A synergic approach to plant pattern generation

A Synergic Approach to Plant Pattern Generation ROGER V. JEAN Universi[v of QuPhec. Rimourki, Received 20 December

Quebec, Canada GSL 7C3

1988; revised 4 July 1989

ABSTRACT The paper shows convergences between the results found in various models of phyllotaxis. It shows that a synergic approach is needed to deal with the problems of phyllotaxis. An algorithm, called the +-model, based on the observation of the meaningful and symmetry-generating presence of the golden ratio $J in all types of spiral patterns, and consequently in all types of regular patterns in phyllotaxis, is proposed. The model is suggested by a property of the allometry-type model for pattern recognition in phyllotaxis. It extends recent morphological models developed around the idea of packing efficiency of plant primordia, models that yield the noble numbers, among which are the divergence angles of spiral patterns. The +-model also gives the noble numbers and moreover orders them in a way that establishes connections with the morphogenetic principles used in models for pattern generation; the order has to do with the relative frequencies of the spiral patterns in nature. The $-model is a link between the two entropy models in phyllotaxis and offers a nice correspondence with the minimal entropy model generated by a systemic and holistic approach. This latter type of approach is put forward as being able to give a general framework in which to organize the concepts, results, and models in phyllotaxis in a way that produces a synergy of efforts. The necessity of doing so is seen clearly when one considers that phyllotaxis-like patterns are encountered in other fields of research, so that the problem appears to transcend the strict botanical substratum.

1.

PRELIMINARIES

A.

BASIC

DEFINITIONS

The surfaces of plants show spiral patterns such as the one easily observed on the capituli of daisies. Such a pattern is made of two families of m and n spirals winding in opposite directions around the center of the capitulum. The pair is denoted by (m, n) and is generally called a contact parastichy pair. This gives rise to the centric representation of phyllotaxis, where the natural pattern becomes a pattern of m and n evenly spaced logarithmic spirals winding in opposite directions around a common pole. MATHEMATICAL

BIOSCIENCES

OElsevier Science Publishing 655 Avenue of the Americas,

98:13-47

13

(1990)

Co., Inc., 1990 New York, NY 10010

00255564/90/$03.50

14

ROGER

V. JEAN

The centers of the primordia (the florets in the capitulum) become points at the intersections of opposed spirals in the centric representation. These points are numbered in succession according to their distances to the pole. The divergence angle d, given as a number between 0 and l/2, or as an angle between 0” and 180”, measures the angular distance at the pole between any two consecutive points p and p + 1 in the regular representation. The contact parastichy pair becomes one of the visible opposed parastichy pairs and is generally a conspicuous parastichy pair. To better understand these latter two concepts, let us consider the cylindrical representation of the patterns. This representation is the one obtained directly from a pine cone showing a cylindrical shape, by cutting its surface along a generator and unfolding it in a plane. By a logarithmic transformation, the pair (m, n) in the centric representation becomes a pair of families of evenly spaced straight lines in the plane. The divergence d becomes the horizontal distance between any two points p and p + 1. In particular, d is the horizontal distance between points 0 and 1, where 0 represents the origin of coordinates and the ordinate of point 1 is called the rise r, the ordinate of point p being pr. The value pd - (pd) is called the secondaty divergence of point p, where (pd) denotes the integer nearest to pd. The points of the centric representation thus become points in an infinite vertical stripe with a basis of length 1 between - l/2 and l/2, centered on point 0, such as the one on the right in Figure 1, by which the generator passes. Depending on the value of d, by moving upward along the vertical axis we meet points alternately on each side of the axis and getting closer to it, such as the points 3, 5, 8, 13, 21, 34, 55, and 89 in Figure 1. These are the neighbors of the vertical axis. This rhythmic alternation comes from the fact that + does not have intermediate convergents in its continued fraction [6]. Joining point 0 to one of the neighbors of the vertical, say m, constitutes a direction and makes it possible to join all the points of the lattice with m parallel straight lines. Doing the same with 0 and the next point, say n, which is on the other side of the vertical, gives n parallel straight lines opposed to the former m lines. The pair (m, n) is called a visible opposed spiralpair, such as the pairs (5,8), (13,21), and (21,34) in Figure 1. The word visible does not refer to the eye but to the fact that any two opposed lines in a visible pair meet at a point of the lattice [l]. There are pairs that are visible but not opposed, such as the pair (21,8), and pairs that are opposed but not visible such as (5,ll). Given that the visible opposed pair (5,8) is made with the two points that are the closest to the origin, the pair is said to be conspicuous. Notice that for d fixed, as the rise r decreases the neighbors of the origin change, so that other visible opposed pairs become conspicuous pairs. In the figure the next pair to become conspicuous by decreasing r is (8713).

PLANT

PATTERN

15

GENERATION

FIG. 1. The normalized cylindrical representation of a pattern where $I = (6 + 1)/Z. Multiplied by 360”. d is seen to represent

with divergence d = I/- *, an angle of about 137.5’.

The length of the base of the stripe, that is, the segment between two consecutive representations of the same point p, is 1. We consider the stripe between - l/2 and l/2. Points 5 and 8 are neighbors of the origin so that the pair (5.8) is conspicuous; the value of the angle of intersection B between intersecting lines in that pair is about 105”. The pairs (3,5),(13,8),(55,89), . are visible and opposed.

The divergence angle in Figure 1 is approximated by the fractions l/2,1/3,2/5,3/8,5/13 ,... called phyllotactic fractions. To better understand that concept, rebuild the cylinder from the stripe by having the two representations of point 0 coincide. Then locate a point approximately superposed to 0, say 21. By definition a phyllotactic fraction is the ratio of the number of turns around the cylinder, by the shortest path from 0 to 1, to the number of primordia met. For the point 21, the fraction is S/21 given that we make 8 turns to the left around the cylinder by going through the points 0,1,2,3 ,..., and we meet 21 points, excluding 0. The approximation gets better with neighbors of the vertical axis closer to that axis. The line following the points 0, 1,2,3,. . is called the genetic spiral. B.

SPIRA LITY

IN PHYLLOTA

XIS

The main patterns observed in phyllotaxis are first and foremost spiral patterns. They are generally expressed by consecutive terms of the Fibonacci sequence (1,1,2,3,5,8,. . .) (also called the main sequence or hauptreihe). When they are not, then they are expressed by consecutive terms of a Fibonacci-type sequence such as (1,3,4,7,11,...) (the first accessory se-

16

ROGER

V. JEAN

quence), or (1,4, 5,9,14,. . . ) (the second accessory sequence), or 2(1,1,2,3,5,8,. . . ) (the bijugate main sequence-the 2 at the beginning of the sequence represents bijugacy and means that the system has two genetic spirals and displays the Fibonacci sequence twice), which are all normal phyllotaxis sequences, or such as (2,5,7,12,19,. .) (the first lateral sequence), which is a case of anomalous phyllotaxis. Spirality expressed by these sequences is the central fact in phyllotaxis. Based on biological data, a recent model [23] shows how all other types of phyllotactic patterns, such as alternating and superposed whorls, derive from spiral patterns. The first thing to do is thus to concentrate on spirality and give conditions by which one spiral pattern or another will be generated. What is the central fact concerning spirality in patterns on plants? It is the overwhelming presence of the golden ratio, the number $I = (6 + 1)/2. Phyllotaxis is a consequence of the occurrence of 9, regardless of the growth of the plant in diameter and in length. Growth determines only the conspicuous pair. And very often growth contributes to stress the presence of $J by the phenomenon of risingphyllotaxis, the process by which the conspicuous pair (m, n) of m and n spirals, n < rn, is replaced by the conspicuous pair (m + n, m), which is replaced by conspicuous pair (m + n, 2m + n), . . . as the rise r decreases (the internode distance), and the apical dome enlarges with respect to the size of the primordia. Indeed, this phenomenon has the effect that the ratios of the two terms in the pairs, of the corresponding rises in each pair, and of their secondary divergences get closer to + as the next paragraph shows. Let us consider a plant growing according to the sequence of normal phyllotaxis J(1, t, t + 1,2t + 1,3t + 2,. .), where J 2 1 (representing the number of genetic spirals in the system) and t 2 2 are integers, and where each term is the sum of the preceding two. Call F(k) the k th term of the Fibonacci sequence. It can be shown that the abscissas (called secondary divergences) of points m=J(F(k)t+F(k-1)) of the sequence

X”Z

and

are respectively

n=J(F(k-l)t+F(k-2))

equal to

(-1)” = &‘J(t#d

+1)

( -l)k-l and

%=

#-2J(cpt+l)

’

that the rises of m and II in the normalized cylindrical representation (see Figure 2) are respectively mr and nr, and that the divergence angle d between consecutively born primordia is in an interval around [ J( t + $- ’ )] ‘. Using the well-known approximation F(k) = +“/6 [precisely, F(k) = 1‘t is easily seen that the ratios (1) of the rises, (2) of the r+k l t(- ~)-“l/m,

PLANT

PATTERN

17

GENERATION

FIG. 2. Parameters involved in the cylindrical representation of phyllotaxis, where it of any can be shown that MW’ - rcot/3 + x,x, = 0, where /3 is the angle of intersection two lines (representing the spirals on plants) in the pair (m, n) [the figure shows the direction of each family in the pair: from (0.0) to mn, and from (1.0) to mn], r is the rise or ordinate of point 1. rntn that of point mn, and x, and x, are the secondary divergences of n and nz. respectively.

