On Growth and Form

elongated fir-cone; or, on the other hand, we may be more attracted by, and regard as of greater importance, the logarithmic spirals which we trace in the curving rows of florets in the discoidal inflorescence of a sunflower. Whether one way or the other be the better, or even whether one be not positively correct and the other radically wrong, has been vehemently debated. In my judgment they are, both mathematically and biologically, to be regarded as inseparable and correlative phenomena.

The helical arrangement (as in the fir-cone) was carefully studied in the middle of the eighteenth century by the celebrated Bonnet, with the help of Calandrini, the mathematician. Memoirs published about 1835, by Schimper and Braun, greatly amplified Bonnet’s investigations, and introduced a nomenclature which still holds its own in botanical textbooks. Naumann and the brothers Bravais are among those who continued the investigation in the years immediately following, and Hofmeister, in 1868, gave an admirable account and summary of the work of these and many other writers*.

Starting from some given level and proceeding upwards, let us mark the position of some one leaf (A) upon a cylindrical stem. Another, and a younger leaf (B) will be found standing at a certain distance around the stem, and a certain distance along the stem, from the first. The former distance may be expressed as a fractional “divergence” (such as two-fifths of the circumference of the stem) as the botanists describe it, or by an “angle of azimuth” (such as dé = 144°) as the mathematician would be more likely to state it. The position of B relatively to A must be determined, not only by this angle ¢, in the horizontal plane, but also by an angle (@) in the vertical plane; for the height of B above the level of A, in comparison with the diameter of the cylinder, will obviously make a great difference in the appearance of the whole system, in short the position of each leaf must be expressed by F(¢.sin@). But this matter botanical students have not concerned themselves with; in other words, their studies have been limited (or mainly limited) to the relation of the leaves to one another in azimuth.

Whatever relation we have found between A and B, let pretisely the same relation subsist between B and C: and so on. Let the growth of the system, that is to say, be continuous and uniform ; it is then evident that we have the elementary conditions for the development of a simple cylindrical helix; and this “primary helix” or “genetic spiral” we can now trace, winding -round and round the stem, through A, B, C, ete. But if we can trace such a helix through 4, B, C, it follows from the symmetry of the system, that we have only to join A to some other leaf to trace another spiral helix, such, for instance, as A, C, EH, etc.; parallel to which will run another and similar one, namely in this case B, D, F, etc. And these spirals will run in the opposite direction to the spiral ABC.

In short, the existence of one helical arrangement of points implies and involves the existence of another and then another helical pattern, just as, in the pattern of a wall-paper, our eye travels from one linear series to another. . A modification of the helical system will be introduced when, instead of the leaves appearing, or standing, in singular succession, we get two or more appearing simultaneously upon the same level. If there be two such, then we shall have two generating spirals precisely equivalent to one another; and we may call them A, B, C, etc., and A’, B’, C’, and so on. These are the cases which we call “whorled” leaves, or in the simplest case, where

the whorl consists of two opposite leaves only, we call them decussate. : Among the phenomena of phyllotaxis, two points in particular have been found difficult of explanation, and have aroused discussion. These are (1), the presence of the logarithmic spirals such as we have already spoken of in the sunflower; and (2) the fact that, as regards the number of the helical or spiral rows, certain numerical coincidences are apt to recur again and again, to the exclusion of others, and so to become characteristic features of the phenomenon.

' The first of these appears to me to present no difficulty. It is a mere matter of strictly mathematical “deformation.” ~The stem which we have begun to speak of asa cylinder is not strictly so, inasmuch as it tapers off towards its summit. The curve which winds evenly around this stem is, accordingly, not a true helix, for that term is confined to the curve which winds evenly around the cylinder: it is a curve in space which (like the spiral curve we have studied in our turbinate shells) partakes of the characters of a helix and of a logarithmic spiral, and which is in fact a logarithmic spiral with its pole drawn out of its original plane by a force acting in the direction of the axis. If we imagine a tapering cylinder, or cone, projected, by vertical projection, on a plane, it becomes a circular disc; and a helix described about the cone necessarily becomes in the disc a logarithmic spiral described about a focus which corresponds to the apex of our cone. In like manner we may project an identical spiral in space upon such surfaces as (for instance) a portion of a sphere or of an ellipsoid ; and in all these cases we preserve the spiral configuration, which is the more clearly brought into view the more we reduce the vertical component by which it was accompanied. The converse is, of course, equally true, and equally obvious, namely that any logarithmic spiral traced upon a circular disc or spheroidal surface will be transformed into a corresponding spiral helix when the plane or spheroidal disc is extended into an elongated cone approximating to a cylinder. This mathematical conception is translated, in botany, into actual fact. The fir-cone may be looked upon as a cylindrical axis contracted at both ends, until

it becomes approximately an ellipsoidal solid of revolution, generated about the long axis of the ellipse; and the semi-ellipsoidal capitulum of the teasel, the more or less hemispherical one of the thistle, and the flattened but still convex one of the sunflower, are all beautiful and successive deformations of what is typically a long, conical, and all but cylindrical stem. On the other hand, every stem as it grows out into its long cylindrical shape is but a deformation of the little spheroidal or ellipsoidal surface, or cone, which was its forerunner in the bud.

