated by a rational. Different
radii have different number of cells and correspond to
different approximations for A, in a smooth and automatic fashion, provided by continued fraction representation for A. Number theory has thereby imposed its
relevance to crystallography and hence to physics and
biology. Self-similarity of the structures generated
=
60
(exhibited in Figs. 6 and 7), is a by-product of number
theory.
Homogeneity and self-similarity are sufficient to
limit the number of possible structures and impose a
crystallography. Indeed, the essence of the problem
was summarised by D’A. W. Thompson, «... and not
the least curious feature of the case (phyllotaxis) is
the limited, even the small number of possible arrangements which we observe and recognize » [6, p. 912].
Acknowledgments.
gratefully acknowledged.
GRAINS,
the
new
set.
GRAIN
etc.
BOUNDARIES,
NORMAL
AND
FALSE
-
A
grain is defined by three
B(2)m+ 1
=
is another intermediate
conver-
gent,
The next outer
sets of
parastichies (spirals), that is by the three integers B
denoting the number of parastichies in each set.
Homogeneity within the grain requires that the B’s
satisfy the Fibonacci or triangle relation 4.6.
A grain boundary is a circular shell containing
2 d + s cells, consisting of d dipoles of neighbouring
pentagonal and heptagonal cells forming d dislocations, with s solitary hexagonal cells. It is defined by the
two integers d and s.
Two out of the three sets of parastichies continue
through the grain boundary, from one grain to the
next. The third set terminates on the solitary hexagons
of the boundary, to be replaced on the next grain by
2 d + s new parastichies (of opposite chirality in
normal phyllotaxis). One of the two sets of parastichies
going through to the next grain has d parastichies,
because the hepta- and pentagon, being neighbours,
belong to the same parastichy. Homogeneity relation
4. 6 on the inner grain
on
as
A.
PHYLLOTAXIS.
and
In general, B(1)m+ 1 is only an intermediate convergent
(4.8). Only if qm+l 1, is Bm + 1 Bm + 1, a principal
convergent, and one has normal phyllotaxis, with Â.
terminating as 1 /-r.
The next outer grain has Bm+ 1 and Bm going
through, with
=
We would like to thank all our colleagues who have
directed us towards exotic references or manifestations
of the Golden section, especially R. Blanc, J.-P. Clerc,
G. Giraud, A. L. Mackay, R. Mosseri, R. Santini,
H. E. N. Stone, J. Yeomans and E. Guazzelli (photograph of a nematic convective structure). Extensive
comments on the manuscript by P. Manneville are also
Appendix
implies that the other set going through the boundary
has d + s parastichies.
Given an arbitrary irrational A, and its continuous
fraction expansion (4.2, 3, 4), one starts with a grain
containing at least two successive principal convergents, B. and Bm-l’ say. Homogeneity (4. 6) within the
grain requires that the third set of parastichies is
grain has Bm2+ 1
and
B. going through,
Conversely, the next inner grain has Bm and Bm -1
going through the grain boundary, with B (M’7- - 1) as
the new set of parastichies, because homogeneity (4.6)
requires
is
an
intermediate convergent
Bm(qm -1) = Bm-2 a principal
1 is
if qm
the
and
convergent,
(only
=
phyllotaxis is normal).
It is only after qm + 1 grains with only one principal
convergent (eqs. 3 and 4), that one recovers a grain
with two principal convergents (Bm and Bm + 1) similar
to that with which one had started. The set of Bm
parastichies continues through qm+1 + 2 grains,
and thus explains the strongly spiralling appearance of
false phyllotaxis (Fig. 5), where these parastichies are
very conspicuous. On the "other hand, parastichies
labelled by intermediate convergents are only carried
through 2 grains,
the outer grain
The structure loses its manifest
self-similarity
if
q. :0 1 (false phyllotaxis). Similarity is only recovered
after qm
grains. The situation is summarized in table I
(normal phyllotaxis) and table II (false phyllotaxis).
These are the only configurations compatible with
homogeneity (4. 6) within each grain, and continuation
of 2 (out of 3) sets of parastichies from one grain to
Fig.
9.
-
Positions and labels of cells
on a
grain boundary.
the next.
61
Appendix B.
CONSTRUCTION OF A, BTNARD CELLULAR STRUCTURE
We start with the daisy algorithm (4.1),
(Fig. 7).
{6,1},
parametrized by the angle Â.6 2/(11
and modify the radial law, r(l) replacing the generative
spiral by concentric circular shells, to eliminate the
spiralling, conspicuous in the daisy but absent in
the Benard structure.