absolute values of the secondary divergences, (3) of u/v, where u/m and u/n are the endpoints of the interval enclosing the divergence angle d (see Section lC), and (4) of the points m and n, are always close to + (and get closer to it as k increases or as the phyllotaxis rises). This can be seen to be true whatever Fibonacci-type sequence (a, b, a + b, a + 2 b, 2a + 3b,. . . ) is involved. In particular it is also true for the sequence of anomalous phyllotaxis (2,2t + 1,2t + 3,4t + 4,. . . ), t 2 2, for which the divergence is given by [2 + (t + 4. ‘)-‘I- ‘, and the secondary divergences are approximately given [17] by

%* =

( -l)k-2 &*(2c#Jt+c$+2)

( -l)k-3 and

xn= qJk-y2c#Bt+(p+2)

for the general terms m=2F(k-l)t+F(k)+F(k-2) ,t=2F(k-2)t+F(k-l)+F(k-3).

and

18 C.

ROGER

THE RELATIONSHIP AND DIVERGENCE

BETWEEN ANGLE

PARASTICHY

V. JEAN

PAIRS

In contrast to mechanistic models in phyllotaxis trying to control the behavior of the parameters r and d, the allometry-type model (Section 2A) proposes a description of the systems on the basis of r and (m, n) because of the close geometrical relationship between (m, n) and d. This relationship will now be presented. Suppose that the conspicuous parastichy pair (m, n) is observed on a plant. In the cylindrical lattice this means that the points m and n are closer to the origin 0 of the coordinates than any other point of the lattice is. The presence of this particular pair conveys the knowledge that a particular divergence angle is concerned. And measurements of divergence angle on plants show that this is indeed the case. The reciprocal is also true. More precisely, Adler [1] showed that the pair (m, n), where m and n are the consecutive terms of the normal phyllotaxis sequence given above is a visible opposed parastichy pair if and only if the divergence angle is equal to F(k)/m or to F( k - 1)/n or is between these two values. Notice that as k increases, F( k)/m and F( k -1)/n approach (t + +p’)p’, the divergence angle of normal phyllotaxis. Thus the visible opposed pair (5,8) can be observed if and only if the divergence is in the interval [3/8,2/5], that is, if it is in [1350,144”]. More generally it has been shown [24] that for the spiral pair (m, n) and the divergence d < + the following properties are equivalent: (a) The pair (m, n) is visible and opposed. (b) There exist unique integers 0 I u < n and 0 I u < m such that Imv - nul =l, and d is in the closed interval whose endpoints are u/m and v/n. Given that un - urn = k 1, it is easily seen that as n < m increases, if m/n approaches (p, then u/u approaches (p. For the pair (m, n) of anomalous phyllotaxis given earlier, u=F(k-l)t+F(k-2)

and

u=F(k-2)r+F(k-3)

[17], so that u/m and v/n both approach the divergence angle of anomalous phyllotaxis given earlier. This means that the parameters to consider first are either d and r, or (m,n) and r. What r specifies in the cylindrical lattice is the conspicuous pair-that is, that pair among the visible opposed pairs for which m and n are the closest points of the lattice to the origin 0 of the coordinates. Section 2A shows an important relationship between r, (m, n), and /3, the angle between opposed parastichies in a pair. Here is an algorithm to determine an interval for d from a visible opposed parastichy pair. It amounts to solving the diophantine equation

PLANT

PATTERN

19

GENERATION

Inu - mu] -1.

For example, if the visible opposed pair is (29,lQ that the possible values for u and u are u=29pf8

it is seen

o=18pf5,

and

where p is any integer. If we are to have 0 < u < 29 and 0 < u < 18, then we mustputp=l(u=21andu=13)orp=O(u=8andv=5).Butonlythe value p = 0 gives a divergence smaller than i, so that the interval for d is [8/29,5/18], or [99.31”,100”]. 2.

THE ALLOMETRY-TYPE

A.

DESCRIPTION

MODEL

AND APPLICATION

IN PHYLLOTAXIS

OF THE MODEL

It has been shown that in the cylindrical

lattice [16]

r = 1-4P)/( m + nJ2,

(1)

where j3 is the angle of intersection of any two opposed straight lines in the visible opposed pair (M, n), the lines in each family being evenly spaced and parallel, r is the ordinate of primordium 1, and p(p)

+

JScotp +(5cot2/3 +4y2 2Js

(2)

By the transformation r = (In R)/2 n, the families of straight lines become two families of logarithmic spirals winding in opposite directions around a common pole, and R > 1 is the ratio of the distances to the common pole of two consecutively born primordia. This is the centric representation introduced earlier. What is the meaning of this model? This is not a model for pattern generation but for pattern recognition. It means that for a given /3, all spiral patterns are on the same straight line of slope -2 in the log-log grid. In other words, if one observes a plant with ratio R and phyllotaxis (m, n), the point (log R, m + n) is on the line of slope - 2 and intercept 2lrp(/?)log(expl). For each given j3, this remarkable symmetry property is true whatever pair (m, n) is observed. Formula (1) is an allometric expression of differential growth. From (1) all the forms of Richards’ famous phyllotaxis index can be deduced (see [16]). Contrary to this index, which is the curious expression P.I. = 0.379-2.3925loglog implicitly

centered

on the Fibonacci

R

sequence, the model expressed by (1)

20

ROGER

V. JEAN

and (2) concerns patterns t+,,essed by any Fibonacci-type sequence and any symmetrical patterns in general, spiral or whorled. What is the practical value of this model? It has proved to be a very reliable tool, more accurate and much more easy to apply than the former models meant for practical pattern assessment, such as van Iterson’s model (see [22]). It uses an automatic pattern determination table in which the botanist can put the observed data to obtain the values of other parameters. All the possible situations involving the parameters (m, n), d, p, and r (or R) are easily dealt with by this pattern determination table. The method allows us to organize phyllotactic data in a more coherent manner than was permitted by the earlier methods, and it considers the diversity of relative sixes and shapes of the primordia in relation to the sizes of the shoot apices. In Section 6B, the allometry-type model will be used to perform an analysis of Adler’s contact pressure model. Section 2C presents mathematical properties of the allometry model that open the way to the formulation of the minimality condition defining the +-model. Let us now use the allometry model to introduce an extension of Coxeter’s formula. B.

COXETER’S

FORMULA

The model in (1) gives an approximation of the value of r as a function of the visible opposed pair (m, n) with angle of intersection j?. It generalizes and approximates what was called [18] Coxeter’s formula. This formula is given by

It was established for the Fibonacci sequence (F(k)) = (1,1,2,3,5,8,...) (for which d = +- *) and orthogonal intersection of the pair (F(k), F( k + 1)) such that F(k) and F( k + 1) are the neighbors of the origin of the cylindrical lattice (thus giving a conspicuous pair). Coxeter [6] is looking for a value of r such that the three points F( k - l), F(k), and F( k + 1) are the closest to 0. He thus also calculates the value of r for which the pair (F( k - l), F(k)) is orthogonal and then chooses the “good value” r=l/F(k)#,

intermediate between the former two values of r. Putting /3 = 90”, m = F(k + l), and n = F(k) gives

in formulas

(1) and (2)

21

PLANT PATTERN GENERATION

by using the approximation F(k) = +“/Js. Putting this approximation of F(k) in Coxeter’s formula (3) gives the same value of r. For the orthogonal pair (F(k -l), F(k)), the two formulas give r =6/d”-‘. The “good value” of r is thus

and approximates the “good value” obtained by Coxeter. The same process of finding a “good value” can be performed by means of formula (l), whatever Fibonacci-type sequence is involved (and thus whatever divergence angle is used), and the “good value” between the two values for which two pairs (m, n) and (m, m + n) are successively orthogonal is given in Proposition 1. In this proposition the term “d corresponding to a sequence” refers to the theorems in Section 1C. It means that in the lattice determined by d, consecutive terms of the sequence constitute visible opposed spiral pairs and the members of the sequence are the denominators of the principal convergents of the continued fraction of d. PROPOSITION

I

Consider the Fibonacci-type sequence a, b, a + b,2 b + a,. . . , n = F( k - l)b +F(k-2)a,m=F(k)b+F(k_l)a,..., and the value of d corresponding to that sequence. When (m, n) is orthogonal, 3 6 r =

whena=l

e(l+

n)’

=

$?k-3(+b

+ a)’ ’

andb=t, r = ( mn)-1/2+-k+V* f#Jt+l

is a formula which generalizes (3) [put t = 11. For three neighbors m - n, n, m of the origin, the value of r is approximately given by r =\/5/(

+a + b)2$2k-4

(the values a = 1 and b = 1 yield an approximation obtained by Coxeter). C.

DISTANCE

PROPOSITION

BETWEEN

TWO POINTS

(4 of the “good value”

ON THE ALLOMETRIC

LINE

2

On each straight line of slope - 2 in the log-log grid, B fixed, the distance from the point (q, a) to (r2, b), to (r3, a + b), to (r4,2b + a), to

ROGER V. JEAN

22 (rs,3b+2a),..., where the ‘; are rises, log (pfi = 0.46731.

around

the vaIue

is thus given by fi{log[(F(k)b+F(k-l)a)/(F(k-l)b+ so that replacing the Fibonacci F(k) =#/fi, in the distance obtained by their approximations, we obtain distance is at the limit equal to \/s log $. n

numbers that this

Proof.

The square of the distance

(logr,

rapidly

stabilizes

is given by

-logr,_,)*+(log[F(k)b+

F(k-l)a]

-log[F(k-l)b+F(k-2)a])’

F(k)b+P’(k-1)a F(k-l)b+F(k-2)a

II. 2

This distance F(k -2)a)]}.

