#  Vascular phyllotaxis transition and an evolutionary mechanism of
phyllotaxis

Takuya Okabe  Faculty of Engineering, Shizuoka University, 3-5-1 Johoku,
Hamamatsu 432-8561,Japan<EMAIL_ADDRESS>
###### Abstract

Leaves of vascular plants are arranged regularly around stems, a phenomenon
known as phyllotaxis. A constant angle between two successive leaves is called
divergence angle. On the one side, the divergence angle $\alpha_{0}$ of an
initial pattern of leaf primordia at a shoot apex is most commonly an
irrational number of about 137.5 degrees, called limit divergence. On the
other side, the divergence $\alpha$ of a final pattern of leaf traces in the
vascular system of a mature stem is expressed in terms of a sequence of
rational numbers, $\frac{1}{2}$, $\frac{1}{3}$, $\frac{2}{5}$, $\frac{3}{8}$,
$\frac{5}{13}$, $\frac{8}{21}$, called phyllotactic fractions. The
mathematical relationship between the initial divergence $\alpha_{0}$, the
final divergence $\alpha$, and the number of internodes traversed by the leaf
traces $n_{c}$ is investigated by means of a theoretical model of vascular
phyllotaxis. It is shown that continuous changes of the trace length $n_{c}$
induce transitions between the fractional orders in the vascular structure.
The vascular phyllotaxis transition suggests an evolutionary mechanism for the
phenomenon of phyllotaxis. To provide supporting evidence for the model and
mechanism, available experimental results for fossil remains of Lepidodendron
and the vascular structure of Linum and Populus are analyzed with the model.

###### keywords:

phyllotaxy; Fibonacci numbers; golden ratio; natural selection; Linum
usitatissimum; Populus deltoides

##  1 Introduction

###  1.1 Review, background, and motivation

Astonishing regularity manifested in plant architecture has fascinated various
fields of scientists for centuries. The regular arrangement of leaves, flowers
and floral organs of higher plants is called phyllotaxis. A constant angle of
rotation between two successive organs is called divergence angle, on which
two apparently irreconcilable concepts have been in general use since the
inception of quantitative investigations on phyllotaxis.

Figure 1:  (a) A typical pattern of leaf primordia (points) on a shoot apex
with the initial divergence of $\alpha_{0}=1/(1+\tau)$, or
$360\alpha_{0}\simeq 137.5$ in degrees. The irrational number $\tau\simeq
1.618$ is the golden ratio defined by the proportion equation
$1:\tau=\tau-1:1$. The primordia are numbered in the reverse order of
production. A solid spiral connecting all the primordia in the numerical order
is the genetic spiral. Three dashed spirals (clockwise inward) and five dotted
spirals (counterclockwise inward) are 3 and 5 parastichies, respectively. This
pattern has a parastichy pair $(3,5)$. (b) A typical pattern of leaves on a
mature stem characterized with a divergence fraction of $\alpha=\frac{2}{5}$
($360\alpha=144^{\circ}$). Oblique strands diverging to leaves 1 and 6 are
leaf traces. A solid spiral surrounding the stem is the genetic spiral.

Braun (1831, 1835) and Schimper (1835) noticed that divergence angle is
various but not arbitrary. It is a fraction, or a rational number, a number
that can be expressed as the quotient $\frac{n}{m}$ of two integers $n$ and
$m$. The most widespread is the helical phyllotaxis, also called spiral or
alternate phyllotaxis, in which stems bear a leaf per node. In the helical
phyllotaxis, the numerator $n$ and denominator $m$ of the fraction normally
are two alternate terms of a Fibonacci sequence, 1, 2, 3, 5, 8, 13, 21, 34,
55, 89, $\cdots$. It is generated by the Fibonacci recurrence relation that
each number after the first two terms is the sum of the previous two numbers.
The phyllotactic fractions $\frac{1}{2}$, $\frac{1}{3}$, $\frac{2}{5}$,
$\frac{3}{8}$, $\frac{5}{13}$, $\frac{8}{21}$, $\frac{13}{34}$, $\cdots$
comprise what is called the main sequence of phyllotaxis. A $\frac{2}{5}$
phyllotaxis is schematically shown in Fig. 1. In multijugate, verticillate or
whorled phyllotaxis, where more than two leaves are borne at each node, the
divergence angle is divided by the number of leaves in a whorl. In general, a
plant stem is partitioned into nodes and internodes. A node is a point at
which a leaf or leaves are attached, and an internode is a section of the stem
between two successive nodes. In the fractional phyllotaxis, there are leaves
aligned vertically above each other along a stem, as represented by leaves 1
and 6 in Fig. 1. A straight line connecting the superposed leaves is called an
orthostichy. In the helical phyllotaxis, the denominator of the phyllotactic
fraction is equal to the number of orthostichies. It is also the number of
internodes between two adjacent leaves on an orthostichy. Thus, the
$\frac{2}{5}$ phyllotaxis in Fig. 1 has five orthostichies, 1-6, 2-7, 3-8, 4-9
and 5-10, and five internodes separate leaves on each orthostichy. An
imaginary spiral connecting all the leaves in the order of production is
called the genetic, fundamental, generative, or ontogenetic spiral. The
numerator of the fraction refers to the number of turns of the genetic spiral
between the two adjacent leaves on an orthostichy. In Fig. 1, a solid spiral
is the genetic spiral. From the leaf 6 to 1, the genetic spiral winds around
the stem twice, the number two being the numerator of $\frac{2}{5}$. As
remarked below, the phyllotactic fraction does not lose its significance even
though vertical alignment is actually not exact but approximate.

In contrast, Bravais and Bravais (1837) suggested that divergence angle is
uniquely and invariably given by an irrational number, that is, a number which
cannot be expressed as a fraction. The most typical angle of $360/(1+\tau)$
degrees is called the golden angle, where the irrational number $\tau$, called
the golden ratio, golden mean, golden section, or extreme and mean ratio, is
defined by the proportional relation $1:\tau=\tau-1:1$. As the positive
solution of the quadratic equation $\tau(\tau-1)=1$, it is given by

$\tau=\frac{\sqrt{5}+1}{2}\simeq 1.61803399\cdots.$ (1)

The defining equation is transformed to $\tau^{-1}=1/(1+\tau^{-1})$. Recursive
substitution of $\tau^{-1}$ in the left-hand side to the right-hand side gives
an infinite continued fraction representation,

$\tau^{-1}=\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{1+\ddots}}},\qquad\tau=1+\tau^{-1}.$

The golden angle $360/(1+\tau)=360/\tau^{2}$ is approximately $137.50776$
degrees. By definition, the golden angle is the smaller angle created by
sectioning the circumference of a circle (360 degrees) according to the golden
ratio $1:\tau$, the golden section. A phyllotactic pattern with divergence
equal to the golden angle is shown in Fig. 1. The ratio of the angle subtended
by 1 and 2 to the angle between 1 and 3 is $\tau$, or $\angle 1O2:\angle
1O3=1:\tau-1=\tau:1$, where $O$ is the origin. Similarly, $\angle 1O4:\angle
2O4=\angle 1O9:\angle 4O9=1:\tau$, and so on. Thus, phyllotactic patterns with
constant divergence equal to the golden angle have harmonious proportions.
Patterns with an irrational divergence angle have no orthostichy in a strict
sense, as no two leaves align vertically or radially. Instead, therefore,
attention is directed to secondary spirals connecting positionally nearby
leaves, called parastichies. Like an orthostichy, a parastichy is
characterized by a difference in number of leaves on it. In Fig. 1, the
genetic spiral, three parastichies and five parastichies are drawn with a
solid curve, dashed curves and dotted curves, respectively. Each of three
parastichies 1-4-7-10, 2-5-8 and 3-6-9, is called a 3-parastichy. Hence there
are three 3-parastichies and five 5-parastichies in Fig. 1, and the pattern in
Fig. 1 is said to have a parastichy pair of $(3,5)$, which is also denoted as
$(3+5)$ or $3:5$. As a remarkable fact, parastichy numbers are almost always
given by Fibonacci numbers. This is a mathematical consequence of the fact
that divergence angle is almost always the special irrational number, the
golden angle. The golden angle is also called the Fibonacci angle, for it is
the limit angle of divergence for the phyllotactic fractions belonging to the
main sequence;

$\begin{array}[]{c}\textstyle 360\times\frac{2}{5}=144,\\\
360\times\frac{3}{8}=135,\\\ 360\times\frac{5}{13}\simeq 138.46,\\\
360\times\frac{8}{21}\simeq 137.14.\end{array}$

The rational angles beyond $\frac{5}{13}$ are practically indistinguishable
from the ‘ideal’ irrational angle of 137.507764$\cdots$ degrees. Therefore, it
is argued that what appear to be different rational angles are nothing but a
single irrational angle disturbed by inevitable random errors.

The seemingly conflicting views on the divergence angle, whether rational
numbers or an irrational number, are a source of inspiration and confusion. In
effect, they are not only compatible but both indispensable. On the one hand,
the irrational number applies to the divergence angle of phyllotactic patterns
of undifferentiated tissues at shoot tips or apical meristems (Church (1904);
Hirmer (1922, 1931)). Let us call it the initial divergence angle. It is
commonly referred to as the ideal or limit divergence angle for the reason
mentioned above. On the other hand, the rational (fractional) divergence
applies to phyllotaxis of leaves, or primary vascular architecture on a mature
stem (Lestiboudois (1848); Nägeli (1858)). In the literature, the majority of
studies discuss the former, i.e., the process of organ initiation, positioning
of the leaf primordia from which leaves will develop, and transitions of
patterns at the shoot apical meristem. In recent years, substantial progress
has been made in understanding plant hormonal factors that influence or
control the formation of leaf primordia and their arrangement on the apical
meristem (Reinhardt (2005); Kuhlemeier (2007)). In striking contrast, the
fractional phyllotaxis of the mature stem have received less scholarly
attention, unfortunately. This is not because the latter is less important
than the former. As a matter of fact, experimental findings on the close
relationship between phyllotactic fraction and vascular organization have been
accumulated without being theorized from a general perspective (Sterling
(1945); Girolami (1953); Jensen (1968); Namboodiri and Beck (1968); Larson
(1977); Beck et al. (1982); Kirchoff (1984)).

Figure 2:  Schwendener’s causal model. Right: Contiguous circles with
decreasing radius are stacked on an unrolled surface of a stem cylinder.
Contact parastichy numbers, or differences in the numbers of the circles in
contact, change from (1,1) at the bottom to (5,8) at the top. Left: A
mathematical relation between the divergence angle (the horizontal axis) and
the radius of the contiguous circles (the vertical axis) is indicated with a
solid zigzag curve starting from the top left corner (divergence of 180∘,
corresponding to the bottom part of the right figure) down to the golden angle
137.5∘ (the top of the right figure). The zigzagging is due to shifts in the
contact parastichy numbers from $(1,1)$ through $(1,2)$, $(2,3)$, $(3,5)$,
$(5,8)$, $(8,13)$, $(13,21)$ to $(21,34)$. The top branch for $(1,2)$ extends
from 180∘ to $128^{\circ}34^{\prime}$. Adapted from Schwendener (1883).

Since the influential text by Hofmeister (1868), research into causal or
dynamical mechanisms of primordia initiation has been the central pillar in
the study of phyllotaxis. The empirical observation that new leaf primordia
arise in the largest space between the older primordia is called Hofmeister’s
rule. What was originally a rule of thumb of botanists has been refined and
developed into causal or dynamical models. Airy (1873) speculated on a causal
mechanism in terms of geometrical objects in mechanical action. Schwendener
(1878) put a similar idea on a more solid mathematical basis by regarding
leaves on a stem as solid disks contiguously covering a cylinder surface of
infinite length (Fig. 2). In Schwendener’s model, contiguous circles of a
constant radius are arranged in a periodic pattern characterized with a given
set of contact parastichy numbers. Then it is a purely geometrical problem to
derive various mathematical relations for the divergence angle, the radius of
the circles, the girth of the cylinder, and the set of parastichy numbers. The
radius of the circles is regarded as an independent variable, or a control
parameter of the model. By letting the radius change continuously along the
stem cylinder, the divergence angle varies concomitantly with the contact
parastichy numbers according to the mathematical relations. As a remarkable
result, the divergence angle converges toward the golden angle $137.5^{\circ}$
by decreasing the radius sufficiently slowly from a large initial value to a
small constant value. Before attaining to the golden angle, the model predicts
that the divergence angle oscillates with decreasing amplitude (Fig. 2). The
decrease in the radius corresponds to decrease in relative size of leaf
primordia on the stem or apex. This is a brief summary of Schwendener’s causal
mechanism for the golden angle. The model is referred to as a mechanical or
causal model of phyllotaxis.

Related causal models were discussed in depth by Delpino (1883) and van
Iterson (1907). In recent decades, models of Schwendener and van Iterson have
been elaborated on and developed further mathematically (Adler (1974); Rothen
and Koch (1989); Levitov (1991); Kunz (2001); Atela et al. (2002)) or
numerically (Williams and Brittain (1984); Hellwig et al. (2006)), and even
realized dynamically in a physics laboratory experiment (Douady and Couder
(1996)). The causal models are founded on the basic assumption of causal
determinism that a phyllotactic pattern is a result of causal interaction of
pattern units. In particular, the position of an initiated leaf primordium is
determined by the position of the older primordia according to (supposedly
simple) causal rules. The manner in which the units are arranged depends on
the dynamic history of growth, or particularly on the course of changes in
size of leaf primordia. Thus, a common key factor of the causal models is a
gradual change in size of leaf primordia under mutual repulsion. Accordingly,
there is a variety of causal models in which the repulsive interaction is
ascribed not to the mechanical contact pressure as supposed by Airy and
Schwendener, but to a chemical diffusion process (Schoute (1913); Thornley
(1975); Mitchison (1977); Veen and Lindenmayer (1977); Young (1978); Marzec
and Kappraff (1983); Schwabe and Clewer (1984); Chapman and Perry (1987);
Roberts (1987); Steeves and Sussex (1989); Yotsumoto (1993); Koch and
Meinhardt (1994); Meinhardt et al. (1998)). There are another causal models
based on physical (Hofmeister (1868); Green et al. (1996); Newell et al.
(2008)) and chemical (Cummings and Strickland (1998)) instabilities. Recently,
more intricate models based on molecular-genetic experiments have been
discussed (Smith et al. (2006a, b); Jönsson et al. (2006); Shipman et al.
(2011)), while geometrical models have been used to interpret patterns of real
systems (Malygin (2006); Hotton et al. (2006); Zagórska-Marek and Szpak
(2008)). All these causal models are based on the assumption that divergence
angle is intrinsically variable and determined causally.