This identity of the helical spirals around the stem with spirals projected on a plane was clearly recognised by Hofmeister, who was accustomed to represent his diagrams of leaf-arrangement either in one way or the other, though not in a strictly geometrical projection*. According to Mr A. H. Church}, who has dealt very carefully and elaborately with the whole question of phyllotaxis, the logarithmic spirals such as we see in the disc of the sunflower have a far greater importance and a far deeper meaning than this brief treatment of mine would accord to them: and Sir Theodore Cook, in his book on the Curves of Life, has adopted and has helped to expound and popularise Mr Church’s investigations.

Mr Church, regarding the problem as one of “uniform growth,” easily arrives at the conclusion that, 7f this growth can be conceived as taking place symmetrically about a central point or “pole,” the uniform growth would then manifest itself in logarithmic spirals, including of course the limiting cases of the circle and straight line. With this statement I have little fault to find; it is in essence identical with much that I have said in a previous chapter. But other statements of Mr Church’s, and many theories woven about them by Sir T. Cook and himself, I am less able to follow. Mr Church tells us that the essential phenomenon in the sunflower disc is a series of orthogonally intersecting logarithmic spirals. Unless I wholly misapprehend Mr Church’s meaning, I should say that this is very far from essential: The spirals

intersect isogonally, but orthogonal intersection would be only one particular case, and in all probability a very infrequent .one, in the intersection of logarithmic spirals developed about a common pole. Again on the analogy of the hydrodynamic lines of force in certain vortex movements, and of similar lines of force in certain magnetic phenomena, Mr Church proceeds to argue that the energies of life follow lines comparable to those of electric energy, and that the logarithmic spirals of the sunflower are, so to speak, lines of equipotential*. And Sir T. Cook remarks that this “theory, if correct, would be fundamental for all forms of growth, though it would be more easily observed in plant construction than in animals.” The parallel I am not able to follow.

Mr Church sees in phyllotaxis an organic mystery, a something for which we are unable to suggest any precise cause: a phenomenon which is to be referred, somehow, to waves of growth emanating from a centre, but on the other hand not to be explained by the division of an apical cell, or any other histological factor. As Sir T. Cook puts it, “at the growing poimt of a plant where the new members are being formed, there is simply nothing to see.” But it is impossible to deal satisfactorily, in brief space, either’ with Mr Church’s theories, or my own objections to themt. Let it suffice to say that I, for my part, see no subtle mystery in the matter, other than what lies in the steady production of similar growing parts, similarly situated, at similar successive intervals of time. If such be the case, then we are bound to have in

* “The proposition is that the genetic spiral is a logarithmic spiral, homologous with the line of current-flow in a spiral vortex; and that in such a system the action of orthogonal forces will be mapped out by other orthogonally intersecting logarithmic spirals—the ‘parastichies’”; Church, op. cit. 1, p. 42. + Mr Church’s whole theory, if it be not based upon, is interwoven with, Sachs’s theory of the orthogonal intersection of cell-walls, and the elaborate theories of the symmetry of a growing point or apical cell which are connected therewith. According to Mr Church, “the law of the orthogonal intersection of cell-walls at a growing apex may be taken as generally accepted” (p. 32): but I have taken a very different view of Sachs’s law, in the eighth chapter of the present book. With regard to his own and Sachs’s hypotheses, Mr Church makes the following curious remark (p. 42): ‘‘ Nor are the hypotheses here put forward more imaginative than that of the paraboloid apex of Sachs which remains incapable of proof, or his

construction for the apical cell of Pteris which does not satisfy the evidence of his own drawings.” consequence a series of symmetrical patterns, whose nature will depend upon the form of the entire surface. If the surface be that of a cylinder we shall have a system, or systems, of spiral helices: if it be a plane, with an infinitely distant focus, such as we obtain by “unwrapping” our cylindrical surface, we shall have straight lines; if it be a plane containing the focus within itself, or if it be any other symmetrical surface containing the focus, then we shall have a system of logarithmic spirals. The appearance of these spirals is sometimes spoken of as a “subjective ” phenomenon, but the description is inaccurate: it is a purely mathematical phenomenon, an inseparable secondary result of other arrangements which we, for the time being, regard as primary. When the bricklayer builds a factory chimney, he lays his bricks in a certain steady, orderly way, with no thought of the spiral patterns to which this orderly sequence inevitably leads, and which spiral patterns are by no means “subjective”? The designer of a wall-paper not only has no intention of producing a pattern of criss-cross lines, but on the contrary he does his best to avoid them; nevertheless, so long as his design is a symmetrical one, the criss-cross intersections inevitably come.