Now

Example (a). Considering the divergence angle corresponding to the sequence 3,4,7,11,18,29,. . . , the distance between the points (r,,(4,7)) and (r,,(11,7)) in the log-log grid is given by filog(18/11) = 0.47825, for any fixed 8. Example (b). One goes from the orthogonal conspicuous pair (5,3) to the orthogonal conspicuous pair (5,8) by a jump of fi log(13/8) = 0.47148 in the log-log grid, on the line r =1.8944(m + TI-~.

This means that on the straight line of slope - 2 corresponding to a given /I, when a system experiences rising phyllotaxis along a given Fibonacci-type sequence, at each step the system jumps on the line by a factor approximately equal to Js log +. PROPOSITION

3

Given the integer m, the value of n < m for which the distance between (rm9 m) and (r,, n) in the log-log grid, for any given 8, is the closest to fi log+ is such that n < m I 2n and is given by n = (m+-‘), the parentheses denoting the closest integer to rn+-‘. Proof. We are looking for the value of n for which the function decreasing JS[log(m/n)-log+] is th e c1osest to 0. This is a monotonically function of n, and the logarithm is approximately equal to 0 for n = (m$-I), and then n < m I 2n. n

PLANT PATTERN GENERATION PROPOSITION

23

4

Given the integer m and the angle 8, if point n = k, is such that IS c1osest to 0, then for the integer m + n and for /I, the filog(m/k,+) number 6 log[( m + n)/k,$] is closest to 0 for k, = m. Proof. Using the relation (p’ = $I + 1, if m/k,+ is the nearest k, = n = (m$-‘) then m/n4 + (p-’ is the nearest to 1+ (m + n)/n$ is the nearest to +, (m + n)/n+* is the nearest to 1, (n+ + n) is the nearest to 1, and (m + n)/(k, + n) is the nearest k, = m = (n+). n 3.

THE +-MODEL

A.

ON THE FOUNDATIONS

to 1 for +-‘= $I, (m + n)/ to 1 for

OF THE MODEL

We underlined the presence of the mrmber $J in spiral patterns. Section 3B presents a minimality condition defining the +-model suggested by a property of the allometry formula. This model has to do with basic mathematical deductions about the morphology of phyllotaxis, by which noble numbers are generated that are functions of $, as we will now see. The +-model is also able to generate these numbers, and moreover it is able to order them in a way that establishes connections with the morphogenetic principles used in modeling the phyllotactic patterns, as we will see in Section 6. Out of Bravais and Bravais’s (a botanist and a crystallographer-mathematician who gave us the Bravais lattice) biologically plausible mathematical assumptions [d is an irrational number, and by moving upward along the vertical axis the neighbors of this axis on the two infinite poiygonal lines are alternately on each side of the axis (cf. Figure l)], Coxeter [6] deduced the sets of angles and sequences expressing normal phyllotaxis. Some mathematical approaches (e.g., Rivier [35], Ridley [32], Marzec [28]) stress that phyllotaxis is the solution of a structural variational principle giving maximal uniformity to the patterns and requiring divergence angles whose continued fractions terminate as l/+ = [O; l,l, 1,. . .], like those of normal and anomalous phyllotaxis. As a generative spiral, Rivier uses Vogel’s cyclotron parabolic spiral defined by r=a& and 8=2nnd, n-1,2,3,..., and where d is the Ridley divergence angle. This spiral is also used by Ridely to show that d = (P- 2 gives the best packing efficiency of the seeds in the sunflower head. The three families of parastichies determined by triple contact points (n, m, m + n) in phyllotaxis correspond to the lattice planes of conventional crystallography, and phyllotaxis then appears to be a special case of two-dimensional crystallography. Crystallographic requirements deliver what Bivier and others call noble numbers, for possible values of d, precisely those

24

ROGER V. JEAN

whose continued fractions terminate as l/+ (among which is c#-* observed in most cases in plants). The cyclotron spiral gives uniform floret density, and the noble numbers optimize the local homogeneity and self-similarity of the patterns. The set of noble numbers is precisely what Maizec and Kappraff [29] obtained. It is their set $c of divergence angles that distribute the leaves around the stem of a plant more uniformly than the neighboring angles can do. Outside this set the pattern becomes very strongly spiralling, because it is dominated by a set of very long parastichies, and the structures generated are not self-similar. Rothen and Koch [38,39] clearly illustrated what this uniformity and efficiency is about. They showed that when the value of the rise r changes in the cylindrical lattice (the divergence angle being constant), the Voronoi polygon (that is, the Dirichlet domain or Wigner-Seitz cell) does not change in form, exclusively when the divergence d is a noble number. Given a point p of the cylindrical lattice, such a polygon encloses by definition all points of the plane that are nearer to p than to any other point of the lattice. The scales of a pineapple are Voronoi polygons. In other words, the noble numbers give shape invariance or self-similarity of the Voronoi polygons around the points of the lattice. When the divergence angle is not a noble number, the shape of the polygons changes when r changes. It is particularly important this this polygon stays similar to itself when one considers the natural transitions observed in daisies or sunflowers, which show a system (m, n) in the center of the capitulum, (m, m + n) in the middle portion of the capitulum, and (2m + n, m + n) in the outer portion, corresponding to changes in r (rising phyllotaxis). In such capituli there are two circular defect lines having a quasi-crystalline structure. With self-similarity, the same plant needs only one algorithm to produce both small and large capituli, that is, the same code to make their phyllotaxes rise. The golden ratio (p = (1 + fi)/2 expresses the symmetrical proportions involved in the generation of such patterns. The problem of pattern generation thus consists in devising models that will put the systems on tracks defined by the golden ratio. This is what the minimality principles found, for example, in the +-model and Adler’s model do. Now why does nature prefer self-similar structures and efficient packing? This is a question that brings us deeper into the causes. It has to do with Zimmermann’s telome (primary cylindrical axis) theory, according to which higher plants would come from Propterydophytes of the Rhynia type, via six elementary morphogenetic (branching) principles resting on empirical material (this theory, introduced in Jean [20], gives a biological basis for the hierarchical representation of phyllotaxis presented in Section 4A). It has to do with Lyell’s principle of evolutionary continuity according to which a large part of the cellular organization is very remote and is common to all organisms. It has to do with the principle of minimal entropy production, a form of which has been

PLANT

PATTERN

25

GENERATION

operationally formulated by Rosen [36] under the name principle of optimal design, used in Section 4B. But let us not inquire further here along these lines of causality, and take the importance of I$ in the morphogenesis of spiral patterns for granted. The very simple minimality condition that will be introduced, an algorithm which at first sight does not seem to have much to do with morphogenesis, will allow us to obtain and to order the noble numbers generated from the previous packing efficiency considerations. This order will be compared to the orders generated by two entropy models of phyllotaxis (Sections 5 and 6A), orders corresponding to the relative frequencies of the spiral patterns in nature. The minimality condition is suggested, as we said, by a property of the allometry model, presented in Section 2C. The condition incorporates a general notion of rhythm. Rhythm in nature is essential, and we consider it a primitive term of physiology. But it can be given a mathematical meaning as we will see. If this rhythm is established at plastochrone m (when m primordia exist), then a member of +c will be selected by the minimality condition, and this member will m aximize the packing efficiency of the pattern for this value of 112. B.

MINIMALITY

PRINCIPLES

THE MINIMALITY

CRITERION

Given M, for any 8, a point n < m is said to satisfy the minimality criterion if the function of k, \/3 log( m/k+), k < m, is closest to 0 for k = n. The value M,,, =6log(m/n9) is then called the minimum of the function. The point n is unique. By Proposition 4, if n satisfies the minimality criterion for m, then m satisfies the minimality criterion for m + n, and m + n satisfies the minimality criterion for 2m + n, and so forth. PROPOSITION

5

Given a point m of the Fibonacci-type sequence a, b, a + b, 2b + a,. . . , if the minimality criterion is satisfied for the adjacent member n < m in the sequence, then the criterion is satisfied for any point p greater than m in the sequence by the number q -Cp and adjacent to p in the sequence. The consecutive minima +l,h+ rCkJU are alternately positive and negative and decrease in absolute M rC‘(k value toward 0. Proof. m x=

Suppose for definiteness F(k+l)b+F(k)a F(k)b+F(k-l)a<

that F(k+2)b+ F(k+l)a F(k+l)b+F(k)a

m+n =m’

ROGER V. JEAN

26 Then

for the other members of the sequence larger than m + n, the fractions (2m + n)/(m + n),(3m +2n)/(2m + n),... are inside the interval [m/n, (m + n)/m], as consecutive alternate mediants. We recall that the mediant of the fractions p/q and r/s is the fraction (p + r)/( q + s). We thus have a net of closed intervals whose intersection contains only one member, which is 4. It follows that the consecutive fractions are alternately smaller and larger than 4; that is, the values m/n+, (m + n)/m+, (2m + n)/( m + n)$, . . . are alternately smaller and larger than 1. The logarithm is then alternately positive and negative. Given that the length of the consecutive intervals decreases toward 0, the consecutive minima decrease toward 0. n THE MJNJMALITY

CONDITION

At plastochrone m, when m primordia exist, the minimality condition states that nature requires the primordia to show the visible opposed parastichy pair (m - n, n) such that n satisfies the minimality criterion (recall that the plastochrone is the unit of time elapsed between the emergence of two primordia). It is known [24] that each of the two extensions (a, a + b) and (a + b, b) of a visible opposed pair (a, b) is a visible pair, and that one of them is opposed. Given that the consecutive minima in Proposition 4 are decreasing toward zero while alternating in sign, the consecutive alternate extensions of the conspicuous pair (m-n, n), that is, (m, n),(m,m + n),(2m + n, m + n),..., are visible and opposed. This means that if the system starts to experience the minimahty condition at plastochrone m, the system enters into a loop for the first time; that is, a rhythm of growth is installed, generating conspicuous pairs along the Fibonacci-type sequence (m - n, n, m, m + n, . . . ). Table 1 gives the sequences obtained for the various values of m. Consider now the value m = 9 and the sequence (1,4,5,9,14,23,. . . ). For consecutive numbers greater than 5 of the sequence, the minimality condition is satisfied by this sequence, but for the first four it is not. Indeed, for m = 9, n = 6 gives the minimality condition, for m = 5 the condition is satisfied for n = 3, and for m = 4 it is satisfied for n = 3 (the sequences then obtained can be read in Table 1). Hence the parentheses in the sequence on line 14 of Table 1. Of course these parentheses do not mean that the visible opposed parastichy pairs (1,4) and (5,4) are excluded by the algorithm, given that when the sequence (1,4,5,9,14,. . .) arises in a plant, at whatever plastochrone m it may be-say m = 14 for example-then all consecutive extensions of (5,9) are visible opposed pairs, as we have seen, but also (see [24] and [l]) the contractions of (5,9), which are (1,4) and (5,4), are visible opposed pairs. These pairs are observed on plants such as Strangeria paradoxa, Cereus pasacana, Lycopodium selago, and Echinocactus williamsii (see [23], p. 295).