In the recent literature, we have had few opportunities of finding
phyllotactic fractions in use. Most theoretical and experimental works attach
importance to parastichy numbers instead, and the phyllotactic fraction is not
mentioned or regarded merely as an approximation even if mentioned (Williams
(1974); Steeves and Sussex (1989); Lyndon (1990); Jean (1994)). The fractional
phyllotaxis is the original problem. There are some reasons for this trend.
First, early researchers did not appreciate the structural significance of the
phyllotactic fraction (Hofmeister (1868); de Candolle (1881); Church (1920);
Hirmer (1922); Richards (1951); Snow (1955)). Second, the studies of vascular
structure organization are comparatively so few in number that they are
overshadowed by intensive research interests directed towards the shoot apical
meristems. Third, Schwendener’s causal model and its descendants are at
variance with the fractional divergence. According to the model, the
divergence angle varies depending on size of leaves, the vertical coordinate
of Fig. 2. Schwendener (1883) guessed that a fractional pattern would be made
secondarily as a result of mechanical straightening of parastichous bundles
connecting initiated leaves. Teitz (1888) confirmed indeed that the fractional
phyllotaxis is accomplished by secondary torsion occurring in the vascular
system during growth of the stem. When there is little or no secondary
distortion for lack of subsequent growth or internodal elongation, the
original pattern at the apex may grow to a similar pattern of mature organs.
This holds true for the most eye-catching patterns of closely packed
reproductive organs, which, therefore, are often compared favorably with
numerically simulated outputs of causal models. Even then, the basic concept
of the phyllotactic fraction may remain significant internally in vascular
connections (Watson and Casper (1984)). In fact, a stem with short internodes
takes a high-order fraction, which can be indistinguishable from the limit
divergence of the undistorted stem. Thus, the phyllotactic fraction may not be
judged by the external appearance alone.

In vascular plants, each leaf is connected to the main stem vascular system
through a strand of fluid-carrying vascular tissue called a leaf trace. A leaf
may have several to many leaf traces. Leaf traces diverge from the stem
vascular system some distance below or very near the nodes at which they enter
the leaves (Lestiboudois (1848); Nägeli (1858); Beck (2010)). At the level of
the shoot apex, leaf traces form parastichous strands winding obliquely round
the stem axis. As the stem elongates, the leaf traces align up along the stem
to make orthostichous bundles by forcing the whole stem to twist slightly from
the original pattern (Fig. 1), thereby a phyllotactic pattern characterized by
a phyllotactic fraction is established. In the final pattern, there is a
definite relationship between the denominator of the phyllotactic fraction and
the number of vascular orthostichies (Kirchoff (1984)). The term orthostichy
might be misleading, because the straightened bundles still may maintain their
tilted course. The fraction neatly represents geometrical arrangement of leaf
traces, and the fractional order need not mean that leaves are positioned
exactly vertically. Developmental sequences in differentiation and
vascularization of leaf primordia are numerically correlated with the
phyllotactic fraction of the shoot (Priestley and Scott (1936); Girolami
(1953); Esau (1965)). Seemingly irregular rhythmical variations in various
lengths of the external structure of a mature plant may be understood as a
consequence of a hidden phyllotactic order in the vascular system (Unruh
(1950); Kumazawa and Kumazawa (1971)). There is evidence for restricted
pathways of translocation of photosynthetic assimilates related to phyllotaxis
(Watson and Casper (1984)). The patterns of translocation are sectorial, or
the phyllotactic fraction has biological significance. The observation most
pertinent to the present work is the significant correlation existing between
the phyllotactic fraction and the number of internodes traversed by leaf
traces: The higher phyllotactic fractions are associated with the longer leaf
traces (Girolami (1953); Esau (1965)). While leaf traces of plants with
helical phyllotaxis typically traverse more than one internode, in distichous
phyllotaxis, a $\frac{1}{2}$ phyllotaxis of two-ranked leaf arrangement, and
in verticillate phyllotaxis, leaf traces are approximately one internode or
less (Beck et al. (1982)). Accordingly, low-order systems of a $\frac{1}{2}$
and $\frac{1}{3}$ phyllotaxis are seen on the stems of plants with long
internodes, while plants with short internodes show high-order fractions, as
remarked above.

The stem vascular bundles, or axial bundles, and associated leaf traces
comprise sympodia, on the nature of which there are two perspectives (Beck
(2010)). In one view, the sympodia are of cauline origin, or derived from stem
vascular tissue (Beck et al. (1982)). In the other view, they are of foliar
origin, or derived from leaf traces (Esau (1965)). There are two different
views on the causal relation between initiation of primordia and development
of leaf traces or procambial strands, the strands differentiating into
vascular bundles of xylem and phloem. In one view, the initiation of leaf
primordia brings about the differentiation of the leaf traces. Hence the
initiation of the leaf traces occurs basipetally, or in the direction from the
leaf primordia toward the vascular system of the stem. In the opposing view,
phyllotaxis of leaf primordia is dictated by the vascular organization that
has been established before the leaf primordia are initiated (Larson (1977,
1983)). The latter is consistent with the observation that an incipient leaf
trace develops acropetally, or in the direction toward the leaf primordium it
serves (Esau (1965); Nelson and Dengler (1997)). Priestley and Scott (1933)
was criticized by Snow and Snow (1934). They both do not cast doubt on
Hofmeister’s rule, that is, they share the causal view that phyllotaxis is a
natural consequence of growth and development of an individual plant. They
differ in what they regard as a basic unit of phyllotaxis. The former adopts
growth units including leaf traces, while the latter places primary emphasis
on leaf primordia at the apex. Accordingly, the former and the latter attach
little importance to the irrational and rational divergence, respectively.
Thus, the causal view has been the paradigm of phyllotaxis.

On the whole, causal models are successful in deriving indefinitely continuing
stable systems, resembling actual phyllotactic patterns. From a computational
point of view, they are particularly appealing in that they provide us with
programmable protocols leading to the golden angle. Irrespective of detailed
mechanisms, however, realistic phyllotactic patterns are derived based on the
following observational facts (Vogel (1979); Rivier et al. (1984);
Prusinkiewicz and Lindenmayer (1991)): (i) Divergence angle is constant. (ii)
The constant is the golden angle. For the sake of argument, the former is
often taken so broadly that the constant may take any value. On this premise,
phyllotaxis is rendered to a geometrical playground of mathematics. There are
mathematical arguments for (ii) based on the generalized hypothesis (i) (de
Candolle (1881); Coxeter (1972); Leigh (1972); Ridley (1982); Marzec and
Kappraff (1983)). It is often stated in this regard that the golden angle is a
special angle at which optimal packing is achieved. As a matter of fact, this
is not true literally, for it is only under the constraint (i) that the golden
angle may be said optimal and there is no a priori reason for the constancy.
For living organisms, the property (i) is far from obvious and no less
astounding than (ii), especially because the angular regularity may persist in
spite of temporal irregularity. A time interval between the formation of
successive leaves is called a plastochron, which is used as a morphological or
developmental time scale. Plants grown in different environmental conditions
may be compared in plastochron units but not in physical time units. The fact
that unit of time is a plastochron and duration of a plastochron is not
constant in physical time poses a problem for realistic causal models based on
physical time.

Phyllotactic patterns at a shoot apex are more regular than those on a mature
stem, because internodes tend to be elongated less regularly on the mature
stem. As a matter of fact, the exact level in the stem at which a bifurcation
or recombination of vascular bundles takes place is not an important
morphological constant (Dormer (1972)). Accordingly, trace lengths may vary
arbitrarily along the stem. For this reason, it is often argued that one
should devote oneself exclusively to the study of the growing apex (Church
(1904); Snow (1955)). Nonetheless, further mathematical relations for the
spiral patterns at the apex can be derived by assuming a stronger mathematical
constraint of exponential growth, according to which the leaf primordia are
arranged on logarithmic spirals in a centric representation (Fig. 1) (Church
(1904); Richards (1951); Thomas (1975); Jean (1994)). In the exponential
growth, the ratio of the distances from the center of the apex to two
successively numbered primordia is a constant, called the plastochron ratio.
For a fixed value of divergence angle, Richards (1951) has advocated the use
of a phyllotaxis index defined in terms of the plastochron ratio (cf. (4)).
The index is used to designate two sets of parastichies intersecting
orthogonally. For instance, for Fig. 1, the plastochron ratio is 1.2, the
phyllotaxis index is 3, and $(3,5)$ parastichies cross at right angles. A
fractional value of the index, such as 2.7, means that no two parastichies are
orthogonal. In this geometrical model, a shift in parastichy numbers, e.g.
from $(3,5)$ to $(5,8)$, a phenomena called rising phyllotaxis, is related to
a variation of the plastochron ratio, or the exponential growth rate. The
model has been generalized to allow for other constant divergence angles than
the golden angle (Richards (1951); Thomas (1975); Jean (1994)). In contrast to
these geometrical models based on constant divergence angle, there exist
geometrical causal models in line with Schwendener’s model, which aim at
deriving the limit divergence angle by assuming variable divergence angles
depending on plastochron, the plastochron ratio, and their own rules (van
Iterson (1907); Williams (1974); Erickson (1983); Williams and Brittain
(1984)). For a vegetative shoot, a plastochron index is defined in terms of
length of leaves, and a leaf on a shoot is labeled with a leaf plastochron
index (Erickson and Michelini (1957)). A developmental index of this kind is
indispensable for the systematic study of plant development (Meicenheimer
(2006)). The exponential growth is a practically useful approximation in
dealing with young organs and early stages of development, although it is
neither essential nor peculiar to phyllotaxis.

Despite the apparent success of causal models, neither their intrinsic
mechanisms nor predictions have yet been subjected to experimental tests
specifically. To name several problems on a descriptive level, existing causal
models that explain all types of observed patterns cannot help predicting also
a multiplicity of unreal or too rare patterns. Even when they are capable of
deriving normal patterns, they are not free from instabilities apparently
irrelevant to living organs. Causal models in general are confronted with a
subtle trade-off. Normal phyllotactic patterns must be stable enough to
account for the current prevalence in nature, while they cannot be quite
stable in order to allow for many other exceptional ideal angles just to such
a degree that they are actually existent. In short, rare patterns should be
neither too common nor too rare. It is not clear how and why this subtle
balance between stability and instability is maintained universally, since
fine control of relative size of phyllotactic units depends not only on
species but individual plants or even on parts of the individual plant. We get
puzzled all the more by the observations of more frequent occurrence of rare
patterns among fossil plants.

Causal models, whether physical or chemical, provide dynamical schemes of
self-adjusting the system under the influence of the older leaf primordia. On
the premise that divergence angles between successive leaves are freely
variable by nature, they aim to derive a special angle, normally the golden
angle, toward which the variable divergence angles tend ultimately. They do
not assume any special constant divergence a priori. For the very reasons,
they are likely to be beset with a fundamental difficulty in protecting the
system against disturbance. In this regard, Hofmeister’s empirical rule is
often overestimated. Observed patterns satisfy Hofmeister’s rule, but
Hofmeister’s rule is not sufficient for observed patterns. Hofmeister’s rule
does not imply the periodic appearance of new primordia (Kirchoff (2003)), nor
does it ensure precise regulation of the divergence of 137.5∘ (cf. Fig. 12).
It is not difficult to draw an unreal pattern according to Hofmeister’s rule.
The remarkable empirical fact is rather that divergence angle during steady
growth seems always regulated stably to one of special angles closely related
to the golden ratio. In fact, if a causal rule is to be strictly applied
throughout, fluctuations in size of the domain of influence of a leaf
primordium should inevitably leave behind everlasting irregularities
propagated in the developing pattern (Snow and Snow (1962)). Mathematically,
the instability is a general consequence of the fact that the number of
possible phyllotactic configurations proliferates as relative size of
phyllotactic units decreases. According to causal interpretations, higher
phyllotaxis becomes more vulnerable. The difficulty may not be obvious if one
were interested only in low-order patterns like a $(2,3)$ and $(3,5)$
phyllotaxis, but it should become conspicuous when dealing with a higher order
pattern which requires higher precision maintenance. Besides this stability
problem for high-order patterns, causal models have another difficulty for
low-order patterns (Sec. 3).

There are apparent geometric correlations between parastichy numbers and
relative size of primordia on the apex (Church (1904); Richards (1951);
Kirchoff (2003)) and between leaf arcs and the plastochron ratio (Rutishauser
(1998)). Causal models implement them as causal relationships with the
intention of proving that a phyllotactic pattern, especially the divergence
angle of 137.5∘, is a necessary consequence of changes in the causal agent,
relative size of leaf primordia. According to this interpretation, divergence
angles and contact parastichies must depend not only on the shape of primordia
but on the geometry of the surface on which they are located. The dependence
has been investigated by van Iterson (1907) on the assumption that all the
primordia keep a common shape while they are allowed to change their sizes. So
far, however, no direct evidence has been provided to support the presumed
causal relationship. As a matter of fact, there are very few studies in which
sufficiently detailed data are obtained to make a close comparison with the
models possible or useful (Erickson (1983)). In particular, the prediction of
causal models that rising phyllotaxis, or change in parastichy numbers, should
accompany wide variations and abrupt turns of divergence angle, as indicated
in Fig. 2, has not been supported unequivocally. On the contrary, the success
of Richards’ model indicates the exponential growth with constant divergence
angle irrespective of whether parastichy numbers rise or fall. Church (1904)
refuted Schwendener’s model by counterexamples showing normal spiral patterns
of circular primordia whose positions are widely separated. In comparing
treated plants, Maksymowych and Erickson (1977) found no significant change in
divergence angle in a correlation diagram for the plastochron ratio and
divergence angle. Statistical analysis of Fujita (1939) has revealed that
divergence angles do not depend so much on parastichy numbers as expected from
causal models (Jean (1986)). There is clear evidence against the basic
assumption that the primordia size is the causal factor of divergence angle. A
plant appears to accomplish geometrical correlations in a phyllotactic pattern
by adapting the size and shape of leafy organs as if it knows the end pattern
at which it aims. Snow and Snow (1962) observe that the secondary extension of
a leaf base adjusts itself so that divergence angle is little affected in
spite of artificial disturbances. This observation, despite the authors’
claim, undermines their space-filling mechanism that the leaf base extension
regulates the phyllotactic pattern. To the contrary, apparent causal changes
in the position, size and shape of leaves or scales in chemical or physical
contact may be just incidental phenomena (Church (1904); Richards (1948); Marc
and Hackett (1991)). No doubt there are cases in which physical or chemical
contact pressure may induce secondary displacement of compactly packed lateral
organs.

Natural selection plays no role in causal interpretations of phyllotaxis. If
one supposes to the contrary that natural selection holds the key to
understanding the golden angle at the shoot apex, then one should investigate
a special effect of the special angle, instead of its cause. In other words,
one should look for the distal or ultimate cause of the special angle, instead
of the proximity cause. This sort of theory intends to explain special traits
not in terms of immediate physiological factors, but in terms of evolutionary
forces acting on them. It aims at a full understanding of the phenomena at a
phenomenological level, independently of whatever physiological mechanisms may
be involved. There is a long history of investigations into selective
advantage of the observed divergences based on the external structure. It goes
as follows: common phyllotactic patterns distribute leaves as evenly as
possible and maximize exposure of leaves to enhance the capacity to intercept
sunlight (Wright (1873)). Such an argument is unpromising because leaves are
aligned in vertical ranks. Indeed, changes in leaf shape and stem length can
compensate for the negative effects of leaf overlap (Niklas (1988, 1998)). For
this obvious reason, it is often argued to the contrary in favor of the ‘most
irrational’ divergence angle; no two leaves lie precisely under one another
when divergence angle is equal to the golden angle (de Candolle (1881);
Wiesner (1875, 1907); Coxeter (1972); Leigh (1972); Takenaka (1994); Pearcy
and Yang (1998); Valladares and Brites (2004); King et al. (2004); Bryntsev
(2004)). There is also a long history of criticism of this view (Thompson
(1917)). In the first place, the golden angle is not a general rule for mature
shoots, and the light-capture mechanism deepens the riddle of the common
occurrence of a $\frac{2}{5}$ phyllotaxis. In general, existing theories tend
to argue for the uses of irrational angles without regard to the uses of
rational angles or vice versa.

