Principle of Minimax and Rise Phyllotaxis
(Mechanistic Phyllotaxis Model)
Dmitriy Weise
Keywords: Botany, Mathematics, Phyllotaxis,
Morphogenesis, Fibonacci numbers.
1.1. Introduction
Pattern formation in organisms is one of the most common
phenomena observed in nature. The arrangement of repeated units, such as
leaves around a stem, florets in the head of a daisy, scales on a pine
cone or on a pineapple, and seeds in a sunflower is known as phyllotaxis.
These repeated units are called in their young stages primordia
[6].
1.2. History
The study of phyllotaxis is traced from the first primitive
observations in ancient times to sophisticated studies of today
[1].
Little is known about the Ancient Period which goes back
at least to Theophrastus (370 B.C.-285 B.C.) and Pliny (23 A.D.-79
A.D.). Theophrastus, in his Inquiry into Plants, says about plants
that "those that have flat leaves, have them in a regular series". Pliny,
in his Natural History, gives more details. In his description of
oparine he says that it is a ramose, hairy plant with five or six leaves
at regular intervals, arranged circularly around the branches.
The Modern Period (from the fifteenth century to 1970)
is marked by the observations of Leonardo da Vinci (1452-1519), J. Kepler
(1571-1630), by the works of C. Bonnet (1754), C.F. Schimper (1830), A. Braun (1831,
1835), the Bravais brothers (1837), M.T. Lestiboudois (1848), W. Hofmeister (1868),
S. Schwendener (1878), A.H. Church (1904), D'Arcy W. Thompson (1917), M. Snow
and R. Snow (1962).
In the Contemporary Period (from 1970 onwards), there
were studies of H.S.M. Coxeter (1972), I. Adler (1974), L.V. Beloussoff
(1976), G.W. Ryan, J.L. Rouse and L.A. Bursill, S. Douady and Y. Couder (1992),
H. Meinhardt (1984), R.V. Jean, where in the lists are mentioned only the
most remarkable scientists.
1.3. Patterns
Regardless of the overwhelming multiformity of plants structure,
there are common patterns that link a wide range of species. There are
two large categories of patterns that could be recognized: the
whorled and the spiral patterns.
1.3.1. Whorled pattern
In a number of common species, the leaves are arranged
in whorls at the level of the stem (Fig. 1). The number n of leaves in a whorl varies from species to
species, in the same species, and can even vary in the same specimen (
Fig. 2). In the whorled patterns, the leaves at any node are generally
inserted above the gaps of the preceding ones (
Fig. 3).
1.3.2. Spiral pattern
The most common pattern, the spiral pattern, involves
an insertion of a single primordium (Fig.
4, Fig. 5, Fig.
6) at each node. In this case it is possible
to trace the spirals (Fig. 7)
which in the botanical literature are called parastichies.
The primordia in parastichy could be or not in contact
(Fig. 8). Those segments of parastichies
which are visible, thanks to the contacts, are termed contact parastichies.
The parastichies running in the same direction with respect
to the axis of the plant with the same pitch constitute a family of
parastichies, and two obvious families winding in opposite directions
are called a parastichy pair.
1.4. Direction and numbers as characteristics of a parastichies
family
We can characterize the patterns according two criteria:
by determining the direction of a parastichies winding, and by counting
the number of the parastichies in the family.
1.4.1. Spirals winding direction
Spirals winding doesn’t have a strictly determined direction.
The shoots of plants can be both left-handed and right-handed enantiomorphs
(Fig. 9, Fig.
10, Fig. 11).
1.4.2.1. Fibonacci numbers
A parastichy pair formed by a family of m spirals
in one direction and n spirals in the opposite direction is denoted
(m, n).
The numbers m and n in the parastichy pair
on pineapples, cones, and sunflowers are consecutive Fibonacci numbers.
The first few Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, ... , where the
sum of two consecutive numbers is the next number.
1.4.2.2. Fibonacci-type sequences
When visible opposed parastichy pairs do not contain numbers
that are consecutive values in the Fibonacci sequence, they often contain
those from another sequence, derived in the same manner as the Fibonacci
sequence, but with other initial terms, for instance 1 and 3 (the Lucas
sequence) (Fig. 12).
1.4.2.3. Rising phyllotaxis
The numbers of visible opposed parastichies rise along
the Fibonacci sequence. In the inner part we have few visible opposed parastichies,
and in the outer part more visible opposed parastichies (
Fig. 13). With an increase of the shoot radius, a change of parastichy
pair takes place, for example, from (8, 13) to (21, 13), from (21,
13) to (21, 34) (Fig. 14) etc.