PLANT

PATTERN

27

GENERATION TABLE

1

Values of n Satisfying the Minimahty Criterion for Given Values of m > n, and for p Fixeda

m

n

3’ 4 5*+ 6’ 7*’

2 3 3 4 4

8* 9* 10* 11*+ 12 13* 14*+ 15 16* 17*+ 18 19*+ 20

5 6 6 7 7 8 9 9 10 11 11 12 12

21 22* 23 24

13 14 14 15

25 26 27

15 16 17

Visible Opposed Pair(n,m-n) (291) i3.1) (3.2) 2(2,1) (4.3) (533) 3(2,1) 2(3,2) (7.4) (7>5) (8.5) (9.5) 3(3,2) 2(5,3) (11.6) (11.7) (12,7) 4c3.2) (1378) 2f7.4) (14.9) 3C5.3) 5(3,2) 2f8,5) (17,lO)

Divergence 137.51” 99.5” 137.51” 68.75” 99.5” 137.51” 45.84” 68.75” 99.5” 151.1” 137.51” 77.96” 45.84” 68.75” 64.08” 99.5O 151.1° 34.38” 137.51” 49.75” 77.96” 45.84” 27.5’ 68.75” 106 =

Sequence (1,2.3,5.8,...) ((1),3,4,7,11,...) (1,2.3,5,8,...) 2((1),2,3.5,8,...) ((1),3,4,7.11,...) (1,2,3.5,8....) 3((1),2,3,5,8 >...) 2((1),2,3,5.8,...) ((1),3,4,7,11,...) ((2.5),7,12,19 . ...) (1.2.3,5,8....) ((1,4,5),9,14 . ...) 3((1),2,3.5,8 ,...) 2((1),2,3,5.8,...) ((1,5,6),11,17 ,...) ((1),3,4,7,11,...) ((2,5).7,12,19 ,...) 4((1,2),3,5.8....) (1.2,3,5,8,...) 2((1,3,4),7,11,...) ((1,4,5),9,14 >...) 3((1),2,3,5,8,...) 5((1,2),3,5,8,...) 2((1).2,3,5,8,...) ((3,7,10),17,27 ,...)

“The divergence angle and the sequence are derived. The numbers in parentheses at the beginning of a sequence are those for which the minimality criterion is not satisfied in this sequence. The * denote values of m for which the G-model and the minimal entropy model (Section 4) give the same value of n. The consecutive ’ correspond to the ordered values of WI for which the divergence angles obtained from these two models are the consecutive divergence angles obtained from Marzec’s entropy model [29] (Section 6A).

Of course the ordering of the noble numbers corresponding to the angles given in Table 1 (divide these angles by 360 to obtain approximations of these numbers whose exact values are given in Section 3) has to do with the elements at the beginning of the continued fraction of those numbers. For example, the continued fraction corresponding to the sequence (1,4,5,9,14,23 ,...) is [0;4,1,1,1,... 1. By Section 1C we know that a divergence angle corresponds to each such sequence. And before the appearance of l/$ in the continued fraction-that is, before the rhythm starts-the

28

ROGER

V. JEAN

minimality principle does not apply to that sequence. Of course the criterion is universal-for each m there is a unique n-but the sequence obtained varies with m.

4.

JEAN’S ENTROPY

MODEL

A.

THE HIERARCHICAL

STRUCTURE

OF PHYLLOTACTIC

PATTERNS

In Jean’s entropy model [12, 251 the problem of pattern generation in phyllotaxis is treated according to an interpretive approach, in contrast to a mechanistic approach where it is assumed that a given mechanism is supposedly responsible for the generation of the patterns. An entropylike function is defined, and it is shown that when the principle of minimal entropy production is used to discriminate among the phyllotactic lattices it succeeds in finding the angles and sequences of normal and anomalous phyllotaxis. The model offers an abstract understanding and a physical basis called the hierarchical structure of phyllotaxis, in which the fact of negentropy in living organisms can be given a meaning. In this model, phyllotaxis is interpreted in terms of its ultimate effect, which is assumed to be the minimization of the entropy of plants. A formula for bioentropy acts as a functional cost that allocates a cost to every possible spiral pattern. Then we look in the set of costs for minimal costs. This function is defined in Section 4B. First let us briefly consider the hierarchical (or multilevel, or branched) representation of the patterns of primordia of plants. All hierarchies of simple (no bifurcation) and double (bifurcation) nodes are considered. The one in Figure 3 represents the pattern (2,3) (two and three nodes, respectively, in the first two levels) rising along the sequence (2,3,5,8,13 ,... ). Th e or d er of the points in each row corresponds to the order generated by the projection, on the d axis of the cylindrical lattice, of the same points of the lattice as can be seen from Figure 1. For example, notice that the order of the points in the cycle 5,2,4,1,3 in Figure 3 corresponds in Figure 1 to the phyllotactic fraction 2/5 and to the order of the corresponding points when going from left to right. Biological and formal evidence of hierarchies in phyllotaxis have been discussed and exemplified in earlier papers [15,20] showing the hierarchical structure of spiral patterns, and in particular of vascular systems. Hierarchies are fractals expressing the self-similarity of phyllotactic branching structures; their common dimension is 2, given in the allometric formula (1) (see Rigaut [33]). The relevance of this ramified structure to represent the functionality of the phyllotactic spiral patterns is furthermore underlined by considering the question of sectorial translocation of assimilates in plants, in contrast with uniform distribution. This phenomenon is clearly seen in Figure 4, showing an average divergence angle of 136.7”. The figure illustrates the restricted

PLANT cycle

PATTERN

phyllotactlc fraction

2

l/2

3

l/3

5

215

a

29

GENERATION hierarchy

318

13

5113

21

8121

21 8 16 3 11 19 6 14 1 9 17 4 12 20 7 15 2 10 18

5

13

FIG. 3. The hierarchy (2.3) of bifurcated and simple induction lines. The order of the points in each row expresses phyllotactic fractions. In row 4, for example, we go from point 1 to point 8 along the consecutive points by moving three times from left to right. Each row is linked to adjacent rows according to Bolle’s law of induction: If there are n leaves in a cycle, those leaves x = 1,2,3,. , k that were in the previous cycle bifurcate and give leaves x and x + n in the next cycle.

movement of labeled carbon applied to a source leaf, into specific sectors of the growing stem. The dashed lines indicate three sets of leaves that are assumed to have the most direct vascular connections with leaf 0 (3/8 vascular phyllotaxis: 135”). Only the dominant pattern found in each leaf is shown. Given that in Figure 4 the source leaf is marked 0, to establish the comparison between Figures 3 and 4 we have to increase every number in Figure 4 by 1. Doing so, notice then that leaves 2, 5, 7, 8, 10, 13, 15, 16, 18, and 21 have been only partially reached by the 14C02. This can be correlated with the right-hand part of the hierarchy in Figure 3, developing under point 2 (linked to points 5, 7, 10, 13,15, 18) and representing a sector of the whole hierarchy that has been only partially touched by the assimilate. Moreover, the primordia on the dashed lines (most direct vascular connections with the source leaf 1)-that is, 4, 6, 9, 12, 14, 17, 20, and 22-correspond to the sector of the hierarchy in Figure 3 developing under point 1. The patterns displayed by 14C assimilates are thus predictable on the basis of the ramified structure of phyllotaxis. My previously reported papers on hierarchies show how, formally, we can transform the pattern in Figure 4 to obtain a hierarchy such as the one in Figure 3. Ho and Peel [ll] also demonstrated a marked correlation between the phyllotactic configuration of leaves in shoots of Salix viminalis and the transport of tracers between these leaves. They present a figure (their figure 1) almost identical to Figure 4 but with only 10 primordia and numbered

30

ROGER

V. JEAN

FIG. 4. Fibonacci phyllotaxis of leaves around a stem of apple (from Barlow [3]). The source leaf is marked 0. and successive younger leaves are numbered 1 to 24. The source leaf was fed with 14C02 for l-2 h. The pattern of transport may be followed on a leaf-by-leaf basis. Whole leaf autoradiography one to several days after treatment showed the distribution pattern of 14C. Leaves that lie superimposed in vertical ranks and that are located closest to the fed leaf take up label over their entire surface. Leaves on the opposite side of the stem from the fed leaf received no labeled assimilate. Leaves in adjacent ranks acquire label in a small sector on one side of the midrib or another depending on their positions with respect to the fed leaf. Leaf 24 was not examined.

differently. It can be verified, by marking their leaf a, as leaf 1 and the other leaves as leaves 2, 3, 4,. . . along the genetic spiral, that when 14C is applied to leaf 6, leaves 2, 5, 7, and 10 show almost no 14C activity. These leaves correspond again to the right-hand part of the hierarchy in Figure 3, representing the (2,3) phyllotaxis of the plant. The assimilate was transported from leaf 6 to leaves 3, 4, 8, and 9 via leaf 1 on the left-hand side of the hierarchy. That is why the authors noticed two main channels in the transport of tracers. As another example, in the sunflower the pattern of assimilate movement is also highly sectorial. In the domestic sunflower, which derived from branched ancestors relatively recently, “perhaps the pronounced sectorialization reflects an architectural constraint that is remnant of this branched ancestry” (Watson and Casper [43]; cf. also Zimmermann’s telome theory mentioned earlier). B.