###  1.2 Aim and scope of this paper

Let us direct attention to the internal structure, the vascular system.
Mathematical interrelationship between the initial (apical) and the mature
(vascular) phyllotactic pattern seems to have not been discussed
experimentally nor theoretically. This paper develops a theory of vascular
phyllotaxis to fill in the gap between the two distinct but intimately related
phenomena. A physical model has been described mathematically in the previous
paper (Okabe (2011)). However, the model was abstract and its relevance to
real phenomena was not clearly elucidated. The aim of this paper is to develop
the model to show its experimental validity and relevance. This is relevant to
a fundamental problem of phyllotaxis: Is phyllotaxis determined causally or
genetically? In contrast to numerous models holding the causal view, the
present model is based on the genetic perspective that special numbers in
phyllotaxis are primarily of genetic origin, so that it is assumed that
constant primordial divergence angle during steady growth is genetically
determined. According to the model, the effect of constant divergence angle is
investigated, and what value of the constant is advantageous is settled. This
work is not concerned about transient fluctuations of divergence angle during
ontogeny. Therefore, the model is compatible with any physical or chemical
causal models for the positioning of leaf initiation at the shoot apex,
although the limit divergence angle at the apex is interpreted totally
differently. The special angle is not an inevitable consequence of ontogenetic
dynamics, whether physical or chemical. It is regarded as a heritable trait of
a plant. It is supposed that once there was a wide variation in the traits of
individuals, or there have formerly been wide variations of divergence angles.
The special limit divergences found in nature have survived natural selection.
This conforms with the traditional view that biological features that are
under tight genetic control and that have very narrow ranges of variation are
believed to be adaptive (Niklas (1997)). Although the author believes that the
premise of the model, divergence angle as a trait of a plant, is not only
plausible but supported by circumstantial evidence, it has not been
unanimously accepted at present. It may be verified or refuted experimentally
in the future.

For the efficient transport of materials throughout an indefinite number of
leaves attached to a stem of a finite cross section, the leaves should be
aligned along a finite number of ‘orthostichious’ bundles. At this point, a
whole number enters the theory. There are modes of orthostichous order
depending on the initial arrangement and length of leaf traces. The number of
vascular orthostichies may be increased or decreased, but not arbitrarily. By
regarding a leaf primordium and the leaf trace(s) associated with it as the
fundamental unit of phyllotactic patterns, a mathematical correspondence is
derived between the divergence angle of the initial phyllotactic pattern,
$360\alpha_{0}$ degrees, and the phyllotactic fraction $\alpha$ of a mature
pattern, where the number of internodes traversed by the leaf traces, $n_{c}$,
plays a pivotal role. As a general rule, it has been known that phyllotactic
fraction of a vascular plant may vary sequentially during growth (Braun
(1835); Skutch (1927); Allard (1942); Puławska (1965); Larson (1977)). By
means of the mathematical relation between $\alpha_{0}$ (an irrational number)
and $\alpha$ (rational numbers), it is shown that changes in $n_{c}$ cause the
phyllotactic transitions in $\alpha$. As a natural consequence, an
evolutionary mechanism for the phenomenon of phyllotaxis is suggested.
Supporting evidence for the model and the evolutionary mechanism is presented
by analyzing experimental results.

In Sec. 2, a model and results used in the following sections are presented by
means of figures and tables without using mathematics. Tables 1$\sim$18 have
not been presented before.

In Sec. 3, observed precision of the initial divergence $\alpha_{0}$ is
explained by means of a correlation predicted between the range of
$\alpha_{0}$ and the highest-order fraction $\alpha$. In short, divergence
angle $\alpha_{0}$ of a system with a high phyllotactic fraction $\alpha$
should be accurately controlled in order to avoid unnecessary changes in
vascular structure.

In Sec. 4, phyllotaxis of Lepidodendron by Dickson (1871) is analyzed.
Diversity of phyllotaxis is discussed as a result of ineffective selective
pressures.

In Sec. 5, the vascular structure of Linum usitatissimum by Girolami (1953) is
investigated. Various relations between phyllotactic fraction and parastichy
numbers, the phyllotactic fraction $\alpha$ and the length per internode of
leaf traces $n_{c}$, and directions of parastichies and the genetic spiral are
pointed out.

In Sec. 6, the phyllotactic transition of Populus deltoides by Larson (1977)
is analyzed. It is shown that a continuous change in length of leaf traces
causes the discontinuous effect of the phyllotactic transition in the vascular
structure.

In the appendix, a relation between the trace length $n_{c}$ and the
plastochron ratio $a$ is discussed to indicate that the former serves as a
useful developmental index for the mature stem as the latter is used for the
apex.

##  2 Model

Figure 3:  Phyllotactic patterns of leaf traces with a length of $n_{c}=4$
before and after secondary torsion are arranged side by side. A dotted and
dashed line of each figure represent a vertical cut of a cylinder surface
unrolled. (a) A pattern with initial divergence of $360\alpha_{0}\simeq
137.5^{\circ}$ ($\alpha_{0}=1/(1+\tau)\simeq 0.382$). (b) The final pattern of
a fractional divergence $\alpha=\frac{2}{5}$ resulting from (a). Leaf traces
in the upper part move rightward while the pattern (a) becomes (b), thereby
five 5-parastichies in (a), such as 1-6-11-16-21, align themselves to make
five orthostichies in (b).

A regular helical pattern of leaf traces is schematically plotted as a lattice
of line segments on an unrolled surface of a cylinder. The divergence angle of
the initial pattern is denoted as $360\alpha_{0}$ in degrees, which is assumed
to be less than 180 degrees, i.e., $0\leq\alpha_{0}\leq\frac{1}{2}$ without
loss of generality. Fig. 3 presents a typical pattern for $360\alpha_{0}\simeq
137.5^{\circ}$ ($\alpha_{0}\simeq 0.382$). The length of leaf traces measured
in internodes is denoted as $n_{c}$ in accordance with the previous notation
(Okabe (2011)). As in Fig. 1, $n_{c}=4$ in Fig. 3. The trace length $n_{c}$
need not be an integer; $n_{c}$ is the average number of leaf traces cut by a
transverse section (Fig. 4). As the number in a section is an integer, this
method gives a good estimate of $n_{c}$ particularly for $n_{c}\gg 1$. The
model comprises two parameters $\alpha_{0}$ and $n_{c}$. For the sake of
argument, patterns with constant values of them are considered below. Effects
of their fluctuations may be discussed based on results to be obtained.

Figure 4:  For three transverse sections of a pattern of leaf traces with a
length of $n_{c}=4.3$, the number of the traces in each section is indicated
on the right-hand side below the cut line. The number averaged over sections
should approach $n_{c}$.

The leaf traces repel with each other laterally to arrange themselves in an
orthostichous pattern. The mutual interaction is likely to be regulated by the
plant hormone auxin (Beck (2010)). Fig. 3 is the final pattern resulting from
Fig. 3. Divergence of the final pattern is expressed in terms of the
phyllotactic fraction $\alpha$. The pattern of Fig. 3 is characterized with
$\alpha=\frac{2}{5}$. In Fig. 3, there are five parastichies of 1-6-11-16-21,
2-7-12-17-22, 3-8-13-18-23, 4-9-14-19-24 and 5-10-15-20-25, each of which is
called a 5-parastichy. The five 5-parastichies are lined up vertically to make
five orthostichies of the $\frac{2}{5}$ phyllotaxis in Fig. 3. In the patterns
of Fig. 3, the next visible parastichies are 3-parastichies
(1-4-7-10-13-16-19-22-25, 2-5-8-11-14-17-20-23 and 3-6-9-12-15-18-21-24) and
2-parastichies (1-3-5-7-9-11-13-15-17-19-21-23-25 and
2-4-6-8-10-12-14-16-18-20-22-24). As these parastichies remain conspicuous in
the two patterns, both patterns may be referred to as having a parastichy pair
of $(2,3)$. Thus, for $n_{c}=4$, there is one-to-one correspondence between
$\alpha_{0}\simeq 0.382$ (angle of $360\alpha_{0}\simeq 137.5^{\circ}$) of the
initial pattern and $\alpha=\frac{2}{5}$ of the final pattern. In a similar
manner, $\alpha$ is obtained for arbitrary values of $\alpha_{0}$ and $n_{c}$.
Indeed, we get $\alpha=\frac{2}{5}$ insofar as $3\leq n_{c}<5$ and
$\frac{1}{3}<\alpha_{0}<\frac{1}{2}$ ( see Okabe (2011) for the mathematical
implementation). Below we discuss phyllotactic changes in $\alpha$ that occur
when $n_{c}$ and $\alpha_{0}$ are set out of their respective ranges.

(a)

Figure 5:  Change in a phyllotactic pattern of leaf traces with a length of
$n_{c}=7$ (cf. Fig. 3). (a) The initial pattern with
$\alpha_{0}=1/(1+\tau)=1/(2+\tau^{-1})$ ($360\alpha_{0}\simeq 137.5^{\circ}$).
(b) The final pattern with $\alpha=\frac{3}{8}$.

For a fixed value of $\alpha_{0}\simeq 0.382$, Fig. 5(a) is for $n_{c}=7$ in
comparison with Fig. 3 for $n_{c}=4$. As the traces of length longer than five
internodes cannot be aligned in five orthostichies, we obtain
$\alpha=\frac{3}{8}$ for Fig. 5(a), while $\alpha=\frac{2}{5}$ in Fig. 3.
Thus, it is explained that a higher phyllotactic fraction is obtained for a
longer length of leaf traces. Phyllotactic transition from
$\alpha=\frac{2}{5}$ to $\alpha=\frac{3}{8}$ occurs when $n_{c}$ increases
past a threshold value of $n_{c}=5$. Experimental evidence of this transition
is presented below in Fig. 17.

(a)

Figure 6:  Change in a phyllotactic pattern of leaf traces with a length of
$n_{c}=4$ (cf. Fig. 3). (a) The initial pattern with
$\alpha_{0}=1/(3+\tau^{-1})\simeq 0.276$ ($360\alpha_{0}\simeq 99.5^{\circ}$).
(b) The final pattern with $\alpha=\frac{2}{7}$.

For a fixed value of $n_{c}$, the phyllotactic fraction $\alpha$ depends on
the initial divergence $\alpha_{0}$. For $n_{c}=4$, Fig. 6 is for
$\alpha_{0}\simeq 0.276$ (angle of $99.5^{\circ}$), which is compared with
Fig. 3 for $\alpha_{0}\simeq 0.382$ $(137.5^{\circ})$. The former leads to a
final pattern of $\alpha=\frac{2}{7}$ in Fig. 6, while the latter gives
$\alpha=\frac{2}{5}$ in Fig. 3. In fact, there are three fractional patterns
conceivable for $n_{c}=4$, namely (a) $\alpha=\frac{1}{5}$ for
$0<\alpha_{0}<\frac{1}{4}$, (b) $\alpha=\frac{2}{7}$ for
$\frac{1}{4}<\alpha_{0}<\frac{1}{3}$ and (c) $\alpha=\frac{2}{5}$ for
$\frac{1}{3}<\alpha_{0}<\frac{1}{2}$.

Figure 7:  Tree diagram for the phyllotactic fraction $\alpha$. The horizontal
axis is the initial divergence $\alpha_{0}$, and the vertical axis is the
number of internodes traversed by leaf traces $n_{c}$. Numbers in parentheses
below each fraction are the parastichy pair corresponding to the fraction. By
way of explanation, let us take $\frac{3}{8}$ in the right-bottom quoter as an
example. The fraction $\frac{3}{8}$ with the parastichy pair $(3,5)$ is in
three different positions at $n_{c}=5,6$ and 7. By means of lower order
fractions lying below them, the fraction $\frac{3}{8}$ is bracketed between
$\frac{1}{3}$ and $\frac{2}{5}$. Therefore, we obtain $\alpha=\frac{3}{8}$
with the parastichy pair $(3,5)$ insofar as
$\frac{1}{3}\leq\alpha_{0}<\frac{2}{5}$ and $5\leq n_{c}<8$ (Table 1).
Similarly, we find $\alpha=\frac{5}{12}$ with $(5,7)$ for
$\frac{2}{5}\leq\alpha_{0}<\frac{3}{7}$ and $7\leq n_{c}<12$ (Table 6).

Every phyllotactic fraction for $\alpha$ has its own ranges of values for
$\alpha_{0}$ and $n_{c}$. The mathematical correspondence is presented
succinctly as a tree diagram in Fig. 7. For instance, Fig. 7 gives the
conditions $\frac{1}{4}<\alpha_{0}<\frac{1}{3}$ and $4\leq n_{c}<7$ for
$\alpha=\frac{2}{7}$. For the former inequalities, the boundary fractions
$\frac{1}{4}$ and $\frac{1}{3}$ lie below $\frac{2}{7}$ in Fig. 7. The latter
condition $4\leq n_{c}<7$ is reasoned from the vertical coordinate $n_{c}=4,5$
and $6$ of three $\frac{2}{7}$’s in Fig. 7. The phyllotactic sequence of
fractions derived from an arbitrary value of initial divergence $\alpha_{0}$
may be traced by climbing up the tree of Fig. 7 along the vertical line at
$\alpha_{0}$. For $\alpha_{0}\simeq 0.382$ $(137.5^{\circ})$, the main
sequence
$\frac{1}{2},\frac{1}{3},\frac{2}{5},\frac{3}{8},\frac{5}{13},\frac{8}{21},\cdots$
is obtained in the increasing order of $n_{c}$. The tree diagram extended for
all values of $n_{c}$ includes all conceivable phyllotactic fractions.

For the sake of convenience, let us introduce shorthand notation for the
irrational numbers found in nature,

$\displaystyle\ [n]$ $\displaystyle\equiv$
$\displaystyle\frac{1}{n+\tau^{-1}},$ $\displaystyle\ \qquad[n,m]$
$\displaystyle\equiv$ $\displaystyle\frac{1}{n+\dfrac{1}{m+\tau^{-1}}},$
$\displaystyle\ [n,m,l]$ $\displaystyle\equiv$
$\displaystyle\frac{1}{n+\dfrac{1}{m+\dfrac{1}{l+\tau^{-1}}}},$ (2)

and so on, where $n$, $m$, $l$ are positive integers. With this notation,
$\alpha_{0}=[2]=1/(2+\tau^{-1})=1/(1+\tau)$ gives the main sequence. The last
equality holds by the definition of $\tau$ in (1). Note that the pattern with
$\alpha_{0}=[1]$ ($360\alpha_{0}=222.5^{\circ}$) is nothing but the mirror
image of $\alpha_{0}=[2]$ ($137.5^{\circ}$), because $[1]=1-[2]$ or
$222.5^{\circ}=360^{\circ}-137.5^{\circ}$. For future reference, Tables
1$\sim$18 are provided for the initial divergence $\alpha_{0}$ given by
typical irrational numbers. These are not exhaustive, but they include almost
all phyllotactic fractions observed in nature. The main sequence is presented
in Table 1. In the second column for $\alpha=\frac{2}{5}$, ‘$(2,3)$’ in the
second row represents the parastichy pair corresponding to the fraction
$\frac{2}{5}$, ‘$3\sim$’ in the third row abbreviates $3\leq n_{c}<5$, where 5
for the upper limit is taken from the next column, and
‘$\frac{1}{3}\sim\frac{1}{2}$’ in the fourth row indicates
$\frac{1}{3}<\alpha_{0}<\frac{1}{2}$. As an example, let us take a fraction
$\alpha=\frac{21}{76}$. It is found in the eighth column of Table 2, from
which the conditions $47\leq n_{c}<76$ and
$\frac{8}{29}<\alpha_{0}<\frac{13}{47}$ are read. These results are used in
the next section (Table 20).