As explained in section 5, the radial law r(1 ) follows
the perfect triangular lattice for the first 4 shells,
and continues with concentric parts of staggered
half-shells, with ’ B. cells each (or t(Bm ± 1) if B.
is odd). We list below the numbers of parastichies
{ Bm }, the labels of the first cell in every shell { 1m },
lm Y Bi + 1, and the radial law r(1 ) for the B6nard
-
(U
08 80
C
S xt
c
i
a
^
0
ts
f
c
im
&#x26;
i!!
parametrized by A6 6, 1 }. The only
arbitrariness is in the choice of t(Bm 1: 1) cells per
half-shell if Bm is odd, indicated in parenthesis beside
Bm. We also list the label I’ of the last cell of a given
grain in the Daisy [22].
structure
0 04
U
c
nc
ii
e
"
CIS CIS
0
4-
a
C
e
0 ar
S
U
00
0
r
n
0
0
.H
i
sp
ain
.
N
r
nn
c
y
M
+1
+)5) =
=
+
e
ar
=
62
References
A. L., Sov. Phys. Crystallogr. 26 (1981)
517-22.
PENROSE, R., Bull. Inst. Math. Appl. 10 (1974) 266;
MOSSERI, R. and SADOC, J. F., in Structure of NonCrystalline Materials II, P. H. Gaskell, E. A. Davis
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[2]
[3]
L-143 ;
RIVIER, N., in Structure of Non-Crystalline Materials II,
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of Visual Illusions, Routledge and Kegan Paul
(1982), 2.2 and 3.2.
[18] DAVENPORT, H., The Higher Arithmetic (Hutchinson)
1952, ch. 4.
[19] ITARD, J., Arithmetique et Théorie des Nombres (PUF,
Que sais-je) 1963.
and
[20] Am
Bm are relatively prime, because Am Bm-1 -
Bm Am-1 = (-1)m-1.
(Taylor and Francis), 1983, p. 517-40.
[4] PANTALONI, J., CERISIER, P., BAILLEUX, R. and GERBAUD, C., J. Physique-Lett. 42 (1981) L-147 ;
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[5] BRAY, R. and LOUGHEAD, R. E., The Solar Granulation
(Chapman and Hall), 1967.
[6] D’ARCY W. THOMPSON, On Growth and Form (Cambridge UP, 2nd edition), 1952, ch 14.
[7] JEAN, R., Phytomathématique (Presses Univ. du Québec)
(which
is
Am/B - Am-1/Bm-1= (-1)m-1/Bm -1
has sign alternating with the parity of m, and
magnitude decreasing as m increases, since the
numbers Bm increase with m. Eventually
$$ (Am/Bm),
03BB
so
that
| 03BB - Am/Bm|
=
1/Bm Bm+1. The intermediate convergents A(03B1)m,
B(03B1)m, defined by B(03B1)m 03B1Bm-1 + Bm- 2, 0 03B1 qm,
also satisfy A(03B1)m Bm -1 - B(03B1)m-1
( - 1)m-1,
But then, A(03B1)m/B(03B1)m - 03BB, has the same sign as
AmlBm - 03BB with | 03BB - A(03B1)m/B(03B1)m| 1/B(03B1)m Bm+1.
=
1978.
=
[8] CERISIER, P. and PANTALONI, J., IVth International
Physico-Chemical Hydrodynamics Conference,
N.Y., 1982, to appear in the congress Proceedings.
PANTALONI, J., BAILLEUX, R., SALAN, J., VELARDE,
M. G., J. Non. Equilib. Thermodyn. 4 (1979)
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[9] DREYFUS, J. M. and GUYON, E., J. Physique 42 (1980)
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GUAZZELLI, E. and GUYON, E., C. R. Hebd. Séan.
Acad. Sci. 292 II (1981) 141.
GUAZZELLI, E., Ordre et Désordre de structures convectives dans
une
Instabilité convective d’un Néma-
tique, Thèse, Orsay 1981.
[10] HOWELL, P. R., KILVINGTON, I. T., WILLOUGHBY,
A. and RALPH, B., J. Mat. Sci. 9 (1974) 1823.
FEYNMAN, R. P., LEIGHTON, R. B. and SANDS, M.,
The Feynman Lectures in Physics (Addison-Wesley)
1963, II-30.
[11] PANTALONI, J. and CERISIER, P., Les Embiez Conference
This relation
proved easily by induction, using (4.4)), implies
that successive convergents are alternatively
smaller and larger than 03BB, namely
(1980).
[12] DUFFY, D. M. and RIVIER, N., J. Physique Colloq.
43 (1982) C9-475.
[13] WEYL, H., Symmetry (Princeton UP) 1952, figs.