THE COST

OF EACH

HIERARCHICAL

STRUCTURE

Consider the sequence (1,2,3,5,8,13,...). A plant may show a conspicuous pair (5,3) and another plant the pair (5,8), but both patterns will display

PLANT

PATTERN

31

GENERATION

a divergence angle close to the angle of 137.5” corresponding to that sequence. The same plant may show the pair (3,5), then the pair (8,5), and then the pair (8,13), a phenomenon called rising phyllotuxis. This suggests the following particular set of hierarchies. To each sequence of positive integers H(l), H(2), H(3), . . , where for k = 1,2,3,. . . , H( k + 2) = H( k + 1) + H(k), and H(k) -c H( k + 1) I 2H(k), correspond the hierarchies denotedby(H(k+l),H(k+2)),whereforagivenk, H(k+l)and H(k+2), respectively, represent the numbers of nodes in levels 0 and 1 of the hierarchy. The first inequality means that level 0 contains at least one double node. The second inequality means that the hierarchy is made up with simple and double lines only. This requirement is in agreement with the statement of the botanist Church ([5], p. 13) that “in no case is one number in the spiral pair (m, n) more than twice the other, ranging from (n, n) to (n,2n).” This requirement also comes out from Proposition 3 for the +-model. In the minimal entropy model the hierarchies are generated by irreducible square matrices of order w and by L-systems (this is developed in length in Jean [12,14]). The irreducibility or strong connectedness of the matrices indicates that the process by which a hierarchy is generated follows a rhythmic pattern, coming back at every w levels of the hierarchy. When w levels of the hierarchy are generated, the other levels are generated according to the same rule defined by the matrix, so that the system begins to experience a rhythm. An L-system is a language or interface used to link the matrices and the hierarchies so as to be able to generate those hierarchies from the matrices. Let us consider an example: The square matrix of order w = 2, having l’s as entries except for an entry 0 in the right-hand bottom comer, gives an alphabet with w = 2 symbols a, and u2. The rule of production P of the words of the L-system is, according to the entries of the matrix, given by P(q)

= u1u2

P(u,)

and

=a,.

Starting with the word u1u2 and applying the rule P, notice that you generate the skeleton of the hierarchy in Figure 3 if point 1 in the hierarchy is identified with a, and point 2 with u2. At the time T, at which the rhythm of growth w is established, when there are T, primordia, and when a new cycle of growth is about to start, and to repeat the same generative pattern (the generation of primordia enters into a recursive loop), the hierarchy chosen is the one that minimizes the entropy function given by E = -

i t-1

s(t) 1

log [ X(r)

.

32

ROGER

V. JEAN

Here S(t) is the relative frequency of duplications (double nodes) up to level and X(t) is the complexity of the hierarchy at level t. In the t=1,2,3,... hierarchy of Figure 3, we have S(1) =1/2, S(2) = 3/S, S(3) = 6/10, S(4) = 11/l&. . . . It has been argued that duplications have to do with stability. And X(t) was chosen so as to increase very rapidly with t, that is,

X(t)

= fi

k/(k),

k=l

where f(k) is the number of nodes on level k = 1,2,3,. (a smaller function could be chosen to give the same results). It has been shown that the hierarchies giving minimal entropy are to be sought among those for which w = 2, and that they are of the type (H( k + l), H( k + 2)) for T. = H( k + 3)

WI. Let us make a general remark on entropy. The word “entropy” corresponds today to an adisciplinary concept. The physical meaning of the concept of entropy is much disputed; it is still considered as not very clear. According to Poincare it is a “prodigiously abstract concept.” Wiener indicated the need to extend the notion of physical entropy when he stated that information is negative entropy. The literature-in physics, communication theory, information theory, social sciences and the life sciences, in probability theory, graph theory, and Lebesgue’s integration theory-proposes many concepts and formulas of entropy and quantity of information, such as the topological structural information content of Rashevsky-Trucco, the Shannon-Weaver and Hartley-Nyquist formulas, the chromatic information content of Mowshowitz, the entropy of measurable functions, and epsilon-entropy. There is a whole list of authors who have developed their own concepts of entropy by specifying a function and applying it to a particular context. More often, unfortunately, the form of the mathematical function is not even specified. Schrodinger initiated the use of a concept of entropy in the life sciences by developing a literal interpretation of the well-known formula E = K log D, where D is a measure of the disorder and K is Boltzmann’s constant. But as it is currently being recognized, entropy in living organisms, that is, bioentropy, must be a sum of two factors, one of which brings in negentropy, such as in formula (5) the factor -logS( t). Formula (5) is related to the usual formulas for entropy. For example, the entropy of a single event of probability p has been defined by Renyi by the formula log(l/p). With p = l/X( t), log(l/p) = log X(t) = - log[l/X( t)], one of the two factors in (5), which also represents the entropy of a set of X(t) elements (see for example, Jean [13], which proposes a development along these lines of comparison). We will come back to formula (5) in Section 6A.

PLANT PATTERN GENERATION C.

A PROPERTY

PROPOSITION

33

OF THE MODEL 6

Given a value of T,, then among all the Fibonacci-type sequences of integers (H(k -l), H(k), H(k +l), H(k +2), H(k +3) ,...) such that T, = H(k +3), the hierarchy (H(k+l)+[-H(k-1)/3],H(k+2)-[-H(k-1)/3]) gives minimal entropy, where [ - H(k - 1)/3] larger than - H(k - 1)/3.

means the smallest integer

Proof. The hierarchy we are looking for, giving minimal entropy, is given by (m + n, 2m + n), where m > 0 is the number of double nodes and n 2 0 is the number of simple nodes in level 0 of the hierarchy. The sequence we are looking for starts with the terms

n,m,

m+n,

2m+n,

3m+2n=H(k+3)

It starts also with the terms H(k-1)+3p,

H(k)-2p,

H(k+l)+p,

where p is an integer to be determined. and n=H(k-1)+3p>O,sothat

H(k+2)-p,

H(k+3),

We have that m = H(k) - 2p > 0

_ H(k-1) 3 It can be calculated that the entropy (H(k+l)+p,H(k+2)-p)isgivenby

H(ki3)

for the hierarchy

(m + n,2m + n) =

[H(k+l)+p]H(k+3) [H(k)

-2p][H(k

$2) - p]

1’

This is an increasing function of p in the interval - H(k - 1)/3 I p < H(k)/2. Indeed it is sufficient to consider the derivative of the expression inside the log since everything inside the log is positive. This derivative is positive given that each of its terms is positive. It follows that the value of p giving minimal entropy is the smallest integer larger than - H( k - 1)/3. n The following hierarchy.

algorithms

can be used to determine

the minimal

entropy

ROGER V. JEAN

34 ALGORITHM

I

Given T,. = H( k + 3), we generate any Fibonacci-type sequence starting with H( k - l), H(k), H( k + l), H( k + 2), H( k + 3). We then calculate p and put its value in (H(k - 1) + 3p, H(k) - 2p, H(k + 1) + p, H(k + 2) - p, H( k + 3), . . . ), which is the sequence giving minimal entropy. ALGORITHM

2

Given T, = H( k + 3), we generate all the Fibonacci-type sequences having H( k + 3) as a member, starting with H( k - 1). Then we choose the sequence for which p = 0. Example I. Let T = H(k + 3) =17. Then the possible sequences such that H( k + 1) < H( k + 2) < 2 H( k + 1) are

Fibonacci-type

7,1,8,9,17 ,..., 4,3,7,10,17 )...) 1,5,6,11,17 ,... The first element in each row represents the value of H( k - 1). The sequence giving [ - H( k - 1)/3] = 0 is (1,5,6,11,17,. . .), giving minimal entropy at r, =17. Example 2. Suppose that T, = 23. Produce any Fibonacci-type sequence (H( k - l), H(k), H( k + l), H( k + 2), . . . ) having 23 as its fifth element, such as (- 2,9,7,16,23,. . . ). Then - H( k - 1)/3 = 2/3, so that p = 1. The sequencewearelookingforisgivenby(H(k-l)+3p,H(k)-2p,H(k+l) + p, H(k + 2) - p,. . . ), that is, (1,7,8,15,23,. ). Notice that if we had started with the sequence (10,1,11,12,23,...), then - H(k -1)/3 = -lo/3 so that p = - 3. This value of p returns the same minimal entropy sequence. 5.