The tables show that the denominator of a fraction $\alpha$ and the parastichy
numbers are correlated with the threshold numbers for $n_{c}$. The numbers
comprise a characteristic sequence of integers. The main sequence in Table 1
is characterized with the Fibonacci sequence of 1, 2, 3, 5, 8, 13, 21,
$\cdots$, while Table 6 has a sequence of 1, 2, 3, 2, 5, 7, 12, 19, $\cdots$,
which is sometimes called the lateral sequence. Three consecutive numbers of a
sequence satisfy the Fibonacci recurrence relation ($2+5=7$, $5+7=12$,
$7+12=19$), except for the first several numbers (like 1, 2, 3 in the lateral
sequence). Therefore, each phyllotactic sequence is referred to by a pair of
seed integers for the Fibonacci recurrence formula. The seed pair of each
table, such as $(2,5)$ in Table 6, is highlighted in boldface.

$\alpha$ | $\frac{2}{5}$ | $\frac{3}{8}$ | $\frac{5}{13}$ | $\frac{8}{21}$ | $\frac{13}{34}$ | $\frac{21}{55}$ | $\frac{34}{89}$ | $\frac{55}{144}$
---|---|---|---|---|---|---|---|---
| (2,3) | (3,5) | (5,8) | (8,13) | (13,21) | (21,34) | (34,55) | (55,89)
$n_{c}$ | $3\sim$ | $5\sim$ | $8\sim$ | $13\sim$ | $21\sim$ | $34\sim$ | $55\sim$ | $89\sim$
$\alpha_{0}$ | $\frac{1}{3}\sim\frac{1}{2}$ | $\frac{1}{3}\sim\frac{2}{5}$ | $\frac{3}{8}\sim\frac{2}{5}$ | $\frac{3}{8}\sim\frac{5}{13}$ | $\frac{8}{21}\sim\frac{5}{13}$ | $\frac{8}{21}\sim\frac{13}{34}$ | $\frac{21}{55}\sim\frac{13}{34}$ | $\frac{21}{55}\sim\frac{34}{89}$
Table 1:  Parastichy numbers (in parentheses) and ranges of $n_{c}$ and $\alpha_{0}$ for the phyllotactic fractions $\alpha$ belonging to the main sequence with the limit divergence of $\alpha_{0}=[2]=1/(2+\tau^{-1})\simeq 0.3820$ ($360\alpha_{0}\simeq 137.5^{\circ}$). The parastichy pairs are generated from the seed pair $(1,2)$ for $\alpha=\frac{1}{3}$ (not shown) by a Fibonacci recurrence relation. The golden angle $\alpha_{0}=[2]$ and the main sequence are called the Fibonacci angle and the Fibonacci sequence.  $\alpha$ | $\frac{1}{4}$ | $\frac{2}{7}$ | $\frac{3}{11}$ | $\frac{5}{18}$ | $\frac{8}{29}$ | $\frac{13}{47}$ | $\frac{21}{76}$ | $\frac{34}{123}$
---|---|---|---|---|---|---|---|---
| (1,3) | (3,4) | (4,7) | (7,11) | (11,18) | (18,29) | (29,47) | (47,76)
$n_{c}$ | 3$\sim$ | 4$\sim$ | 7$\sim$ | 11$\sim$ | 18$\sim$ | 29$\sim$ | 47$\sim$ | 76$\sim$
$\alpha_{0}$ | $0\sim\frac{1}{3}$ | $\frac{1}{4}\sim\frac{1}{3}$ | $\frac{1}{4}\sim\frac{2}{7}$ | $\frac{3}{11}\sim\frac{2}{7}$ | $\frac{3}{11}\sim\frac{5}{18}$ | $\frac{8}{29}\sim\frac{5}{18}$ | $\frac{8}{29}\sim\frac{13}{47}$ | $\frac{21}{76}\sim\frac{13}{47}$
Table 2:  Table for the limit divergence of $\alpha_{0}=[3]=1/(3+\tau^{-1})\simeq 0.2764$ ($360\alpha_{0}\simeq 99.5^{\circ}$). The parastichy pairs are generated from the seed pair $(1,3)$ by a Fibonacci recurrence relation. The sequence 1,3,4,7,11,$\cdots$ is called the Lucas sequence or the first accessory sequence.  $\alpha$ | $\frac{1}{4}$ | $\frac{1}{5}$ | $\frac{2}{9}$ | $\frac{3}{14}$ | $\frac{5}{23}$ | $\frac{8}{37}$ | $\frac{13}{60}$ | $\frac{21}{97}$
---|---|---|---|---|---|---|---|---
| (1,3) | (1,4) | (4,5) | (5,9) | (9,14) | (14,23) | (23,37) | (37,60)
$n_{c}$ | 3$\sim$ | 4$\sim$ | 5$\sim$ | 9$\sim$ | 14$\sim$ | 23$\sim$ | 37$\sim$ | 60$\sim$
$\alpha_{0}$ | $0\sim\frac{1}{3}$ | $0\sim\frac{1}{4}$ | $\frac{1}{5}\sim\frac{1}{4}$ | $\frac{1}{5}\sim\frac{2}{9}$ | $\frac{3}{14}\sim\frac{2}{9}$ | $\frac{3}{14}\sim\frac{5}{23}$ | $\frac{8}{37}\sim\frac{5}{23}$ | $\frac{8}{37}\sim\frac{13}{60}$
Table 3:  Table for the limit divergence of $\alpha_{0}=[4]=1/(4+\tau^{-1})\simeq 0.2165$ ($360\alpha_{0}\simeq 78.0^{\circ}$). The parastichy pairs except (1,3) are generated from the seed pair (1,4) by a Fibonacci recurrence relation. The sequence 1,4,5,9,14,$\cdots$ is called the second accessory sequence.  $\alpha$ | $\frac{1}{4}$ | $\frac{1}{5}$ | $\frac{1}{6}$ | $\frac{2}{11}$ | $\frac{3}{17}$ | $\frac{5}{28}$ | $\frac{8}{45}$ | $\frac{13}{73}$
---|---|---|---|---|---|---|---|---
| (1,3) | (1,4) | (1,5) | (5,6) | (6,11) | (11,17) | (17,28) | (28,45)
$n_{c}$ | 3$\sim$ | 4$\sim$ | 5$\sim$ | 6$\sim$ | 11$\sim$ | 17$\sim$ | 28$\sim$ | 45$\sim$
$\alpha_{0}$ | $0\sim\frac{1}{3}$ | $0\sim\frac{1}{4}$ | $0\sim\frac{1}{5}$ | $\frac{1}{6}\sim\frac{1}{5}$ | $\frac{1}{6}\sim\frac{2}{11}$ | $\frac{3}{17}\sim\frac{2}{11}$ | $\frac{3}{17}\sim\frac{5}{28}$ | $\frac{8}{45}\sim\frac{5}{28}$
Table 4: $\alpha_{0}=[5]=1/(5+\tau^{-1})\simeq 0.1780$ ($360\alpha_{0}\simeq 64.1^{\circ}$).  $\alpha$ | $\frac{1}{4}$ | $\frac{1}{5}$ | $\frac{1}{6}$ | $\frac{1}{7}$ | $\frac{2}{13}$ | $\frac{3}{20}$ | $\frac{5}{33}$ | $\frac{8}{53}$
---|---|---|---|---|---|---|---|---
| (1,3) | (1,4) | (1,5) | (1,6) | (6,7) | (7,13) | (13,20) | (20,33)
$n_{c}$ | 3$\sim$ | 4$\sim$ | 5$\sim$ | 6$\sim$ | 7$\sim$ | 13$\sim$ | 20$\sim$ | 33$\sim$
$\alpha_{0}$ | $0\sim\frac{1}{3}$ | $0\sim\frac{1}{4}$ | $0\sim\frac{1}{5}$ | $0\sim\frac{1}{6}$ | $\frac{1}{7}\sim\frac{1}{6}$ | $\frac{1}{7}\sim\frac{2}{13}$ | $\frac{3}{20}\sim\frac{2}{13}$ | $\frac{3}{20}\sim\frac{5}{33}$
Table 5: $\alpha_{0}=[6]=1/(6+\tau^{-1})\simeq 0.15112$ ($360\alpha_{0}\simeq 54.4^{\circ}$).  $\alpha$ | $\frac{2}{5}$ | $\frac{3}{7}$ | $\frac{5}{12}$ | $\frac{8}{19}$ | $\frac{13}{31}$ | $\frac{21}{50}$ | $\frac{34}{81}$ | $\frac{55}{131}$
---|---|---|---|---|---|---|---|---
| (2,3) | (2,5) | (5,7) | (7,12) | (12,19) | (19,31) | (31,50) | (50,81)
$n_{c}$ | 3$\sim$ | 5$\sim$ | 7$\sim$ | 12$\sim$ | 19$\sim$ | 31$\sim$ | 50$\sim$ | 81$\sim$
$\alpha_{0}$ | $\frac{1}{3}\sim\frac{1}{2}$ | $\frac{2}{5}\sim\frac{1}{2}$ | $\frac{2}{5}\sim\frac{3}{7}$ | $\frac{5}{12}\sim\frac{3}{7}$ | $\frac{5}{12}\sim\frac{8}{19}$ | $\frac{13}{31}\sim\frac{8}{19}$ | $\frac{13}{31}\sim\frac{21}{50}$ | $\frac{34}{81}\sim\frac{21}{50}$
Table 6: $\alpha_{0}=[2,2]=1/(2+1/(2+\tau^{-1}))\simeq 0.4198$ ($360\alpha_{0}\simeq 151.1^{\circ}$). The parastichy pairs except (2,3) are generated from the seed pair (2,5). The sequence 2,5,7,12,19,$\cdots$ is called the first lateral sequence.  $\alpha$ | $\frac{1}{4}$ | $\frac{2}{7}$ | $\frac{3}{10}$ | $\frac{5}{17}$ | $\frac{8}{27}$ | $\frac{13}{44}$ | $\frac{21}{71}$ | $\frac{34}{115}$
---|---|---|---|---|---|---|---|---
| (1,3) | (3,4) | (3,7) | (7,10) | (10,17) | (17,27) | (27,44) | (44,71)
$n_{c}$ | 3$\sim$ | 4$\sim$ | 7$\sim$ | 10$\sim$ | 17$\sim$ | 27$\sim$ | 44$\sim$ | 71$\sim$
$\alpha_{0}$ | $0\sim\frac{1}{3}$ | $\frac{1}{4}\sim\frac{1}{3}$ | $\frac{2}{7}\sim\frac{1}{3}$ | $\frac{2}{7}\sim\frac{3}{10}$ | $\frac{5}{17}\sim\frac{3}{10}$ | $\frac{5}{17}\sim\frac{8}{27}$ | $\frac{13}{44}\sim\frac{8}{27}$ | $\frac{13}{44}\sim\frac{21}{71}$
Table 7: $\alpha_{0}=[3,2]=1/(3+1/(2+\tau^{-1}))\simeq 0.2957$ ($360\alpha_{0}\simeq 106.4^{\circ}$).  $\alpha$ | $\frac{2}{5}$ | $\frac{3}{7}$ | $\frac{4}{9}$ | $\frac{7}{16}$ | $\frac{11}{25}$ | $\frac{18}{41}$ | $\frac{29}{66}$ | $\frac{47}{107}$
---|---|---|---|---|---|---|---|---
| (2,3) | (2,5) | (2,7) | (7,9) | (9,16) | (16,25) | (25,41) | (41,66)
$n_{c}$ | 3$\sim$ | 5$\sim$ | 7$\sim$ | 9$\sim$ | 16$\sim$ | 25$\sim$ | 41$\sim$ | 66$\sim$
$\alpha_{0}$ | $\frac{1}{3}\sim\frac{1}{2}$ | $\frac{2}{5}\sim\frac{1}{2}$ | $\frac{3}{7}\sim\frac{1}{2}$ | $\frac{3}{7}\sim\frac{4}{9}$ | $\frac{7}{16}\sim\frac{4}{9}$ | $\frac{7}{16}\sim\frac{11}{25}$ | $\frac{18}{41}\sim\frac{11}{25}$ | $\frac{18}{41}\sim\frac{29}{66}$
Table 8: $\alpha_{0}=[2,3]=1/(2+1/(3+\tau^{-1}))\simeq 0.4393$ ($360\alpha_{0}\simeq 158.1^{\circ}$). The sequence 2,7,9,16,$\cdots$ is called the second lateral sequence.  $\alpha$ | $\frac{2}{7}$ | $\frac{3}{10}$ | $\frac{4}{13}$ | $\frac{7}{23}$ | $\frac{11}{36}$ | $\frac{18}{59}$ | $\frac{29}{95}$ | $\frac{47}{154}$
---|---|---|---|---|---|---|---|---
| (3,4) | (3,7) | (3,10) | (10,13) | (13,23) | (23,36) | (36,59) | (59,95)
$n_{c}$ | 4$\sim$ | 7$\sim$ | 10$\sim$ | 13$\sim$ | 23$\sim$ | 36$\sim$ | 59$\sim$ | 95$\sim$
$\alpha_{0}$ | $\frac{1}{4}\sim\frac{1}{3}$ | $\frac{2}{7}\sim\frac{1}{3}$ | $\frac{3}{10}\sim\frac{1}{3}$ | $\frac{3}{10}\sim\frac{4}{13}$ | $\frac{7}{23}\sim\frac{4}{13}$ | $\frac{7}{23}\sim\frac{11}{36}$ | $\frac{18}{59}\sim\frac{11}{36}$ | $\frac{18}{59}\sim\frac{29}{95}$
Table 9: $\alpha_{0}=[3,3]=1/(3+1/(3+\tau^{-1}))\simeq 0.3052$ ($360\alpha_{0}\simeq 109.9^{\circ}$).  $\alpha$ | $\frac{1}{4}$ | $\frac{1}{5}$ | $\frac{2}{9}$ | $\frac{3}{13}$ | $\frac{5}{22}$ | $\frac{8}{35}$ | $\frac{13}{57}$ | $\frac{21}{92}$
---|---|---|---|---|---|---|---|---
| (1,3) | (1,4) | (4,5) | (4,9) | (9,13) | (13,22) | (22,35) | (35,57)
$n_{c}$ | 3$\sim$ | 4$\sim$ | 5$\sim$ | 9$\sim$ | 13$\sim$ | 22$\sim$ | 35$\sim$ | 57$\sim$
$\alpha_{0}$ | $0\sim\frac{1}{3}$ | $0\sim\frac{1}{4}$ | $\frac{1}{5}\sim\frac{1}{4}$ | $\frac{2}{9}\sim\frac{1}{4}$ | $\frac{2}{9}\sim\frac{3}{13}$ | $\frac{5}{22}\sim\frac{3}{13}$ | $\frac{5}{22}\sim\frac{8}{35}$ | $\frac{13}{57}\sim\frac{8}{35}$
Table 10: $\alpha_{0}=[4,2]=1/(4+1/(2+\tau^{-1}))\simeq 0.2282$ ($360\alpha_{0}\simeq 82.2^{\circ}$).  $\alpha$ | $\frac{2}{5}$ | $\frac{3}{8}$ | $\frac{4}{11}$ | $\frac{7}{19}$ | $\frac{11}{30}$ | $\frac{18}{49}$ | $\frac{29}{79}$ | $\frac{47}{128}$
---|---|---|---|---|---|---|---|---
| (2,3) | (3,5) | (3,8) | (8,11) | (11,19) | (19,30) | (30,49) | (49,79)
$n_{c}$ | 3$\sim$ | 5$\sim$ | 8$\sim$ | 11$\sim$ | 19$\sim$ | 30$\sim$ | 49$\sim$ | 79$\sim$
$\alpha_{0}$ | $\frac{1}{3}\sim\frac{1}{2}$ | $\frac{1}{3}\sim\frac{2}{5}$ | $\frac{1}{3}\sim\frac{3}{8}$ | $\frac{4}{11}\sim\frac{3}{8}$ | $\frac{4}{11}\sim\frac{7}{19}$ | $\frac{11}{30}\sim\frac{7}{19}$ | $\frac{11}{30}\sim\frac{18}{49}$ | $\frac{29}{79}\sim\frac{18}{49}$
Table 11: $\alpha_{0}=[2,1,2]=1/(2+1/(1+1/(2+\tau^{-1})))\simeq 0.3672$ ($360\alpha_{0}\simeq 132.2^{\circ}$).  $\alpha$ | $\frac{2}{5}$ | $\frac{3}{7}$ | $\frac{5}{12}$ | $\frac{7}{17}$ | $\frac{12}{29}$ | $\frac{19}{46}$ | $\frac{31}{75}$ | $\frac{50}{121}$
---|---|---|---|---|---|---|---|---
| (2,3) | (2,5) | (5,7) | (5,12) | (12,17) | (17,29) | (29,46) | (46,75)
$n_{c}$ | 3$\sim$ | 5$\sim$ | 7$\sim$ | 12$\sim$ | 17$\sim$ | 29$\sim$ | 46$\sim$ | 75$\sim$
$\alpha_{0}$ | $\frac{1}{3}\sim\frac{1}{2}$ | $\frac{2}{5}\sim\frac{1}{2}$ | $\frac{2}{5}\sim\frac{3}{7}$ | $\frac{2}{5}\sim\frac{5}{12}$ | $\frac{7}{17}\sim\frac{5}{12}$ | $\frac{7}{17}\sim\frac{12}{29}$ | $\frac{19}{46}\sim\frac{12}{29}$ | $\frac{19}{46}\sim\frac{31}{75}$
Table 12: $\alpha_{0}=[2,2,2]=1/(2+1/(2+1/(2+\tau^{-1})))\simeq 0.4133$ ($360\alpha_{0}\simeq 148.8^{\circ}$).  $\alpha$ | $\frac{1}{4}$ | $\frac{2}{7}$ | $\frac{3}{11}$ | $\frac{4}{15}$ | $\frac{7}{26}$ | $\frac{11}{41}$ | $\frac{18}{67}$ | $\frac{29}{108}$
---|---|---|---|---|---|---|---|---
| (1,3) | (3,4) | (4,7) | (4,11) | (11,15) | (15,26) | (26,41) | (41,67)
$n_{c}$ | 3$\sim$ | 4$\sim$ | 7$\sim$ | 11$\sim$ | 15$\sim$ | 26$\sim$ | 41$\sim$ | 67$\sim$
$\alpha_{0}$ | $0\sim\frac{1}{3}$ | $\frac{1}{4}\sim\frac{1}{3}$ | $\frac{1}{4}\sim\frac{2}{7}$ | $\frac{1}{4}\sim\frac{3}{11}$ | $\frac{4}{15}\sim\frac{3}{11}$ | $\frac{4}{15}\sim\frac{7}{26}$ | $\frac{11}{41}\sim\frac{7}{26}$ | $\frac{11}{41}\sim\frac{18}{67}$
Table 13: $\alpha_{0}=[3,1,2]=1/(3+1/(1+1/(2+\tau^{-1})))\simeq 0.2686$ ($360\alpha_{0}\simeq 96.7^{\circ}$).  $\alpha$ | $\frac{1}{4}$ | $\frac{2}{7}$ | $\frac{3}{10}$ | $\frac{5}{17}$ | $\frac{7}{24}$ | $\frac{12}{41}$ | $\frac{19}{65}$ | $\frac{31}{106}$
---|---|---|---|---|---|---|---|---
| (1,3) | (3,4) | (3,7) | (7,10) | (7,17) | (17,24) | (24,41) | (41,65)
$n_{c}$ | 3$\sim$ | 4$\sim$ | 7$\sim$ | 10$\sim$ | 17$\sim$ | 24$\sim$ | 41$\sim$ | 65$\sim$
$\alpha_{0}$ | $0\sim\frac{1}{3}$ | $\frac{1}{4}\sim\frac{1}{3}$ | $\frac{2}{7}\sim\frac{1}{3}$ | $\frac{2}{7}\sim\frac{5}{17}$ | $\frac{2}{7}\sim\frac{5}{17}$ | $\frac{7}{24}\sim\frac{5}{17}$ | $\frac{7}{24}\sim\frac{12}{41}$ | $\frac{19}{65}\sim\frac{12}{41}$
Table 14: $\alpha_{0}=[3,2,2]=1/(3+1/(2+1/(2+\tau^{-1})))\simeq 0.2924$ ($360\alpha_{0}\simeq 105.3^{\circ}$).  $\alpha$ | $\frac{2}{5}$ | $\frac{3}{8}$ | $\frac{4}{11}$ | $\frac{5}{14}$ | $\frac{9}{25}$ | $\frac{14}{39}$ | $\frac{23}{64}$ | $\frac{37}{103}$
---|---|---|---|---|---|---|---|---
| (2,3) | (3,5) | (3,8) | (3,11) | (11,14) | (14,25) | (25,39) | (39,64)
$n_{c}$ | 3$\sim$ | 5$\sim$ | 8$\sim$ | 11$\sim$ | 14$\sim$ | 25$\sim$ | 39$\sim$ | 64$\sim$
$\alpha_{0}$ | $\frac{1}{2}\sim\frac{1}{3}$ | $\frac{1}{3}\sim\frac{2}{5}$ | $\frac{1}{3}\sim\frac{3}{8}$ | $\frac{1}{3}\sim\frac{4}{11}$ | $\frac{5}{14}\sim\frac{4}{11}$ | $\frac{5}{14}\sim\frac{9}{25}$ | $\frac{14}{39}\sim\frac{9}{25}$ | $\frac{14}{39}\sim\frac{23}{64}$
Table 15: $\alpha_{0}=[2,1,3]=1/(2+1/(1+1/(3+\tau^{-1})))\simeq 0.3593$
($360\alpha_{0}\simeq 129.3^{\circ}$).