Let us, however, leave this discussion, and return to the facts of the case. Our second question, which relates to the numerical coincidences so familiar to all students of phyllotaxis, is not to be set and answered in a word. Let us, for simplicity’s sake, avoid consideration of simultaneous or whorled leaf origins, and consider only the more frequent cases where a single “genetic spiral’ can be traced throughout the entire system. It is seldom that this primary, genetic spiral catches the eye, for the leaves which immediately succeed one another in this genetic order are usually far apart on the circumference of the stem, and it is only in close-packed arrangements that the eye readily apprehends the continuous series. Accordingly in such a case as a fir-cone, for instance, it is certain of the secondary Spirals or “parastichies” which catch the eye; and among fir-cones, we can easily count these, and we find them to be

on the whole very constant in number, according to the species. Thus in many cones, such as those of the Norway spruce, we can trace five rows of scales winding steeply up the cone in one direction, and three rows winding less steeply the other way; in certain other species, such as the common larch, the normal number is eight rows in the one direction and five in the other; while in the American larch we have again three in the one direction and five in the other. It not seldom happens that two arrangements grade into one another on different parts of one and the Same cone. Among other cases in which such spiral series are readily visible we have, for instance, the crowded leaves of the stone-crops and mesembryanthemums, and (as we have said) the crowded florets of the composites. Among these we may find plenty of examples in which the numbers of the serial rows are similar to those of the fir-cones; but in some cases, as in the daisy and others of the smaller composites, we shall be able to trace thirteen rows in one direction and twenty-one in the other, or

perhaps twenty-one and thirty-four; while in a great big sunflower we may find (in one and the same species) thirty-four and fifty-five, fifty-five and eighty-nine, or even as many as eighty-nine and one hundred and forty-four. On the other hand, in an ordinary *pentamerous” flower, such as a ranunculus, we may be able to trace, in the arrangement of its sepals, petals and stamens, shorter spiral series, three in one direction and two in the other. It will be at once observed that these arrangements manifest themselves in connection with very different things, in the orderly interspacing of single leaves and of entire florets, and among all kinds of leaf-like structures, foliage-leaves, bracts, cone-scales, and the various parts or members of the flower. Again we must be careful to note that, while the above numerical characters are by much the most common, so much so as to be deemed “normal,” many other combinations are known to occur.

The arrangement, as we have seen, is apt to vary when the entire structure varies greatly in size, as in the disc of the sunflower. It is also subject to less regular variation within one and the same species, as can always be discovered when we examine a sufficiently large sample of fir-cones. For instance, out of 505 cones of the Norway spruce, Beal* found 92 per cent. in which the spirals were in five and eight rows; in 6 per cent. the rows were four and seven, and in 4 per cent. they were four and six. In each case they were nearly equally divided as regards direction ; for instance of the 467 cones shewing the five-eight arrangement, the five-series ran in right-handed spirals in 224 cases, and in left-handed spirals in 243.

Omitting the “abnormal” cases, such as we have seen to occur in a small percentage of our cones of the spruce, the arrangements which we have just mentioned may be set forth as follows, (the fractional number used being simply an abbreviated symbol for the number of associated helices or parastichies which we can count running in the opposite directions): 2/3, 3/5, 5/8, 8/13, 13/21, 21/34, 34/55, 55/89, 89/144. Now these numbers form a very interesting series, which happens to have a number of curious mathematical properties+. We see, for instance, that the denominator of each fraction is the numerator of the next; and further, that each successive numerator, or denominator, is the sum of the preceding two. Our immediate problem, then, is to determine, if possible, how these numerical coincidences come about, and why these particular numbers should be so commonly met with

+ This celebrated series, which appears in the continued fraction Ley D cet: and is closely connected with the Sectio awrea or Golden Mean, is commonly called the Fibonacci series, after a very learned twelfth century arithmetician (known also as Leonardo of Pisa), who has some claims to be considered the introducer of Arabic numerals into christian Europe. It is called Lami’s series by some, after Father Bernard Lami, a contemporary of Newton’s, and one of the co-discoverers of the parallelogram of forces. It was well-known to Kepler, who, in his paper De nive sexangula (ct. supra, p. 480), discussed it in connection with the form of the dodecahedron and icosahedron, and with the ternary or quinary symmetry of the flower. (Cf. Ludwig, F., Kepler iiber das Vorkommen der Fibonaccireihe im Pflanzenreich, Bot. Centralbl. Lxvmm, p. 7, 1896). Professor William Allman, Professor of Botany in Dublin (father of the historian of Greek geometry), speculating on the same facts, put forward the curious suggestion that the cellular tissue of the dicotyledons, or exogens, would be found to consist of dodecahedra, and that of the monocotyledons or endogens of icosahedra (On the mathematical connexion between the parts of Vegetables: abstract of a Memoir read before the Royal Society in the year 1811 (privately printed, n.d.). Cf. De Candolle, Organogénie végétale, 1, p. 534).