COMPARING THE +-MODEL ENTROPY MODEL

AND THE MINIMAL

Given m = T,, the +-model and the minimal entropy model give the same value of n, the same sequence, and the same divergence angle for m = 3,5,6,7,8,9,10,11,13,14,16,17,19,22 ,... (see the starred items in Table 1). Among these divergence angles are the usual values 137.5”, 99.5”, 78”, 68.75”, and 151.1”, corresponding to the vast majority of types of spiral phyllotaxis (more than 98%: see Jean [21], Table 13), and up to m = 26 the two models give the same angles (but in a slightly different order). When T, = 4 the minimal entropy condition gives no value of n given that in the first two levels of a phyllotactic hierarchy it is impossible to have four nodes if level 1 must contain one double node. As a result of his observations, Wright wrote ([44], p. 399) that “each leaf of the cycle is so placed over the space between older leaves nearest in direction to it as always to fall near the middle, and never beyond the middle

PLANT PATTERN

GENERATION

35 TABLE 2

Values of n < m = T giving the Minimal Entropy for Given Values of M, and /? Fixed’ Visible Opposed Pair (m - n, n)

m

n

4 12

-

15 18 20 21

10 12 13 14 15 16 16 17 18

Sequence

8

23 24 25 26 27

Divergence

Condition

4t2.1) 5C2.1) 6t2.1) (1337) 7C2.1) (15.8) 8t2.1) (16,9) (17.9) 9(2-l)

34.38” 27.5” 22.9” 54.4” 19.64” 47.25” 17.19” 158.1’ 41.77O 15.28”

4(1,2,3,5,8 ,...) 5(1,2,3,5,8 ,...) 6(1,2,3,5,8 ,...) (1,6,7,13,20 ,_._) 7(1,2,3,5,8 ,...) (1,7,8,15,23 ,...) 8(1,2,3,5,8 I...) (2,7,9,16,25 ,...) (1,8,9,17,26 ,...) 9(1,2,3,5,8 ,...)

aThe divergence angles and sequences of integers consequently obtained are given in columns 4 and 5. The table considers only the values of m for which the values of n differ

from the one obtained

from the +-model.

third of the space, or by more than one sixth of the space from the middle, until the cycle is completed.” This means that the divergence angle d is generally such that l/3 < d -Cl/2, that is, between 120” and 180’. This case is dealt with in Proposition 8. The restriction l/4 < d < l/3 corresponds to a divergence angle in the interval [90”,120”]. Proposition 9 deals with this case. The well-known divergence angle of 99.5” is in the latter interval. This angle is observed, for example, in Sedum refexum, Monanthes polyphyllu, Echinocactus williamsii, Euphorbia biglandulosa, and Araucaria excelsa (see Jean [23]). Looking at Tables 1 and 2 it is easily seen that the following propositions hold. The sequences in Table 2 are rarely observed in nature (see Jean [21]). PROPOSITION

7

Under the minimal entropy hypothesis a sufficient condition for Fibonacci phyllotaxis is that T, < 6. PROPOSITION

8

If l/3 < d
9

If l/4 < d -Cl/3, then a sufficient condition for the pattern governed by the sequence (1,3,4,7,11,. . ) to arise under the minimality condition of the

36

ROGER

V. JEAN

+-model is that m < 27. Under the minimal entropy condition this pattern always arises whatever the value of m is. Proof. The first part is proved by looking at Table 1. For the second part, it has been shown (Jean [12]) that the only types of spiral phyllotaxis that can arise are governed by the sequence of normal phyllotaxis, J 21,

5(1,2,3,5,8,...), J(a,a+1,2a+1,3a+2 with the corresponding

divergence

and by the sequence of anomalous (2a +1,2a with the corresponding

+3,4a

divergence

,... ),

J=lor2,

a23,

angles (in degrees)

phyllotaxis +4,6a

+7 ,... ),

a 2 2,

angles (in degrees)

The sequence of angles of normal phyllotaxis for J = 1 and a 2 3 tends toward 0” monotonically, starting with 137.51”,99.5”,77.96”, . . . (when J > 1 the sequence of decreasing angles starts with 68.75”), while the sequence of angles of anomalous phyllotaxis starts with 151.1” and increases monotonically toward 180”. It follows that the region [90”,120”] contains only the angle 99.5”, corresponding to a = 3, so that the only pattern that can arise is expressed by the sequence (3,4,7,11,18,. .). W Notice that both models incorporate an idea of rhythm. In facr, II there is no rhythm there is no pattern, and we will see that Marzec’s and Adler’s models (Section 6) have rhythms, although these authors do not speak about such a notion. Church [5] stresses the importance of rhythm in phyllotaxis. The rhythms in both models considered in the present section generate the usual types of patterns, but generally speaking, starting from m = 27, the types of patterns generated are different. For example, the sequence (3,7,10,17,27,. . . ) does not occur in the minimal entropy model, whereas it does occur in the +-model. This particular pattern does not exist in nature, however (see Jean [21]). The patterns predicted by various models are discussed in Section 6C. The idea of comparing models is not to show that one is better than the other but to show the similarities of results under dissimilar approaches.

PLANT

PATTERN

GENERATION

6.

COMPARISONS

A.

MARZEC’S

MODEL

31

WITH OTHER MODELS OF DIFFUSION

Marzec’s study [28] uses a one-dimensional model first developed by Thomley [42] (see Jean [18], pp. 129-135), where the rise r is purged from the problem. In this model the leaves on the stalk are abstractly treated as points on a circle, each point acting as a morphogen source. Phyllotaxis becomes a dissipative structure, that is, a nonequilibrium system sustained by a flow of matter and energy through its boundaries, by the creation, diffusion, and decay of the morphogen according to a diffusion-reaction equation. Thomley’s model is dynamic, setting a new leaf at the position of the absolute minimum of the concentration field established by the existing leaves, leaf by leaf, until a constant divergence Ap emerges. The strength of each leaf as a morphogen source is decreased by a factor /? < 1 after each plastochrone (unit of time between the emergence of consecutive leaves). Then, setting a constant divergence angle for about 100 primordia, and using an idea of perturbation, the model tries to reproduce that angle. Thomley finds bands of angles (78-85’, 97-lOlo, 105llO”, 132-134”, 136-143”, and 146-180”) containing some of the observed divergence angles. He obtained that when /l> $, only the angle $-* is possible, this angle being approached for values of j3 near 1. Marzec’s treatment is first of all static. It establishes a set $J~ of angles whose continued fractions do not have intermediate convergents after a finite number of terms. It is shown that this criterion is sufficient for uniform spacing of leaves. Any angle in the set possesses a purely geometrical minimization property (theorem 9, p. 216, Marzec and Kappraff [29]) producing a more even distribution of points on the circle than any neigh: boring angle, each leaf being set so that all the leaves have the maximum space. The results of this minimum principle are then applied directly to the minimization of a static concentration function. Marzec uses an informational entropy function defined by the formula

where (C) is the mean concentration and p(p) is a morphogen concentration field established by the leaves, depending on the constant divergence Ap. The entropy of the morphogen distribution is considered to measure the spacing of the leaves. Considered as a function of Ap, it is found that S is a maximum and the rates of entropy production are minimal at a Ap that asymptotically approaches, as B approaches unity, one of the members of +o.

ROGER V. JEAN

38

“In rough order of their appearance as /3 increases, the most prominent members of $c are 137.5’, 99.5”, 78.0”, 64.1”, 151.1”, 54.4”, 158.1”, and 106.4”” [28]. This is akin to what happens in Jean’s models. The angles obtained consecutively in Marzec’s model correspond to the increasing values T, = 3 and 5, 7 and 11, 14, 17, 19, 20, 25, respectively, under the minimal entropy condition (see Tables 1 and 2), and to the value m = 27 under the minima&y condition. The three models show the same order of generation for the values 5,7,11,14,17, and 19 (the values marked with+ in Table l), corresponding to the main divergence angles observed. Both the dynamic and static approaches have been considered in Jean’s minimal entropy modeling. The search for a suitable bioentropy formula [formula (5)] was preceded by the consideration of the formula E = - log[S( t)/X(t)] (not incorporating a notion of rhythm; see Jean [25], p. 255). Using this latter formula, when we choose among the hierarchies the one minimizing such a function for each consecutive level t = 1,2,3,. . . , corresponding to a dynamic approach, unfortunately only the pattern defined by the Fibonacci sequence arises (as in Thomley’s model for /? > +). Similarly, for low values of p the only solution of the dynamic problem in Marzec’s model is $-*. Why is that? Matzec conjectures that this robustness of +-* underlies its predominance in plants. The reason seems to be that the dynamic approach does not incorporate any notion of rhythm, so fundamental in growth processes. In Matzec’s static approach the rhythm is settled by the presence of cp in the continued fraction of every member of +o, which produces the alternation of the neighbors of the vertical axis as we have observed for +- ’ in Figure 1. Formula (5) expresses a dynamic approach up to the time T,, and a static one thereafter, the system having become rhythmic and deterministic. It generates the divergences of normal and anomalous phyllotaxis, included in the set $c. In Adler’s model we have a dynamic approach up to q;., and a static one after. B.