$\alpha$ | $\frac{2}{5}$ | $\frac{3}{7}$ | $\frac{4}{9}$ | $\frac{7}{16}$ | $\frac{10}{23}$ | $\frac{17}{39}$ | $\frac{27}{62}$ | $\frac{44}{101}$
---|---|---|---|---|---|---|---|---
| (2,3) | (2,5) | (2,7) | (7,9) | (7,16) | (16,23) | (23,39) | (39,62)
$n_{c}$ | 3$\sim$ | 5$\sim$ | 7$\sim$ | 9$\sim$ | 16$\sim$ | 23$\sim$ | 39$\sim$ | 62$\sim$
$\alpha_{0}$ | $\frac{1}{3}\sim\frac{1}{2}$ | $\frac{2}{5}\sim\frac{1}{2}$ | $\frac{3}{7}\sim\frac{1}{2}$ | $\frac{3}{7}\sim\frac{4}{9}$ | $\frac{3}{7}\sim\frac{7}{16}$ | $\frac{10}{23}\sim\frac{7}{16}$ | $\frac{10}{23}\sim\frac{17}{39}$ | $\frac{27}{62}\sim\frac{17}{39}$
Table 16: $\alpha_{0}=[2,3,2]=1/(2+1/(3+1/(2+\tau^{-1})))\simeq 0.4356$ ($360\alpha_{0}\simeq 156.8^{\circ}$).  $\alpha$ | $\frac{2}{5}$ | $\frac{3}{7}$ | $\frac{5}{12}$ | $\frac{7}{17}$ | $\frac{9}{22}$ | $\frac{16}{39}$ | $\frac{25}{61}$ | $\frac{41}{100}$
---|---|---|---|---|---|---|---|---
| (2,3) | (2,5) | (5,7) | (5,12) | (5,17) | (17,22) | (22,39) | (39,61)
$n_{c}$ | 3$\sim$ | 5$\sim$ | 7$\sim$ | 12$\sim$ | 17$\sim$ | 22$\sim$ | 39$\sim$ | 61$\sim$
$\alpha_{0}$ | $\frac{1}{3}\sim\frac{1}{2}$ | $\frac{2}{5}\sim\frac{1}{2}$ | $\frac{2}{5}\sim\frac{3}{7}$ | $\frac{2}{5}\sim\frac{5}{12}$ | $\frac{2}{5}\sim\frac{7}{17}$ | $\frac{9}{22}\sim\frac{7}{17}$ | $\frac{9}{22}\sim\frac{16}{39}$ | $\frac{25}{61}\sim\frac{16}{39}$
Table 17: $\alpha_{0}=[2,2,3]=1/(2+1/(2+1/(3+\tau^{-1})))\simeq 0.4100$ ($360\alpha_{0}\simeq 147.6^{\circ}$).  $\alpha$ | $\frac{2}{5}$ | $\frac{3}{8}$ | $\frac{5}{13}$ | $\frac{7}{18}$ | $\frac{12}{31}$ | $\frac{19}{49}$ | $\frac{31}{80}$ | $\frac{50}{129}$
---|---|---|---|---|---|---|---|---
| (2,3) | (3,5) | (5,8) | (5,13) | (13,18) | (18,31) | (31,49) | (49,80)
$n_{c}$ | $3\sim$ | $5\sim$ | $8\sim$ | $13\sim$ | $18\sim$ | $31\sim$ | $49\sim$ | $80\sim$
$\alpha_{0}$ | $\frac{1}{3}\sim\frac{1}{2}$ | $\frac{1}{3}\sim\frac{2}{5}$ | $\frac{3}{8}\sim\frac{2}{5}$ | $\frac{5}{13}\sim\frac{2}{5}$ | $\frac{5}{13}\sim\frac{7}{18}$ | $\frac{12}{31}\sim\frac{7}{18}$ | $\frac{12}{31}\sim\frac{19}{49}$ | $\frac{31}{80}\sim\frac{19}{49}$
Table 18: $\alpha_{0}=[2,1,1,2]=1/(2+1/(1+1/(1+1/(2+\tau^{-1}))))\simeq 0.3876$ ($360\alpha_{0}\simeq 139.5^{\circ}$).  $\alpha$ | $\frac{1}{4}$ | $\frac{1}{5}$ | $\frac{1}{6}$ | $\frac{1}{7}$ | $\frac{2}{13}$ | $\frac{3}{19}$ | $\frac{5}{32}$ | $\frac{8}{51}$
---|---|---|---|---|---|---|---|---
| (1,3) | (1,4) | (1,5) | (1,6) | (6,7) | (6,13) | (13,19) | (19,32)
$n_{c}$ | 3$\sim$ | 4$\sim$ | 5$\sim$ | 6$\sim$ | 7$\sim$ | 13$\sim$ | 19$\sim$ | 32$\sim$
$\alpha_{0}$ | $0\sim\frac{1}{3}$ | $0\sim\frac{1}{4}$ | $0\sim\frac{1}{5}$ | $0\sim\frac{1}{6}$ | $\frac{2}{13}\sim\frac{1}{6}$ | $\frac{2}{13}\sim\frac{3}{19}$ | $\frac{5}{32}\sim\frac{3}{19}$ | $\frac{5}{32}\sim\frac{8}{51}$
Table 19: $\alpha_{0}=[6,2]=1/(6+1/(2+\tau^{-1})\simeq 0.1567$
($360\alpha_{0}\simeq 56.4^{\circ}$).

Having prepared the mathematical relationship between the initial divergence
$\alpha_{0}$, the final divergence $\alpha$ and the trace length $n_{c}$, we
are in a position to give an account of what is special about the golden
angle. As shown below in Figs. 16 and 19, discontinuous change in phyllotactic
fraction $\alpha$, or phyllotactic transition, involves reconstruction of the
vascular structure. Therefore, it is advantageous for a plant to suppress the
transitions as few as possible. As internodes vary in length during growth,
the trace length per internode $n_{c}$ may change accordingly. For instance,
$n_{c}$ may depend on the plastochron ratio $a$ (A). Patterns with a fraction
that appears in many places of Fig. 7 are stable against occasional changes in
$n_{c}$. The lowest fraction that appears more than once is $\frac{2}{5}$.
Thus, systems with initial divergence angle giving rise to stable fractions
are most likely to survive. Among all possible values of $\alpha_{0}$, the
initial divergence angle which suffers the least number of phyllotactic
transitions is the golden angle $\alpha_{0}=[2]$ (137.5 degrees). This is a
summary of the evolutionary mechanism for the golden angle (Okabe (2011)).
Fig. 8 shows phyllotactic fractions resulting from various representative
values of $\alpha_{0}$ while $n_{c}$ increases up to eleven. The number of
phyllotactic transitions is indicated by a dashed line. In this example,
initial divergence angles from 135∘ to 154∘ are most likely to be naturally
selected.

Figure 8:  Phyllotactic fractions resulting while $n_{c}$ increases to eleven
are arranged vertically for eight representative values of the limit
divergence $\alpha_{0}$. A dashed line is the number of phyllotactic
transitions counted from a $\frac{1}{2}$ phyllotaxis. Initial divergences
within a narrow range around the golden angle $\alpha_{0}=[2]\simeq 0.382$
($137.5^{\circ}$) are most likely to survive.

##  3 Precision of initial divergence angle

As the evolutionary mechanism relies on statistical screening processes, it
does not predict a limit divergence angle with unlimited precision. It is an
empirical fact that divergence angles at the level of the shoot apex are
regulated toward a mean value comparable with an ideal angle given by the
formula (2) after some transient fluctuations (Davies (1939); Snow and Snow
(1962); Barabé et al. (2010)). Excepting initial fluctuations, the precision
with which leaves are organized on the apical meristems is remarkable. It is
undoubtedly controlled by genetics, though it may be slightly affected by
light stimuli depending on the orientation (Kumazawa and Kumazawa (1971)).
Twenty samples of young shoots of Erigeron sumatrensis (Sumatran fleabane)
show mean divergence angles from 137.23∘ to 137.97∘ with the sample average of
$137.499\pm 0.212^{\circ}$ (Kumazawa and Kumazawa (1971)). The mean divergence
angle of the individual plant may deviate statistically significantly from the
ideal limit angle (Maksymowych and Erickson (1977)). Sometimes there occur
other ideal divergence angles than the normal golden angle of 137.5∘.
Phyllotaxis of Musa sapientum (banana) changes with the age of the plant from
$\frac{2}{5}$ through $\frac{3}{7}$ to $\frac{4}{9}$ (Skutch (1927)). This is
consistent with a unique initial divergence of $\alpha_{0}=[2,3]$ (Table 8),
which seems to be true for all species of Musa propagated vegetatively.
Rutishauser (1998) has presented a remarkably exotic pattern of Picea abies
(Norway spruce) showing a $(6,13)$ phyllotaxis (Table 19).