45
and 55.
[14] DORMER, K. J., Fundamental Tissue Geometry for
Biologists (Cambridge UP) (1980).
[15] « ... qualitative hypotheses which are both biologically
plausible and mathematically interesting ». COXETER, H. S. M., J. Algebra 20 (1972) 167-75.
[16] COXETER, H. S. M., Scripta Math. 19 (1953) 135-43.
[17] Conversely, concentric black and white circles on a
chequered background of spirals, look themselves
like a spiral. This optical illusion is attributed to
FRASER, J. (Brit. J. Psychol. 2 (1908) 307-20),
and is illustrated in COOK, T. A., The Curves of
Life, Constable, London (1914), Dover (1979),
[21] See the Classic : COOK, T. A., The Curves of Life [17]
(especially Plate VIII).
PEDOE, D., Geometry and the Liberal Arts, Penguin
(1976). Typical construction can be found in :
FUNCK-HELLET, C., Composition and Nombre d’Or dans
les 0152uvres Peintes de la Renaissance (Vincent
et Cie, Paris) 1950, (in a series under the mildly
inflationnary motto « le Très Noble et Très Droit
Réseau Fondamental »). Whether such constructions were actually used by artists is debatable,
but A. Dürer is credited with an approximate
construction of the Golden section using one single
opening of the compass (« rusty compass »)
(CLEYET-MICHAUD, M., Le Nombre d’Or (PUF,
Que Sais-Je) p. 46).
[22]
The
boundary between grain m : (Bm, Bm-1, Bm-2 )
and grain m + 1 : (Bm+1, Bm, Bm-1 ) is given by
the label lcm of the first isolated hexagonal cell on
the grain boundary. lcm, the last cell of the Bm-2
parastichy, is the smallestl such that the distance
on the old parastichy Bm- 2, d (l + Bm- 2, l) exceeds
that on the new parastichy Bm+1, d(l + Bm,
l - Bm-1), thus
d(l+Bm-2,l) ~ d(l+Bm,l-Bm-1)
(Fig. 9). Equality
lcm = 1/2 {
-
holds for
Bm-2 + [(B2m+103B32m+1 - B2m-2 03B32m-2)/
(03B32m+1 - 03B32m-2)]½}
cos (203C0 ||03BBBm - Am |). This relation
where 03B3m
is obtained by using trigonometry and the recursion
relation 4.4. For 03BB6
{ 6,1 }, one obtains, by
rounding lcm to the next higher integer, { lcm }
=
=
=
(8,15), 44, 113, 302, 797, 2102, corresponding
to
63
(1), 33 (2), 53 (4), 86 (6), 139 (9). This
exactly where transition between successive
grains occur in the large daisy of figure 6, as
confirmed by tedious enumeration. Approximate
knowledge of the critical numbers { lcm } is essen-
Bm
=
20
is
[23]
tial to construct the convective structure of section 5, i.e. to ascertain the number of circular
shells containing the same number of cells and
belonging to the same grain.
It is possible to encode any integer qm in the continuous
fraction (4.3) by using binary digits 0 and 1.
Indeed, a sequence of three consecutive q’s, the
middle one being 0, qi
a, qi+ = 0, qi+2 = b,
is identical to the single q
a + b in (4.3). Thus
2
(1, 0, 1), 3 = (1, 0, 1, 0, 1), etc. The binary
representation 03BB =1/(q1 + 1/q2 + + 1/qm + ···)),
qi 0, 1, is unique if successive q 0 are forbidden (since (1, 0, 0)
1).
=
=
=
···
=
=
=
[24] PERCIVAL, I. C., Physica D 6D (1982) 67.
[25] WEAIRE, D., in Topological Disorder in Condensed
Matter, T. Ninomiya and F. Yonezawa, eds
(Springer) 1983, p. 51.
[26] OCCELLI, R., GUAZZELLI, E., PANTALONI, J., IVth
International Physico-Chemical Hydrodynamics
Conference, N.Y. 1982, to appear in the Proceedings.
[27] GREENE, J. M., J. Math. Phys. 20 (1979) 1183.
ARNOL’D, V., Chapitres Supplémentaires de la Théorie
des Equations Différentielles Ordinaires (Mir, Moscov) 1980, §§ 11-12.
[28] SHENKER, S. J., Physica D 5 (1982) 405.
[29] MACKAY, R. J., Physica D (1983) to appear.
[30] POMEAU, Y. and MANNEVILLE, P., J. Physique 42
(1981) 1067.
CROQUETTE, V., MORY, M. and SCHOSSELER, F., J.
Physique 44 (1983) 293.