ADLER’S

MODEL

OF CONTACT

PRESSURE

Adler [l] elaborated a model in the cylindrical lattice, based on the idea of mutual pressures of primordia. It is assumed that the system grows until the primordia experience contact pressure. Then the centers of the primordia move by growth so as to maximize the minimal distance between these primordia. More precisely, Adler’s maximin principle states that if contact pressure starts at T = T,, then for T 2 T, and for the rise r(T), d has this uniquely determined value for which the distance d(0, n) (a function of d) from leaf 0 to its nearest neighbor n is m aximized. In the absence of contact pressure, d(T) and r(T) are independent of one another, but at T = q.;.,for r given d = d( r( T)). When r >a/38 (arbitrary T) or T < 5 (arbitrary r), max d(0, n) has only one maximum for which we can have phyllotaxis (2,l) or (3,2). The

PLANT

PATTERN

GENERATION

39

main result in Adler’s model states that a sufficient condition to have Fibonacci phyllotaxis is that T, < 5 or r(T,) 26/38. This means that Fibonacci phyllotaxis is unavoidable if contact pressure begins before leaf 5 appears (when there are 5 leaves: 0,1,2,3,4; this compares nicely with Proposition 7) or before r becomes smaller than fi/38 = 0.0456. We have phyllotaxis (2,l) if r is in the interval [fi/14,fi/6] = [0.1237,0.2888], and phyllotaxis (3,2) if r belongs to the interval [0.0456,0.1237]. For T = 5 and r 5 and r -CG/38, two or more maxima, and many types of phyllotaxis are possible. This model does not explain the initial placement of the primordia, and it assumes a spiral arrangement. It explains rising phyllotaxis in general terms, by proving that the system will experience rising phyllotaxis along the sequence (a,b,a+b,2b+a,3b+2a,...) if at time T, the system is expressed by any two consecutive terms of this sequence, and if contact pressure is maintained. Thus we have the normal phyllotaxis path, expressed by the sequence of normal phyllotaxis, and another phyllotaxis path, expressed by the sequence (t,2t+1,3t+1,5t+2,...), which coincidewith the sequence of anomalous phyllotaxis only when t = 2. When contact pressure starts, a rhythm of growth is thus established. This contact pressure rhythm can be considered a representation of the rhythm postulated in the minimal entropy model. Adler’s results can be compared with Propositions 7 and 8. Moreover, assuming orthogonal intersection of the conspicuous pair and a divergence angle in the interval [l/3,1/2], Fibonacci phyllotaxis is unavoidable in the +-model if T, < 6 (when five primordia exist), for which r 2 1.8944/(2+ 3)2 = 0.0758 [a value obtained from the allometry formula (1) close to the mean of the endpoints of the interval given for the (2,3) pattern in Adler’s model], and under the same conditions the phyllotaxis (1,2) is obtained if r 2 1.8944/(1+ 2)* = 0.2105 [a value approximately equal to the mean of the endpoints of the interval given for the (1,2) pattern in Adler’s model]. Adler [l] gives a set of conditions by which the system (2,5) can become (7,5) under contact pressure. Among these conditions is that r be in the interval [o/78,0/84] = [0.0222,0.0315] [to avoid the possibility of the system (3,5) when r < fi/38 > fi/84, and to end the system (2,5) by requiring that r 0.03151, and the system (2,5) is replaced by the system (7,5) for r =fi/78. This latter system is replaced by the system (7,12) for the value of r =fi/218 = 0.0079. It follows that in Adler’s model the system (7,5) is orthogonal for r = (0.0079 + 0.0222)/2 = 0.0151. In the +-model, at m = 12 the system is (5,7), and it is orthogonal when r = 0.0132, a value close to

40

ROGER

V. JEAN

0.0151. The value 0.0222 of r corresponds to an intersection /?, of the conspicuous pair (7,5), of about 63” [from formula (1) again]. I obtained the same angle of 63” in a discussion of Mitchison’s computer simulation (see Jean [22], p. 75). This angle corresponds to the point where a system (m, n) is replaced by a system (m + n, m). In the +-model, given m, a small interval of values for r = rm/m is obtained from formula (l), giving r,,, = p(/?)/m2 for /3 fixed. Indeed, formula (1) is an approximation of the relations taking place in the normalized cylindrical lattice (such as the one in Figure 1). An n being also given by the minimality condition such that the pair (m - n, n) is conspicuous, an interval of values for the divergence angle d is obtained (by a theorem of Section 1C). In the (d, r) phase space, the point (d, r) is thus confined in a small rectangle. As the minimality condition is maintained, the system experiences rising phyllotaxis, and the rectangle shrinks and moves toward the point (d,O), where d is the limit divergence angle corresponding to the sequence (m- n,n,m,m+ n,2m+ n,...). Adler’s model gives a conspicuous pair (m - n, n) for which d(0, m - n) = d(0, n), so that the point (d, r) is confined to an arc of circle inside the rectangle mentioned above (such a functional relation between d and r has not been proved yet for plants experiencing contact pressure). Contact pressure being maintained, the point (d, r) moves on a sequence of arcs of circles, phyllotaxis rises, and r decreases on the contact pressure path. The +-model and the contact pressure model are thus similar in some respects. But in contrast to the +-model the contact pressure model gives only particular conditions (at least in its actual form) for the systems (1,2), (2,3), (3,5), (2,5), and (7,5) to arise, and general formal conditions for which a system is on one contact pressure path or another. No conditions are given in Adler’s model for the emergence under contact pressure of the system expressed by the sequence (1,3,4,7,11,. . ), given that the model assumes a divergence angle between l/3 and l/2-that is, between 120” and 180” -while the preceding sequence requires a divergence between l/4 and l/3, that is, between 90” and 120” (giving Proposition 9). In the case of the +-model the system ruled by the sequence (2,5,7,12,19,. ) will arise at m =12 and not before (in Adler’s model, at 5 I T,. < 6 it can arise), and in the case of the minimal entropy model it will arise only at T, = 19 (see Table 1). C.

FUJITA’S

SPIRAL

PATTERNS

According to Fujita [8], the types of spiral patterns are, a priori, expressed by the sequences J(l,r,r+1,2r+1,3t+2

,... ),

t21,

to be found in nature

J21,

PLANT

PATTERN

with a divergence

GENERATION

angle of [J(t + +-‘)I-‘,

J(p,pa+1,(1+a)p+1,(1+2a)p+2 with a divergence

41

)... ),

angle of [J( p + ( u + c#-‘)-~)]-~,

J(p,up-l,(l+a)p-1,(1+2a)p-2,...),

a22,

~22,

J21,

a23,

522,

and ~23,

withadivergenceangleof {J((p-l)+[(a-1)+~-‘](a+~-‘)-‘)}-’.The first sequence is the already known sequence of normal phyllotaxis. This sequence was deduced by Coxeter [6] from Bravais and Bravais’ assumptions as was mentioned in Section 3A. The minimal entropy model produces all the angles of normal phyllotaxis (see Section 1B and the proof of Proposition 9 for the list of angles permitted by the model). The minimal entropy model also brings the sequence of anomalous phyllotaxis J(2,2t + 1,2t +2,4t +4,. ..), corresponding to Fujita’s second sequence with p = 2 and a = t. For unspoken reasons Adler considers the sequence J( t, 2t + 1,3t + 1,5t + 2,. . . ), t 2 2, corresponding to Fujita’s second sequence with p = t and a = 2. In the latter sequence, putting n = F(k+l)t+ F(k-1) and m=F(k+2)t+F(k), the divergence is in the bounds F( k + 1)/n and F( k + 2)/m so that at the limit the divergence is d = [ J( t + c#- ‘)I- ‘. This divergence decreases monotonically toward zero as t increases, just like the divergence corresponding to the sequence of normal phyllotaxis. Notice that the divergence corresponding to anomalous phyllotaxis in the minimal entropy model increases toward 180”, 90”, 60”, . . . as t increases, depending on the values J = 1,2,3,. . . , respectively, making possible the existence of alternating whorls (distichy: 180”, decussation: 90”, . . .) and of Spiro-alternating whorls. The sequence of normal phyllotaxis makes possible the existence of superposed whorls, such as monostichy, via spiromonostichy (see Jean [23]). Adler’s a priori sequence does not allow the existence of alternating whorls, which are more common than superposed whorls. The $-model generates any one of the sequences of anomalous know that the sequence phyllotaxis. For example, we already ((2,5),7,12,19,. . .) is generated by the model. The value m = 41 gives n = 25, thus giving the sequence ((2,7,9,16), 25,41,66,. . . ). The value M = 82 gives n = 51 and the sequence ((2,9,11,20,31), 51,82,. . .). The value m = 98 gives n = 61 and the sequence ((2,11,13,24,37),61,98,. . .), and so on (see Section 3B for the meaning of the parentheses in the sequences). Fujita’s a priori sequences permit the existence of a wide variety of patterns. Fujita’s divergence angles form a proper subset of Marzec’s set +c or set of noble numbers. It seems that the +-model produces all of Marzec’s set. But we obviously do not need that many angles and thus that many sequences in phyllotaxis. This possibility of so many angles might be an

42

ROGER

V. JEAN

inherent limit for a model. For Popper [31], “every ‘good’ scientific theory is a prohibition: it forbids certain things to happen.” But a model not able to reproduce patterns that are known to exist shows another type of limitation, The angles permitted by the minimal entropy model form a proper subset of the set of Fujita’s angles. From a recent survey of the spiral patterns observed over the past 160 years [21], it can be said that the types of patterns permitted to exist by the minimal entropy model (normal and anomalous sequences) are sufficient to cover all observable cases. 7.

CONCLUSION

A.