The evolutionary mechanism predicts a correlation between the range of values
of the initial divergence $\alpha_{0}$ and the highest-order fraction $\alpha$
attained in evolutionary or phylogenetic processes. The correlation may seem
strange at first glance, as it appears as an advanced correlation in
developmental or ontogenetic processes of a plant; the precision of divergence
angles on a young shoot is determined by the phyllotactic form at its
maturity. This phylogenetic correlation is contrasted with the instantaneous
correlation that causal models predict between the divergence angle and
parastichies of the standing pattern. In general, divergence angles of a
$(3,5)$ phyllotaxis are widely variable within
$\frac{1}{3}<\alpha_{0}<\frac{2}{5}$, whereas the range is narrowed to
$\frac{3}{8}<\alpha_{0}<\frac{2}{5}$ when the parastichy pair is raised to
$(5,8)$. Remark that these are general results drawn from regularity of
phyllotactic patterns. The ranges may be restricted further depending on
specific assumptions of models. For instance, consider a regular pattern with
a parastichy pair $(1,2)$, which is realized for any value of divergence
angle. According to Schwendener’s model, however, $(1,2)$ patterns for
$0<\alpha_{0}<0.36$, i.e., from 0 to 128.6 degrees, are not realized, for a
transition to a $(2,3)$ phyllotaxis intervenes at $\alpha_{0}=0.36$ (Adler
(1974); Levitov (1991); Douady and Couder (1996)). See the top branch of the
zigzag path in Fig. 2. The threshold angle $\alpha_{0}=0.36$ specifically
depends on geometrical assumptions, e.g. the circular shape of ‘leaves’ on the
stem cylinder surface. Accordingly, the divergence angles for the parastichy
pair $(1,2)$ is predicted to vary continuously within
$0.36<\alpha_{0}<\frac{1}{2}$, i.e., from 128.6∘ to 180∘. The range is
narrowed substantially but still so wide that it is incompatible with
observations that divergence angles are very close to 137.5∘ even in systems
of low phyllotaxis. Causal models attain a target pattern with 137.5∘ by way
of an almost opposite $(1,2)$ pattern with divergence of about 180∘.
Therefore, they cannot but allow the wide latitude of divergence angles for
the $(1,2)$ pattern, in disagreement with precise control of actual systems
(cf. Fig. 12). This is a very old problem which van Iterson (1907) (p. 247)
was well aware of. Nonetheless, it has been left unnoticed despite a marginal
rise of various causal models in recent years. With reference to experimental
evidence, Church (1904) (p. 340) remarks that already at a $(2,3)$ system the
ideal angle is attained within an error of about one degree. The present model
explains the non-correspondence between divergence angle $\alpha_{0}$ and
parastichy numbers by relating the allowed range of $\alpha_{0}$ not with the
parastichy numbers but with the highest order fraction $\alpha$ that the plant
would attain in its mature state.

By measuring initial divergence angles for thirty species of plants, Fujita
(1939) found that frequency distributions of the divergence angles are almost
independent of the parastichy numbers. The divergence angles cluster in a
narrow range. The width of the range quantifies the remarkable constancy of
the divergence angle (Fig. 12). This results look puzzling from a causal
viewpoint (Jean (1986, 1994)). By contrast, they are consistent with the
evolutionary mechanism in that the initial angle $\alpha_{0}$ is independent
of the parastichy numbers. According to Fujita (1939), initial divergence
angles for the main sequence fall within $138\pm 7^{\circ}$ (Fujita (1939)),
irrespective of the parastichy pair. This corresponds to
$\frac{3}{8}<\alpha_{0}<\frac{2}{5}$ (135 to 144 degrees), which is as
expected for the highest-order fraction of $\alpha=\frac{5}{13}$ (Table 1).
Similarly, an estimate of $99\pm 4^{\circ}$ for Cunninghamia lanceolata (China
fir) (Fujita (1939)) is consistent with $\frac{1}{4}<\alpha_{0}<\frac{2}{7}$
for $\alpha=\frac{3}{11}$ in Table 2, and a narrow scattering of $151\pm
3^{\circ}$ for $(2,5)$ phyllotaxis at the apex of Cephalotaxus drupacea
(Japanese plum yew) (Fujita (1937)) is consistent with
$\frac{5}{12}<\alpha_{0}<\frac{3}{7}$ for $\alpha=\frac{8}{19}$ (Table 6).

Let us make a general remark that parastichy does not substitute for
divergence angle. The former depends on size and shape of the pattern unit or
on a radial or internodal length scale. Therefore, several different
parastichy pairs may be arbitrarily related to a single divergence angle.
Parastichy numbers given in Tables 1$\sim$18 are the simplest pairs, which
normally represent contact parastichies.

Large fluctuations in the initial divergence $\alpha_{0}$ may cause the
phyllotactic transition in the vascular structure, even if the trace length
$n_{c}$ is fixed constant. To suppress the transition that could happen, the
divergence $\alpha_{0}$ has to be restricted within one of the ranges
determined by $n_{c}$. For a fixed length of $n_{c}=5$, the fraction $\alpha$
is plotted against the initial divergence $\alpha_{0}$ in Fig. 9. To maintain
a $\frac{3}{8}$ phyllotaxis, the initial divergence $\alpha_{0}$ must stay
within $\frac{1}{3}<\alpha_{0}<\frac{2}{5}$ (from $120^{\circ}$ to
$144^{\circ}$); otherwise one would observe occasional excursions to
$\frac{2}{7}$ (for $\alpha_{0}<\frac{1}{3}$) or $\frac{3}{7}$ (for
$\frac{2}{5}<\alpha_{0}$) in the midst of a steady course of the $\frac{3}{8}$
phyllotaxis. Similarly, to maintain a $\frac{5}{13}$ phyllotaxis, the initial
divergence $\alpha_{0}$ has to be kept within
$\frac{3}{8}<\alpha_{0}<\frac{2}{5}$ (from $135^{\circ}$ to $144^{\circ}$);
otherwise one would find $\frac{4}{11}$ (for $\alpha_{0}<\frac{3}{8}$) or
$\frac{5}{12}$ (for $\frac{2}{5}<\alpha_{0}$) within the mature state of the
$\frac{5}{13}$ phyllotaxis (cf. Fig. 8). Thus, it is explained why the initial
divergence angle has to be ‘quantized’ or fixed around a special constant with
precision determined by the length of leaf traces. For this mechanism to work,
stepwise changes in the fraction $\alpha$ of the vascular order, which are
presumed to occur if the initial divergence angle $\alpha_{0}$ were not
optimum, should incur penalties of extra energy. Thus, efficiency of the
mechanism depends on the energy cost per transition, which should depend on
species. By and large, however, the number of transition may be used as a good
measure of the total cost, at least as a first approximation (Fig. 8).

Figure 9:  Phyllotactic fraction $\alpha$ versus initial divergence angle
$\alpha_{0}$ for a fixed length $n_{c}=5$ of leaf traces. The fractional order
changes discontinuously while $\alpha_{0}$ changes continuously. There are
plateaus for five phyllotactic orders with $\frac{1}{6}$, $\frac{2}{9}$,
$\frac{2}{7}$, $\frac{3}{8}$ and $\frac{3}{7}$. The initial divergence
$\alpha_{0}$ is ‘quantized’ within a plateau to avoid the discontinuous
transition. In other words, leaves are initiated regularly with a given
angular precision. The wide plateau for $\alpha=\frac{1}{6}$ is the most
unstable against changes in $n_{c}$, while the plateau at $\alpha=\frac{3}{8}$
is the most stable.

##  4 Fossil record and diversity of phyllotaxis

Dickson (1871) found that nine among thirteen specimens of fossil remains of
Lepidodendron (scale tree) show helical phyllotaxis, of which only three
belong to the main sequence. This is in striking contrast to the current
dominance of the main sequence in existing species (Fujita (1938); Zagórska-
Marek (1985); Jean (1994)). Therefore, Dickson concluded that the phyllotaxis
of Lepidodendron is extremely variable, as much so as that of those most
variable plants like cacti. His results provide us with important information
when they are analyzed in terms of the model.

In the second and third line of Table 20, the phyllotactic fractions and the
parastichy pairs for the nine specimens are presented after Dickson. The
fourth and fifth line are the corresponding ranges of $n_{c}$ and $\alpha_{0}$
according to Tables 1, 2, 3, 6, and 9. The last line is the limit divergence
in terms of the bracket notation defined by (2) in the last section. For
instance, in the second column, the specimen No. 1 has a $\frac{13}{34}$
phyllotaxis $(\alpha=\frac{13}{34})$ with the parastichy pair of $(13,21)$,
for which $21\leq n_{c}<34$ and $\frac{8}{21}<\alpha_{0}<\frac{5}{13}$. The
limit divergence of $\alpha_{0}=[2]$ (137.5∘) satisfies the latter condition.
The specimens Nos. 1-3 belong to the main sequence $\alpha_{0}=[2]$. Fig.
10(a) represents graphically the parameter regions allowed for $n_{c}$ and
$\alpha_{0}$. By comparison, Fig. 10 gives a theoretical result for the most
favored regions in which the number of phyllotactic transitions is minimal
(Okabe (2011)).

No. | 1 | 2 | 3 | 8 | 9,10 | 11 | 12 | 13
---|---|---|---|---|---|---|---|---
$\alpha$ | $\frac{13}{34}$ | $\frac{21}{55}$ | $\frac{55}{144}$ | $\frac{21}{76}$ | $\frac{13}{60}$ | $\frac{21}{50}$ | $\frac{18}{59}$ | $\frac{47}{154}$
| (13,21) | (21,34) | (55,89) | (29,47) | (23,37) | (19,31) | (23,36) | (59,95)
$n_{c}$ | [21, 34) | [34,55) | [89,144) | [47,76) | [37,60) | [31,50) | [36,59) | [95,154)
$\alpha_{0}$ | $\frac{8}{21}\sim\frac{5}{13}$ | $\frac{8}{21}\sim\frac{13}{34}$ | $\frac{21}{55}\sim\frac{34}{89}$ | $\frac{8}{29}\sim\frac{13}{47}$ | $\frac{8}{37}\sim\frac{5}{23}$ | $\frac{13}{31}\sim\frac{8}{19}$ | $\frac{7}{23}\sim\frac{11}{36}$ | $\frac{18}{59}\sim\frac{29}{95}$
| [2] | [2] | [2] | [3] | [4] | [2,2] | [3,3] | [3,3]
Table 20:  Ranges of $n_{c}$ and $\alpha_{0}$ for the phyllotactic fraction
$\alpha$ and the contact parastichy pair $(n,m)$ of the nine specimens of
Lepidodendron by Dickson (1871). Abbreviations $[21,34)$ and
$\frac{8}{21}\sim\frac{5}{13}$ mean $21\leq n_{c}<34$ and
$\frac{8}{21}<\alpha_{0}<\frac{5}{13}$. The bracket notation in (2) is used
for the limit divergence in the last row. Only the first three specimens
belong to the main sequence $\alpha_{0}=[2]$ (137.5∘).

(a)

Figure 10:  (a) Ranges of $\alpha_{0}$ and $n_{c}$ for the specimens of
Dickson (1871) (Table 20) are painted black in the $\alpha_{0}$-$n_{c}$ plane.
(b) A theoretical result for the regions in which the number of phyllotactic
transitions is minimal (adapted from Fig. 13 of Okabe (2011)). The golden
angle $\alpha_{0}\simeq 0.382$ (137.5∘) is singled out for $n_{c}$ below
Fibonacci numbers such as 34, 55, 89 and 144.

According to Fig. 10(a), the trace length $n_{c}$ appears to be independent of
the initial divergence $\alpha_{0}$. Moreover, $n_{c}$ is not as variable as
$\alpha_{0}$. As the order of phyllotaxis is very high, there is considerable
uncertainty in $n_{c}$, while $\alpha_{0}$ is quite accurate. The specimens
may be divided into two groups in terms of $n_{c}$, i.e., one with $n_{c}\sim
50$ and the other with $n_{c}>100$. The fact that the fossil specimens show
various but accurate values of $\alpha_{0}$ strongly suggests the evolutionary
origin of the special divergence angles. It is impossible to tabulate all
phyllotactic fractions for such a large value as $n_{c}=50$ due to lack of
space, but it is mentioned only that the number of possible phyllotactic
fractions at $n_{c}=50$ amounts to 387 $(\simeq 3n_{c}^{2}/\pi^{2}/2)$. Among
them, only the single fraction $\alpha=\frac{21}{55}$ of the main sequence
falls in the optimum regions depicted in Fig. 10, while the specimens Nos.
8-12 do not meet the optimum condition. Nevertheless, all the reported
specimens possess the irrational numbers expressed in the form of (2), as
expected in the evolutionary mechanism (cf. Table 2 of Okabe (2011)).
Phyllotactic patterns for $\alpha_{0}=[3,2]$, $[2,3]$, $[2,1,2]$ and others
are not reported, presumably because of lack of enough samples. Thus,
anomalous patterns are regarded as relics of evolutionary processes.

Figure 11:  Three-dimensional fitness landscape. The vertical axis
representing ‘fitness’ is $n_{c}$ minus the number of phyllotactic transition.
The variables on the base plane are $\alpha_{0}$ and $n_{c}$. The number of
transition increases with $n_{c}$. The fitness has a flat bottom minimum in
the worst case of $\alpha_{0}\simeq 0$, whereas there are ‘fitness peaks’ at
$\alpha_{0}=[2]\simeq 0.38$ (the main sequence), $\alpha_{0}=[3]\simeq 0.28$
(an accessory sequence) and others, whose widths decrease as $n_{c}$ increases
(Okabe (2011)).

Figure 12:  (a) Frequency distribution of initial divergence angles for a
$(1,2)$ phyllotaxis of Lysimachia clethroides by Fujita (1939). (b) An
enlarged view of a relative ‘fitness’ in Fig. 11 is plotted against the
divergence angle in degrees, $360\alpha_{0}$. The peak plateau extends from
135 to 144 degrees at $n_{c}=12$, and from 135 to 138 degrees at $n_{c}=19$.

It has been an unresolved problem in what quantitative terms normal and
anomalous phyllotaxis are differentiated. The number of phyllotactic
transition during a steady growth provides us with a numerical measure of
relative fitness in evolution. The most fit divergence angles are indicated in
Fig. 10. They are peaks of a ‘fitness landscape’ (Niklas (1997)), shown in
Fig. 11 (Okabe (2011)). A close inspection of the frequency distribution
curves of Fujita (1939) indicates that a primary peak accompanies small
subsidiary peaks at anomalous angles. In Fig. 12, Fujita’s result for
Lysimachia clethroides (gooseneck loosestrife) is arranged along with
transections of the fitness landscape in Fig. 11. Roberts (1984) has discussed
that his chemical contact pressure model explains the anomalous subsidiary
peaks. However, his conclusion is based on circular reasoning that anomalous
systems are less frequent because they are anomalous. Similar fitness curves
are obtained for light absorption efficiency of rosette plants (Niklas (1988,
1998); Pearcy and Yang (1998); King et al. (2004)).