The explanation of this phenomenon, called rising phyllotaxis,
is the key for the understanding the origin of patterns in plants.
The prevalence of the Fibonacci sequence in phyllotactic
patterns is often referred to as "the mystery of phyllotaxis", and "the
bugbear of botanists".
2. Model
Model is based on suitable analogy primordia with soap bubbles. The similar
assumption has been made by a few authors, for instance, by Van der Linden.
Soap bubble motions are agreed with mechanical laws (3, 4, 11).
But it is only an analogy.
2.1. Principle of minimax
The spherical soap-bubble-like primordia arise from the liquid
in the center of the cylinder top (Fig. 15),
one by one, according to the rule: every primordium moves in
the largest available space. (This rule reminds of Hofmeister’s rule
(1868) [5]: every primordium arises
in the largest available space.)
The primordia move radially and simultaneously, with the equal rate, and
grow in diameter until they experience contact pressure. By
the way, the paths of horizontal motion of primordia are not rectilinear.
Increasing amounts of contact parastichy pairs realize
due to the rearrangement of primordia, during their movement from center
towards the rim. Parastichy becomes contact and visible to the naked eye
when primordia touch one other.
2.2. Mathematical description
The model is formulated in centric representation,
where each family of parastichies is a set of identical Archimedean
spirals [9].
We have centric vector spiral lattice (
Fig. 16). The primordia stand in the nodes of this lattice. They are
numbered according to their age, that is according to the order in which
they arise on the plant apex, with 0 being the youngest (
Fig. 17).
The numbering of primordia is in agreement with the
Bravais-Bravais theorem (1837) [2]:
in a family containing n parastichies,
on any parastichy, the numbers of each consecutive primordia differ by
n. Numbers on each n-parastichy are congruent mod
n, it means, belong to the same residue class mod n. Each parastichy
is considered as a residue class. Difference of numbers of any primordia
is considered, first, as a lattice vector, and, second, as a module
in its residue class.
We can add and subtract integer vectors in agreement with
the parallelogram rule (Fig. 18, Fig.
19, Fig. 20, Fig.
21, Fig. 22).
The origin and replenishment of contact parastichy is
described by addition of vectors at the moment of touch of two (younger
and older) primordia, moving in the opposite corners of the primitive unit
cell (Fig. 23, Fig.
24).
The appearance of new vectors, or moduli, causes the appearance
of new residue classes mod m. It is well
known that there are exactly m distinct residue classes mod m,
consequently, after addition of vectors m and n, (m+n)
residue classes mod (m+n) will appear, this means (m+n)
contact parastichies. A contact parastichy pair (m, n) is replaced
by contact parastichy pair (n, m+n). Thereby, Fibonacci sequence
arises. As it was proved, the rising of spiral phyllotaxis
is isomorphic to increasing of Fibonacci sequence.
Separation and divergence of primordia result in the disappearance
of contact parastichy. In each contact parastichy, addition occurs among
the younger primordia nearest to the centre, but subtraction occurs among
the older primordia farthest from the centre. The primordia move from the
centre to the rim, but location of areas of contact parastichy pairs is
not changed.
The appearance of Fibonacci-type sequences is explained
by misleading of primordia at initial stages of apex development.
2.3. Ornament on the lateral shoot surface
Appearance of the ornament on the lateral cylindrical
surface of shoot is explained by lengthening of internodes along the shoot
axis, after primordia had been arranged on discoid, coniform or domed apex
of shoot. The sliding of the bubbles on the cylinder wall simulates this
lengthening of internodes (Fig.25).
2.4. Explanation of chirality
How do the right and left forms appear? Let us examine the
initial primordia placement (Fig. 26,
Fig. 27). The third primordium can appear
both on one, and on the other side from the reflection line which has the
first two primordia centers lying on it. Solution of this dilemma will
determine hereinafter for each family of spirals its direction. A choice
of position for the third primordium is probably casual, and depends on
external factors, for example, on uneven heating of bud by the sun.
2.5. Versatility of model
The model allows to describe a whorled phyllotaxis, as well
as the spiral one. If the cylinder is sufficiently narrow, and if the primordia
appear as a complexes composed of two, three or more bubbles, we have a
whorled pattern (Fig. 28, Fig.
29, Fig. 30, Fig.
31).
3. Resume
The work shows a possibility of complicated accommodation
on the basis of simple principle of minimax: “It pops up in the biggest
gap” [10].
REFERENCES:
Mail: Weise D. L., Russia, Moscow, 121353, Belovejskaya, 39-2-133.
E-mail: dlweise@dlweise.msk.ru
Received: 29.10.1998
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