CONVERGENCE.7

BETWEEN

MODELS

An analysis of the models available shows that there are convergences and resemblances between them, even when the approaches to the problems are very dissimilar. The preceding sections give many examples of such convergences. Some models have concepts, structures, and results that appear to be more general than others in other models. The +-model shows links and similarities between the usual models (contact pressure, diffusion models) and to the minimal entropy model. which (see [25]) delivers results similar to those of the latter models. But in contrast to them this entropy model is able to predict which types of spiral patterns should exist and which should not, and it orders the patterns according to their relative frequencies of occurrence, that is, according to their increasing entropy costs. For example, the pattern expressed by the sequence (1,2,3,5,8,. . . ) is more frequent than the pattern expressed by the sequence 2(1,2,3,5,8,. . . ), which is more frequent than the one expressed by (1,3,4,7,11,...). This is in a good agreement with the available observations (see [21]). This model, extended recently to superposed and alternating whorls (Jean [23]), predicts, for example, the existence of a presumably impossible type of pattern I called monostichy, where all leaves would be on a single line (orthostichy) parallel to the axis of the plant. I learned then that this type of pattern exists in Utricularia. This extended model has been used recently by Barabe and Vieth [2] to give a more coherent account of the phyllotaxis of Dipsacus than was possible with the earlier interpretations of Plantefol and DeVries. The systemic coherency afforded by the concept of entropy can be the basis for a supermodel that generalizes and unifies the approaches to phyllotaxis. All models, by the very definition of model, are formal, even the usual models that are more or less biologically grounded. Indeed the mechanistic theories have varying amounts of appeal given such principles as contact pressure and diffusion of an inhibitor. Moreover, these postulates can presumably undergo scrutiny in the laboratory, but recent experimental research shows that such scrutiny rather brings discredit to them (see [lo], [26]). Not only is the diffusion theory of an inhibitor not supported by any

PLANT

PATTERN

GENERATION

43

data, but also there are contradictory data. It is well known that the usual theories, built on Hofmeister’s rule according to which the primordia should arise as far away as possible from the existing primordia (thus giving spiral patterns and alternating whorls), are contradicted by the mere existence of superposed whorls. Even if the growth, form, and position of young leaves depend on physical and chemical constraints, we must be careful not to identify these constraints with the postulates just mentioned. Sinnott ([41], p. 161) regarded them as too simple and not logical enough. An emerging trend that is in line with the holistic point of view of the minimal entropy predictive model, and the idea of diffusion of substances at the same time, is that the free energy of the hydrous network is the interface between the genetic and environmental determinisms, and the center of a system of functional coordination controlling the architecture and the distribution of resources of the plant as a whole. The primordia would be harmonic functions of the hydrous potentials diffusing throughout the plants that would translate the influence of the various environmental tensions (gravity, heat, light, radiation, wind, etc.) and would modulate the expressions of a plant genotype. “The lack of consistent evidence of hormonal control of plant development is attributed to the overriding influence of water,. . The fundamental effects of. water on growth and metabolism suggest that water may have played a major role in the evolution of the various physiological mechanisms that regulate the pattern of plant development” (McIntire [30]). B.

SYNERGY

Phyllotaxis has a biological facet and a mathematical facet. Expressing himself generally, Rosen [37] underlines “that no resolution, or even proper formulation, of these problems can be forthcoming unless the mathematician and the physicist can bring their skills, techniques and insights to bear upon them.” This is particularly true for the problems of phyllotaxis, and it is in line with the thoughts of D’Arcy Thompson, for whom “problems of forms are in the first instance mathematical problems, and problems of growth are essentially physical problems.” Mathematics deals not only with descriptive phyllotaxis but also with causal phyllotaxis, via hypothetico-predictive models that must be confirmed or disconfirmed by the biologist. In Hinduism we are told that God is like an elephant surrounded by blind men. Each man touches the elephant from the point where he is-the trunk, the tail, the foot, the head, the forehead, and so on-and believes he holds the whole: thus we have the wars of religions. There are many investigators gathered around the field of phyllotaxis; each has his point of view, and his metaphysics is secretly guiding him. I believe that each perceives a part of the reality and that a sound approach today consists in trying to integrate all the known parts, which, although to some extent antagonistic and contradic-

44

ROGER

V. JEAN

tory, are complementary to each other. I thus agree with Rutishauser and Sattler’s philosophy of perspectivism in phyllotaxis, “according to which complementary models (theories, interpretations, descriptions, etc.) are seen as different perspectives of the same phenomenon” [40]. To complement that point of view, I propose to view the modeling work pyramidally. We should speak about a pyramid of models. Perspectivism seems to refer to a horizontal dimension among selected models or theories viewed as complementary, but the idea of a pyramid refers to both horizontal and vertical dimensions of modeling in phyllotaxis and takes into account every attempt to solve the problem. Theoretically, the main idea is to devise models able to gather under a single view more and more models, so as to finally reach the top of the pyramid of models. The pyramid must be based on reliable data and well-grounded descriptions (see [21, 221). The top of the pyramid represents a comprehensive biomathematical model concerned with general principles incorporating ideas of energy, entropy, and rhythm and having all the explanatory power the other models have. This model might bring a Copemican revolution and will point out the errors of perspective and the limits of validity of the other models in the pyramid. From the beginning of my investigations, I have favored a holistic and synergic point of view from which to look at the problems of phyllotaxis. Future research in this field will have to decidedly give strong emphasis to it, when one considers moreover the phenomenon of spiral pattern formation in other areas of research, as we will now briefly outline. C.

A PROBLEM

REVISITED

The resolution of the problem of phyllotaxis seems to transcend the strict domain of botany. Indeed the crux of the botanical problem of phyllotaxis is the factors determining the position of plant primordia. But this problem is a part of a more general problem concerning arrangements of primordia in general. Indeed there are primordia in such living systems as Hydra and Botryllus “where the over-all process of column growth and bud formation is strictly comparable to that of leaf primordia” (Berrill [4], pp. 228 and 423). Tentacles, canals, and zooids in some medusae conform exactly to (2,3) and (3,5) phyllotaxis [27]. Furthermore, one must not forget that phyllotaxis-like patterns arise at other levels of organization, such as in microorganisms showing 2(2,3) and 2(3,5) patterns [7] and proteins with (7,ll) phyllotaxis and 99.5” divergence [19]. Based on the observation that different substrata generate similar patterns, we are inclined to search for a fundamental principle applicable to all these cases. Beyond the tremendously complex processes involved in the growth of these various organisms, general and simple morphogenetic laws apply, such as the principle of optimal design, the principle of minimal entropy production, the simple laws of packing efficiency, and those of

PLANT PATTERN

GENERATION

45

fractal geometry generating structures such as the hierarchical organization of the vascular system (see [15, 201). With the discovery of quasicrystals in 1984, the discipline of phyllotaxis is becoming a playground for physicists and crystallographers, who now use terms such as parastichy and rising phyllotaxis to describe their lattices. Their own terms, such as cylindrical crystal, dislocation and disclination, scale invariance, inflation and deflation, and quasiperiodic lattices, are begimring to invade the field. For Rivier [34, 351, who used the language of crystallography to describe the phyllotactic pattern of the daisy, it is clear that “phyllotaxis does indeed belong to crystallography.” Rothen and Koch [38, 391 consider that phyllotaxis belongs to exotic crystallography, given that phyllotaxis makes use of concepts from many domains of physics and crystallography. For Hermant [9] (see Jean [20]) phyllotaxis is a purely formal, architectural problem. These remarks bear upon the future of biological approaches in phyllotaxis. A first important consequence is that the models based on diffusion of phytohormones, which have always been favored in botanical circles, have to be revaluated, if not discarded. The application of the methods of phyllotaxis in other fields of research, and the application of the methods of other fields in phyllotaxis, are bound to deeply and irreversibly affect the way researchers look at the problems and the approaches they take to the problems of phyllotaxis. I thank Dr. Sergei V. Petukhov from the USSR Academy of Sciences, Mechanical Engineering Research Institute, Moscow, for his comments on the article while he was a visiting professor in Rimouski. This article has been made possible by the Natural Sciences and Engineering Research Council of Canada, grant A6240.

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in S&x (1969).

2,5,7,12,. __ in der Schraubigen Blattstellung und die Verschiedener Zahlenreihensysteme, Bat. Mug. Tok_ro

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44 35 36 31

38 39 40

41 42 43 44

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G. I. M&tire, The role of water in the regulation of plant development, Can. J. Bot. 65:1287-1298 (1987). K. Popper, The Logic of Scientific Discouety, Harper & Row, New York, 1968. J. N. Ridley, Packing efficiency in sunflower heads, Math. Biosci. 58:129-139 (1982). J. P. Rigaut, Fractals, semi-fractals et biometric, in Fractals, Dimensions Non-Entikes et Applications, G. Sherbit, ed., Masson, Paris, pp. 231-281, 1987. N. Rivier, A botanical quasicrystal, J. Phys. CON. Cj, Suppl. 47:299-309 (1986). N. Rivier, Crystallography of spiral lattices, Mod. Phys. Lerr. B 2:953-960 (1988). R. Rosen, Optimality Principles in Biology, Butterworth, London, 1967. R. Rosen, The generation and recognition of patterns in biological systems, Mathematics and the Life Sciences (Lecture Notes in Biomath. Ser.), Springer-Verlag, New York, pp. 222-341, 1977. F. Rothen and A. J. Koch, Phyllotaxis, or the properties of spiral lattices. I. Shape invariance under compression, J. Phys. (1989), to appear. F. Rothen and A. J. Koch, Phyllotaxis, or the properties of spiral lattices. II. Packing of circles along logarithmic spirals,‘submitted/ 1989. R. Rutishauser/and R. Sattler,‘Complementarity and ,heuristic value of contrasting models in structural botany I. General considerations, Bar. Juhrb. Sysr. 107:415-455 (1985). E. W. Sinnott, Plant Morphogenesis, McGraw-Hill, New York, 1960. J. H. M. Thomley, Phyllotaxis I: A mechanistic model, Ann. Bar. 39:491-507 (1975). M. A. Watson and B. B. Casper, Morphogenetic constraints on patterns of carbon distribution in plants, Ann. Rev. Ecol. Sysf. 15:233-258 (1984). C. Wright, On the uses and origin of the arrangements of leaves in plants, Mem. Am. Acad.

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(1873).