Let us remark incredible precision of the divergence angle. As already
mentioned, it is no less astonishing than the widely noticed fact that
divergence angles converge on one of the special irrational numbers. Let us
take the specimen No. 13 as an example. The divergence angle of the
$\frac{47}{154}$ phyllotaxis is a rational number $360\alpha\simeq
109.870^{\circ}$. This is very close to an irrational, ideal angle of
$\alpha_{0}=[3,3]$, or $360\alpha_{0}\simeq 109.877^{\circ}$. According to
Table 20, the range of $\alpha_{0}$ for the $\frac{47}{154}$ phyllotaxis is
very narrow, that is,

$109.831^{\circ}<360\alpha_{0}<109.895^{\circ},$ (3)

or $360\alpha_{0}\simeq 109.863\pm 0.032$ degrees. The relative precision is
less than about a part per three thousand. For reference, we present results
that would be obtained if $\alpha_{0}$ happens to be off the narrow range of
(3). Instead of $\frac{47}{154}$ and the parastichy pair $(59,95)$ for (3), we
would have obtained $\frac{43}{141}$ and $(59,82)$ if $\alpha_{0}$ were
slightly below the lower limit of (3), or $\alpha=\frac{40}{131}$ and
$(36,95)$ if $\alpha_{0}$ were above the upper limit of (3). Neither of the
last two cases is listed in Tables 1$\sim$18, for they are hardly ever likely
to occur. The plants’ ability to distinguish $\frac{47}{154}$ from
$\frac{43}{141}$ and $\frac{40}{131}$ is due to high precision regulation of
initial divergence angle. The range width of $\alpha_{0}$ depends not so much
on $\alpha_{0}$ as on $n_{c}$. Indeed, we find
$\Delta\alpha_{0}\simeq(\tau/n_{c})^{2}$ according to Eq. (B.39) in Okabe
(2011). The precision as high as the above cannot be attained by a limited
number of cells on the apex (Koch et al. (1998); Meinhardt et al. (1998);
Smith et al. (2006a)). It seems very unlikely that existing causal models can
explain this anomalous phyllotaxis with this precision in this probability of
one out of thirteen specimens.

Diversities of phyllotaxis is considered as a result of selective pressures
being ineffective. In extant plants, the main Fibonacci phyllotaxis is
dominant while some species specifically show very diverse phyllotaxis
(Zagórska-Marek (1994)). In general, the trait diversity will be reduced if
there is selective pressure acting on it. Strength of selective pressure
depends specifically on extra cost required while rearranging phyllotactic
patterns of leaf traces during growth of individual plants. Accordingly, the
diversity may be preserved for some reason or other, e.g., when leaf traces
are so fragile that the energy cost of rearrangement is insignificant. This
view is consistent with recent research on Licopodium revealing a link between
variability of leaf traces and diversity of phyllotaxis (Gola et al. (2007)).
In contrast, the diversity in phyllotaxis of scale trees is considered as a
result of strong selective pressure of insufficient time durations, strong
because divergence angles are highly accurate whereas insufficient because
various angles besides 137.5∘ are still in existence. In discussing diversity
of phyllotaxis, one should make a clear distinction between the variance, or
standard deviation, of divergence angle of an individual and varieties of
divergence angles of individuals. This section was devoted to the latter,
while the former was discussed in the last section.

##  5 Phyllotaxis and vascular organization

Figure 13:  Diagrams of the primary vascular system of $\frac{5}{13}$ (left),
$\frac{5}{18}$ (center), $\frac{8}{21}$ (right) phyllotaxis of Linum
usitatissimum. In the inset (top left), dashed lines mark off five parastichy
sectors in a transverse section of the $\frac{5}{13}$ phyllotaxis stem.
Adapted from Girolami (1953).

Girolami (1953) investigated the relation between phyllotaxis and vascular
organization of Linum (flax), whose vascular structures of a $\frac{5}{13}$,
$\frac{5}{18}$ and $\frac{8}{21}$ phyllotaxis are given in the left, center
and right of Fig. 13, respectively. On the one hand, the genetic spirals of
the $\frac{5}{13}$ and $\frac{5}{18}$ phyllotaxis wind up to the right
(counterclockwise), while it goes to the left (clockwise) for the
$\frac{8}{21}$ phyllotaxis. On the other hand, the main parastichies of the
three patterns run in the same direction. That is to say, 5-parastichies for
$\frac{5}{13}$ (1-6-11-16-21, etc.), 7-parastichies for $\frac{5}{18}$
(1-8-15-22-29, etc.) and 8-parastichies for $\frac{8}{21}$ (1-8-15-22-29,
etc.) run steeply from the bottom right to top left (clockwise). The most
direct vascular connection goes along the main parastichies. The vascular
bundles of these parastichies are recognized as sectioned clusters in a
transverse section of the stem, called parastichy sectors. As shown in the
inset of Fig. 13, the $\frac{5}{13}$ phyllotaxis stem is divided into five
parastichy sectors. In what follows, the following points remarked by Girolami
(1953) are analyzed in terms of the model, whereby some useful general rules
are pointed out:

(G1) The number of the parastichy sectors (5, 7 and 8 for $\frac{5}{13}$,
$\frac{5}{18}$ and $\frac{8}{21}$, respectively) agrees with the numerator of
the phyllotactic fraction for $\frac{5}{13}$ and $\frac{8}{21}$ of the main
sequence, but not for $\frac{5}{18}$ of the accessory sequence.

(G2) The length of leaf traces per internode increases with the number of
parastichy sectors, namely 12 for $\frac{5}{13}$, 17 for $\frac{5}{18}$, and
19 for $\frac{8}{21}$ approximately.

(G3) There is no correlation in the relative directions of the genetic spiral
and the parastichies.

As noted in the first point, there is no easy-to-use general formula between
the parastichy numbers and the phyllotactic fraction (see below however), but
the numerical correspondence is immediately read from Fig. 7 and Tables
1$\sim$18. According to Table 2, the parastichy pair of the $\frac{5}{18}$
phyllotaxis is $(7,11)$. The number of parastichy sectors is the small number
of the parastichy pair. Therefore, the number 7 of the parastichy sectors of
the $\frac{5}{18}$ phyllotaxis is obtained. Unlike the numerator, the
denominator satisfies simple rules. Most notably, the denominator of a
fraction is equal to the sum of the contact parastichy pair corresponding to
the fraction (e.g. $18=7+11$). Mathematical relations between various numbers
in phyllotaxis have been investigated since early times on an empirical ground
based on purely mathematical properties of a regular lattice (Bravais and
Bravais (1837); Naumann (1845); Jean (1994)).

On the second point, Tables 1 and 2 give the conditions $8\leq n_{c}<13$,
$11\leq n_{c}<18$ and $13\leq n_{c}<21$ for the phyllotactic fractions
$\frac{5}{13}$, $\frac{5}{18}$ and $\frac{8}{21}$, respectively. The
predictions of the model are supported by the reported values $n_{c}=12,17$
and 19, which satisfy their respective conditions near their upper limits.
Nevertheless, a close look at Fig. 13 indicates that these figures are not
accurate. As a matter of fact, $n_{c}$ appears not constant but somewhat
larger in the upper part of the stem. Changes in length of the leaf traces are
revealed in a more sophisticated analysis of Meicenheimer (1986), where
progressive transitions from $\frac{1}{3}$ through $\frac{2}{5}$ and
$\frac{3}{8}$ up to $\frac{5}{13}$ have been reported. Phyllotactic transition
caused by changes in $n_{c}$ is discussed in the next section.

Figure 14:  Three branches from Fig. 7 with which to explain spiral directions
of $\frac{5}{13}$, $\frac{5}{18}$, $\frac{8}{21}$ phyllotaxis. As the fraction
$\frac{5}{18}$ and $\frac{5}{13}$ are numerically bigger than their ‘mother’
fraction $\frac{3}{11}$ and $\frac{3}{8}$, their main parastichies of 7 and 5
are contrary in direction to the genetic spiral. On the contrary, 8
parastichies for $\frac{8}{21}$ are in the same direction as the genetic
spiral.  Figure 15:  When the final divergence $\alpha$ is numerically bigger
than the initial divergence $\alpha_{0}$, the stem is twisted in the direction
of the genetic spiral. N.B. Divergence angles are less than 180 degrees.

On the third point, a general rule holding between directions of parastichies
and the genetic spiral is presented based on Fig. 7. To this end, it is
convenient to introduce a ‘mother’ fraction of a fraction $\alpha$, which is
defined as the fraction lying immediately below the fraction $\alpha$ in the
tree of Fig. 7. The mother fractions of $\frac{5}{13}$, $\frac{5}{18}$ and
$\frac{8}{21}$ are $\frac{3}{8}$, $\frac{3}{11}$ and $\frac{5}{13}$,
respectively. It is shown that if and only if a phyllotactic fraction $\alpha$
is numerically bigger than its mother fraction, the main parastichies run in
the direction opposite to the genetic spiral. (The main parastichies are
gentle, long spirals characterized by the small number of the contact
parastichy pair.) The fraction $\alpha=\frac{5}{18}$ and $\frac{5}{13}$ are
bigger than the mother fraction $\frac{3}{8}$ and $\frac{3}{11}$,
respectively, while $\alpha=\frac{8}{21}$ is smaller than the mother fraction
$\frac{5}{13}$. The magnitude relations are schematically shown in Fig. 14
extracted from Fig. 7. Thus, the above rule explains Girolami’s observation
consistently. In practice, this rule may be used to identify the direction of
the genetic spiral of a high order phyllotactic pattern for which parastichies
are far easy to follow visually. Some special cases of this general rule have
been remarked (Church (1904)(p. 96), Namboodiri and Beck (1968)) and
occasionally taken up for discussion (Meicenheimer (1986); Fredeen et al.
(2002)). The directional relations between various spirals of a phyllotactic
pattern are also mathematical consequences of the regularity of the
phyllotactic pattern.

The mother fraction enables us to state general rules for the phyllotactic
fraction and the parastichy number: One of the parastichy pair for a fraction
$\alpha$ is equal to the denominator of the mother fraction of $\alpha$; The
other number in the pair is determined such that the sum of the pair is equal
to the denominator of $\alpha$. Consider $\alpha=\frac{5}{18}$, for instance.
One of its parastichy pair is the denominator 11 of the mother fraction
$\frac{3}{11}$, while the other is the difference of the denominators,
$18-11=7$. As a result, the parastichy pair $(7,11)$ is obtained for
$\frac{5}{18}$. Thus, the rules are used to relate the parastichy numbers and
the phyllotactic fraction.

The vascular systems shown in Fig. 13 form closed networks. In each system,
connections between leaf traces are formed along both the paired parastichies,
so that the vascular bundles are divided into parastichy sectors. Among
dicotyledons with helical phyllotaxis, however, an open vascular system is
rather common (Beck et al. (1982)). Primitive angiosperms and many gymnosperms
have open vascular systems (Beck (2010)). According to Beck et al. (1982),
open systems of five sympodia (a $\frac{2}{5}$ phyllotaxis) characterize 67%
of the species with helical phyllotaxy and are clearly a common type among
dicotyledons. In an open system, leaf traces are connected along one
direction. Although the present model determines the basic architecture of
vascular phyllotaxis, it does not specify detailed structure of the reticulate
pattern, whether it remains open or becomes closed. This is not a shortcoming
of the model, because actual linkages between leaf traces are likely to be
secondary events depending on circumstances (Kang et al. (2003)).

To conclude this section, let us remark another obvious correlation between
the direction of the genetic spiral and the secondary torsion of the stem. The
initial divergence $\alpha_{0}$ is related to the fractional divergence
$\alpha$ of a mature pattern by the angle of twist $\alpha-\alpha_{0}$
undergone in the secondary torsion. The direction of the torsion is the same
as the genetic spiral if and only if $\alpha>\alpha_{0}$. This is shown
schematically in Fig. 15. The direction of the secondary torsion would not be
difficult to check experimentally. In most typical cases, the direction is
reversed, or the sign of $\alpha-\alpha_{0}$ changes, as $n_{c}$ crosses a
threshold of phyllotactic transition. Bravais and Bravais (1837) evaluated the
limit divergence $\alpha_{0}$ from mature shoots by correcting the torsion
angle $\alpha-\alpha_{0}$.

##  6 Phyllotactic transition

Figure 16:  Transition in the primary vascular system of a cottonwood plant
from a $\frac{2}{5}$ to $\frac{3}{8}$ phyllotaxis. Central, right and left
traces are indicated with crosses, filled and open triangles, respectively.
After Larson (1977).

Larson (1977) has investigated phyllotactic transition in the vascular system
of Populus (cottonwood). His result showing transition from a $\frac{2}{5}$ to
$\frac{3}{8}$ phyllotaxis is reproduced in Fig. 16. Each leaf has three
traces; central, right and left traces are indicated with crosses, filled and
open triangles, respectively. The leaf traces are connected with the stem
vascular bundles to make sympodia. The sympodia are separated from each other,
or the vascular system is open. The three traces leading to each leaf
primordium arise on different sympodia. The number of the sympodia changes
from five in the lower portion to eight in the upper portion of Fig. 16. The
number agrees with the denominator of the phyllotactic fraction in each part.
The region of the $\frac{2}{5}$ phyllotaxis occurs in the basal stem above
some primary leaves, while the $\frac{3}{8}$ phyllotaxis occurs at mid and
upper stem levels, principally in the zone of expanding leaves (Larson
(1977)). In Fig. 16, once the transition is initiated at a point IA on a
sympodium number 2, it progresses through the sympodia at points IB through
IE. Three new central traces to establish the three additional sympodia of the
$\frac{3}{8}$ system are derived from left traces in sequence at points IIA-
IIC. Various interrelations between phyllotaxis and leaf development have been
studied (Larson (1980)). In what follows, a correlation between phyllotactic
transition and lengths of the leaf traces is analyzed by means of the model,
whereby supporting evidence of the model is pointed out.

Figure 17:  Length per internode of the leaf traces in Fig. 16 is plotted
against the leaf node number, the vertical axis of Fig. 16. Arrows for a
transition region between the $\frac{2}{5}$ and $\frac{3}{8}$ phyllotaxis and
labels IA-IE and IIA-IIC to indicate initiation of the transition are marked
in accordance with Fig. 16 by Larson. The phyllotactic transition is
consistent with the threshold value of $n_{c}=5$ predicted by the model (Table
1).

The lengths per internode of the leaf traces are optically read from Fig. 16
and plotted in Fig. 17. Arrows indicating the transition region between the
$\frac{2}{5}$ and $\frac{3}{8}$ phyllotaxis in Fig. 17 are marked in
accordance with Fig. 16 after Larson (1977). By comparison, a dashed line at
$n_{c}=5$ is drawn to indicate the theoretical threshold between the
$\frac{2}{5}$ and $\frac{3}{8}$ phyllotaxis (Table 1). In accordance with the
model, the phyllotactic transition is triggered by the increasing length of
the leaf traces crossing a threshold value of five internodes.

According to Table 1, phyllotactic transition is predictable. Transitions of
the main sequence occur whenever the trace length $n_{c}$ crosses Fibonacci
numbers. The trace length, like other parameters of the plant, is predictably
correlated with plant vigor (Larson (1980)). Therefore, in principle, the
model allows us to control phyllotaxis artificially. In Sec. 2, leaf traces
are assumed to have a common length. As noted at the end of the last section,
the direction of the secondary torsion is reversed when $n_{c}$ crosses a
threshold value, so that it may be fixed by a leaf trace of length longer than
the threshold. Fig. 18 schematically shows that long leaf traces 10, 11 and 12
trigger a transition from $\frac{2}{5}$ to $\frac{3}{8}$. In the transition
region of Fig. 17, three left traces (open triangles) of the node number 7, 8
and 9 are the first to cross the threshold at $n_{c}=5$. These are the very
traces labeled with IIA, IIB and IIC by Larson as those from which the three
extra sympodia branch. A close look at Fig. 16 reveals that central traces
below and above the transition region are inclined in the opposite direction.
This is consistent with the prediction of the model, for
$\alpha=\frac{3}{8}<\alpha_{0}<\frac{2}{5}$. Furthermore, five right traces
(filled triangles) striking around $n_{c}\simeq 6$ in Fig. 17 agree with the
special traces labeled with IA through IE. Thus, the observation supports the
special role of the Fibonacci number 5 for the trace length $n_{c}$.

Figure 18:  A phyllotactic pattern with $\alpha_{0}=1/(1+\tau)$ (cf. Fig. 3).
Length of leaf traces is $n_{c}=4$ (solid bars) except for 10, 11 and 12 with
$n_{c}=6$ (bold bars). The longer traces can induce a transition from
$\alpha=\frac{2}{5}$ in the lower portion (cf. Fig. 3) to $\alpha=\frac{3}{8}$
in the upper portion (cf. Fig. 5). At the transition, the longer traces
deflect main parastichies (dotted lines), and the parastichy number increases
from 5 to 8.  Figure 19:  Reconstructed vascular system of a cottonwood plant
showing transition from $\frac{2}{5}$ through $\frac{3}{8}$ to $\frac{5}{13}$
phyllotaxis by Larson (1977). See Fig. 16 for symbols.  Figure 20:  Length per
internode of the leaf traces read from Fig. 19 is plotted against the leaf
index (the vertical axis of Fig. 19). Two arrows at the top indicate where the
$\frac{3}{8}$ phyllotaxis starts and ends according to Larson (1977) (see Fig.
19). According to the theoretical model, stable regions for the $\frac{2}{5}$,
$\frac{3}{8}$ and $\frac{5}{13}$ phyllotaxis are separated by horizontal
dashed lines at Fibonacci numbers 3, 5 and 8 (cf. Table 1). Thus, Larson’s
estimate of the region of the $\frac{3}{8}$ phyllotaxis agrees with the
theory. Left traces (open triangles) reaching a maximum length of about 10
internodes is consistent with an observation that the highest-order
phyllotactic fraction that this plant attains is $\frac{5}{13}$.

Two-step transition from a $\frac{2}{5}$ to $\frac{5}{13}$ phyllotaxis is
shown in Fig. 19 after Larson (1977), where steady increase in length of leaf
traces is more obvious than Fig. 16. Fig. 20 is obtained from Fig. 19 in the
same manner as Fig. 17 is obtained from Fig. 16. Leaf positions at which the
$\frac{3}{8}$ phyllotaxis starts and ends are marked on the right side of Fig.
19 by Larson (1977), according to which the transient pattern of the
$\frac{3}{8}$ phyllotaxis is maintained for the leaves with plastochron index
from 5 to $-7$. Accordingly, the corresponding positions are marked by arrows
in Fig. 20. On the other hand, horizontal lines at Fibonacci numbers 3, 5, and
8 in Fig. 20 theoretically divide the regions for the $\frac{1}{3}$,
$\frac{2}{5}$, $\frac{3}{8}$ and $\frac{5}{13}$ phyllotaxis (Table 1). Thus,
it is confirmed again that continuous changes in length of leaf traces cause
discontinuous transitions in the vascular structure.

Fig. 20 indicates that $n_{c}$ increases steadily up to an upper bound of
about 10. This observation is consistent with the fact that the $\frac{5}{13}$
phyllotaxis was the stable pattern of the old plant (Larson (1980)). According
to the model, the $\frac{5}{13}$ phyllotaxis is stable insofar as $n_{c}$ lies
between 8 and 13, i.e., there is a 5-internode allowance for the trace length
of the $\frac{5}{13}$ phyllotaxis. The main sequence is special for this wide
clearance between successive threshold values. The interval is denoted as
$\Delta n_{c}$ in Okabe (2011). As shown there, the widest clearances are
achieved for Fibonacci numbers, and a sequence of Fibonacci numbers is
realized when the limit divergence angle is one of the special irrational
numbers related to the golden ratio. As shown in Fig. 8, the number of
transitions encountered while $n_{c}$ grows up to above 10 is kept to a
minimum number insofar as the initial divergence is restricted within
$\frac{3}{8}<\alpha_{0}<\frac{3}{7}$ (from 135∘ to 154∘, as noted at the end
of Sec. 2). When $n_{c}$ becomes larger than 12, the range is narrowed to
$\frac{3}{8}<\alpha_{0}<\frac{2}{5}$ (from 135∘ to 144∘). Thus, the normal
phyllotaxis of the main sequence is singled out. Owing to the observation that
the highest-order fraction was $\alpha=\frac{5}{13}$, the model predicts that
the initial divergence $\alpha_{0}$ should be contained within
$\frac{3}{8}<\alpha_{0}<\frac{2}{5}$, just as observed by Fujita (1939) for
other species (Sec. 2). Unfortunately, initial divergences of the cottonwood
plant are not available to us. To support this argument, Puławska (1965) has
reported for Actinidia arguta (hardy kiwi) that initial divergence remains
constant despite changes in the vascular organization between
$\frac{3}{8},\frac{5}{13}$ and $\frac{8}{21}$.

When $n_{c}$ is increased past 8, the model predicts vascular phyllotaxis of
either $\alpha=\frac{5}{13}$ or $\alpha=\frac{5}{12}$ depending on whether the
initial divergence $\alpha_{0}$ is smaller or larger than $\frac{2}{5}$ (angle
of 144∘). Suppose $\alpha_{0}=[3]$ (99.5∘), then one should have five
threshold lines at 2, 3, 4, 7 and 11 (Table 2), instead of four thresholds at
2, 3, 5 and 8 for $\alpha_{0}=[2]$ in Fig. 20. If the initial divergence were
$\alpha_{0}=[5]$ (64.1∘ in Table 4), one should have six threshold lines at
$n_{c}=2$, 3, 4, 5, 6 and 11 separating patterns of
$\alpha=\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},\frac{1}{6},\frac{2}{11}$
and $\frac{3}{17}$ (cf. Fig. 8). The vascular phyllotaxis is very unstable.
The instability is energetically unfavorable. Therefore, $\alpha_{0}=[5]$
(64.1∘) is very improbable to survive natural selection because of the
multiplicity of expected transitions. A general remark should be made when
discussing multiple patterns in sequence. In order for a pattern with a
definite value of $\alpha$ to be distinguished as such, the pattern should
consist of more leaves than the denominator of the fraction $\alpha$. This
holds true if $n_{c}$ varies sufficiently gradually; otherwise phyllotaxis
transition may not be distinctly discernible.

Last but not least, whorled phyllotaxis has not been discussed in this paper.
A $J$-jugate pattern with $J$ fundamental spirals is formed when $J$ leaves
are borne at each node. Compared with a helical phyllotaxis with $J=1$,
divergence angles of a $J$-jugate system are divided by $J$ and the parastichy
pairs $(m,n)$ are multiplied by $J$. Therefore, one obtains
$0<J\alpha_{0}<\frac{1}{2}$ and $J(m,n)$ for the divergence angle and
parastichy pair of a $J$-jugate system. It is known that sometimes vascular
structure may change between helical and whorled phyllotaxis during ontogeny.
This type of ‘anomalous’ phyllotactic transition also appears to be caused by
a decrease in length of leaf traces (Jensen (1968); Beck et al. (1982);
Kwiatkowska (1995)). The present model gives $\alpha=\frac{1}{2}$ for $1\leq
n_{c}<2$ and $\alpha=\frac{1}{3}$ for $2\leq n_{c}<3$ irrespective of
$\alpha_{0}$. Correspondingly, it seems natural to consider that a whorled
phyllotaxis is a variation of the most primitive alternate phyllotaxis and
that a whorled phyllotaxis is triggered as $n_{c}$ becomes less than 1.
However, changes in the vascular structure have to be coordinated with changes
in the positioning of initiated leaf primordia while a whorled pattern is
established (Zagórska-Marek (1994); Meicenheimer (1998); Kelly and Cooke
(2003)). The physiological processes involved are unlikely to be amenable to
simple mathematical analysis. Still, a similar transition rule as a helical
pattern should hold for an established whorled pattern in terms of trace
length redefined with a new internode.

##  7 Conclusions

The present work puts forward an important role of Fibonacci numbers as
critical values of the length per internode of leaf traces played in vascular
phyllotaxis transition.

The regular arrangement of leaves and the regularity in divergence angle of
137.5∘ are a result of selective pressure to reduce possible changes in the
vascular structure during growth, i.e., aperiodic arrangements will
necessitate extra nutrients to reconstruct the sectorial or fractional order
of vascular connections.

The phyllotactic fraction $\alpha$ of mature patterns of leaf traces normally
makes transitions through $\frac{1}{2}$, $\frac{1}{3}$, $\frac{2}{5}$,
$\frac{3}{8}$, $\frac{5}{13}$, $\frac{8}{21}$, $\cdots$, whenever the number
of internodes traversed by the leaf traces, $n_{c}$, crosses Fibonacci
numbers, 1, 2, 3, 5, 8, 13, 21, $\cdots$. The Fibonacci numbers make
appearances because initial divergence angle $\alpha_{0}$ of leaves at the
shoot apex is normally the golden angle of about $137.5^{\circ}$ with a good
precision. The golden angle is prevalent because it is the selectively
advantageous angle at which the number of the phyllotactic transition is the
minimum (Fig. 8). The precision of the initial divergence is determined by the
trace length $n_{c}$.

## Acknowledgement

The author would like to thank Prof. Rolf Rutishauser for valuable comments on
Picea abies and others. He would like to thank Prof. Beata Zagórska-Marek for
informing him about a different view on divergence angle.

##  Appendix A Relation between the trace length $n_{c}$ and the plastochron
ratio $a$

A point on a cylinder surface is located with the angular coordinate $\varphi$
and the height $z$. Leaves on a stem are represented by a lattice of points
given by $\varphi=2\pi\alpha n$ (in radians) and $z=hn$, where $\alpha$ is a
constant angle of divergence, $h$ an internode length, and $n$ is an integer
index. On the other hand, a point on a plane is located in a polar coordinate
system $(r,\varphi)$, where $r$ and $\varphi$ is the radial distance from the
central axis and the angular coordinate about the axis, respectively. Leaf
primordia at a shoot apex are represented by $r=a^{n}$ and $\varphi=2\pi\alpha
n$, where $a$ is a plastochron ratio. In a conformal growth preserving angles,
the two representations are related by ${2\pi}z={\log r}$. Hence, the
internode length $h$ corresponds to the logarithm of the plastochron ratio,
$\frac{1}{2\pi}\log a$. The number of internodes traversed by the leaf traces
is $n_{c}={Z_{\rm lt}}/{h}=2\pi{Z_{\rm lt}}/{\log a}$, where $Z_{\rm lt}$ is a
length of leaf traces in the stem. Therefore, $n_{c}$ may be regarded as
inversely proportional to $\log a$, or the relative growth rate per
plastochron $\frac{{\rm d}r}{{\rm d}n}/r$. The growth rate should depend on
cell types. Accordingly, $n_{c}$ may change during plant growth.

The plastochron ratio may change as a result of alteration in size of the apex
and primordia. Richards (1951) discussed changing phyllotaxis to the effect
that a continuous shift in the parastichy pair of normal Fibonacci phyllotaxis
is linearly correlated with a double logarithm $\log(\log a)$. He defined the
phyllotaxis index (P.I.) by

${\rm P.I.}=0.38-2.39\log_{10}\log_{10}a,$ (4)

where numerical values are chosen such that the index assumes an integral
value whenever two sets of parastichies in the Fibonacci system intersect
orthogonally. The crossing angle between the contact parastichies changes
continuously as a function of the plastochron ratio. In this descriptive
model, the divergence angle $\alpha_{0}$ is fixed at the golden angle.

Changing phyllotaxis due to change in the plastochron ratio is consistent with
the present model of vascular phyllotaxis. In this model, the divergence
$\alpha$ on the stem changes discontinuously, however. To show a
correspondence between changes in phyllotaxis on the apex and the stem, let us
consider the normal phyllotaxis with an initial divergence of the golden angle
$\alpha_{0}=\tau^{-2}$ (Table 1). Let us introduce the Fibonacci sequence
$F_{n}$ generated from initial integers $F_{1}=1$ and $F_{2}=1$ by the
recurrence relation $F_{n+2}=F_{n+1}+F_{n}$. Accordingly, $F_{n}=1,1,2,3,5,8$
and $13$ for $n=1,2,3,4,5,6$ and $7$, respectively. In terms of $F_{n}$, the
phyllotactic fraction $\alpha=\frac{F_{n}}{F_{n+2}}$ and the parastichy pair
$(F_{n},F_{n+1})$ are obtained for $F_{n+1}\leq n_{c}<F_{n+2}$, or for

$(n+1)\log\tau-\log\sqrt{5}\leq\log n_{c}<(n+2)\log\tau-\log\sqrt{5}$

owing to an approximate formula $F_{n}\simeq\tau^{n}/\sqrt{5}$ valid for large
$n$ (see below (B.31) in Okabe (2011)). Therefore, $\log n_{c}$ is
proportional to the integer index $n$.

To put it concretely, we get $\alpha=\frac{1}{3}$ and the parastichy pair
(1,2) for $2\leq n_{c}<3$ or

$0.7\leq\log n_{c}<1.1,$

$\alpha=\frac{2}{5}$ and $(2,3)$ for $3\leq n_{c}<5$ or

$1.1\leq\log n_{c}<1.6,$

$\alpha=\frac{3}{8}$ and $(3,5)$ for $5\leq n_{c}<8$ or

$1.6\leq\log n_{c}<2.1,$

$\alpha=\frac{5}{13}$ and $(5,8)$ for $8\leq n_{c}<13$ or

$2.1\leq\log n_{c}<2.6,$

and so on. Thus, the shift in the parastichy pair is linearly correlated with
$\log n_{c}\propto\log(\log a)$. This is a general property holding also for
other initial divergences found in nature.

For the systematic study of the mature stem, the index $n_{c}$ is more
usefully regarded as a developmental index than $a$, not only because an
internode is a natural unit of length as the plastochron is the developmental
unit of time, but values of $n_{c}$ allowed for a phyllotactic pattern are
delimited by the special integers traditionally familiar to those who are
enchanted by phyllotaxis; Fibonacci numbers. For a given initial divergence,
the numbers comprise a sequence generated by the Fibonacci recurrence relation
$F_{n+2}=F_{n+1}+F_{n}$ from a pair of different seed integers. The main
sequence, 1, 2, 3, 5, 8, $\cdots$ in Table 1, is generated from the simplest
seed pair $(1,2)$. The next simplest seed integers $(1,3)$ give the accessory
sequence 1, 3, 4, 7, 11, 18 $\cdots$ of Table 2. In this manner, any
phyllotactic sequence is characterized by a pair of seed integers, as well as
the limit divergence $\alpha_{0}$. This is in accordance with accumulated
empirical wisdom of phyllotaxis. Traditionally, these special integers have
been remarked in connection with parastichy numbers (cf. Tables 1$\sim$18).
The present work puts emphasis on these numbers as critical values for the
length per internode of leaf traces. This point has never been remarked
before.

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