as to be considered “normal” and characteristic features of the general phenomenon of phyllotaxis. The following account is based on a short paper by Professor P. G. Tait*. Of the two following diagrams, Fig. 325 represents the general case, and Fig. 326 a particular one, for the sake of possibly greater simplicity. Both diagrams represent a portion of a branch, or Q fir-cone, regarded as cylindrical, and unwrapped to form a plane surface. A, a, at the two ends of the base-line, represent the

Fig. 325. same initial leaf or scale: Ois a leaf which can be reached from A by m steps in a right-hand spiral (developed into the straight line AO), and by n steps from a in a left-handed spiral a0. Now it is obvious in our fir-cone, that we can include all the scales upon the cone by taking so many spirals in the one direction, and again include them all by so many in the other. Accordingly, in our diagrammatic construction, the spirals AO and aO must, and always can, be so taken that m spirals parallel to aO, and n spirals parallel to AO, shall separately include all the leaves upon the stem or cone.

If m and n have a common factor, J, it can easily be shewn that the arrangement is composite, and that there are / fundamental, - or genetic spirals, and / leaves (including A) which are situated exactly on the line da. That is to say, we have here a whorled arrangement, which we have agreed to leave unconsidered in favour of the simpler case. We restrict ourselves, accordingly, to the cases where there is but one genetic spiral, and when therefore m and n are prime to one another.

Our fundamental, or genetic, spiral, as we have seen, is that which passes from A (or «) to the leaf which is situated nearest to the base-line da. The fundamental spiral will thus be righthanded (A, P, etc.) if P, which is nearer to A than to a, be this leaf—left-handed if it be p. That is to say, we make it a convention that we shall always, for our fundamental spiral, run round the system, from one leaf to the next, by the shortest way. Now it is obvious, from the symmetry of the figure (as further shewn in Fig. 326), that, besides the spirals running along AO and aO, we have a series running from the steps on aO to the steps on AO. In other words we can find a leaf (S) upon AO, which, hke the leaf O, is reached directly by a spiral series from A and from a, such that aS includes n steps, and AS (being part of the old

spiral line AO) now includes m—n steps. And, since m and n are prime to one another (for otherwise the system would have been a composite or whorled one), it is evident that we can continue this process of convergence until we come down to a 1, 1 arrangement, that is to say to a leaf which is reached by a single step, in opposite directions from A and from a, which leaf is therefore the first leaf, next to A, of the fundamental or generating spiral.

If our original lines along AO and aO contain, for instance, 13 and 8 steps respectively (i.e. m= 13, n = 8), then our next series, observable in the same cone, will be 8 and (13 — 8) or 5; the next 5 and (8 — 5) or 3; the next 3, 2; and the next 2, 1; leading to the ultimate condition of 1, 1. These are the very series which we have found to be common, or normal; and so far as our investigation has yet gone, it has proved to us that, if one of these exists, it entails, ipso facto, the presence of the rest.

In following down our series, according to the above construction, we have seen that at every step we have changed direction, the longer and the shorter sides of our triangle changing places every time. Let us stop for a moment, when we come to the 1, 2 series, or AT, aT of Fig. 326. It is obvious that there is _nothing to prevent us making a new 1, 3 series if we please, by continuing the generating spiral through three leaves, and connecting the leaf so reached directly with our initial one. But in the case represented in Fig. 326, it is obvious that these two series (4, 1, 2, 3, etc., and a, 3, 6, etc.) will be running in the same direction ; i.e. they will both be right-handed, or both left-handed spirals. The simple meaning of this is that the third leaf of the generating spiral was distant from our initial leaf by more than the circumference of the cylindrical stem; in other words, that there were more than two, but less than three leaves in a single turn of the fundamental spiral. .

Less than two there can obviously never be. When there are exactly two, we have the simplest of all possible arrangements, namely that in which the leaves are placed alternately on opposite sides of the stem. When there are more than two, but less than three, we have the elementary condition for the production of the series which we have been considering, namely 1, 2; 2, 3; 3, 5, etc. To put the latter part of this argument in more precise language, let us say that: If, in our descending series, we come to steps 1 and ¢, where ¢ is determined by the condition that 1 and t+ 1 would give spirals both right-handed, or both left-handed ; it follows that there are less than ¢ + 1 leaves in a single turn of the fundamental spiral. And, determined in this manner, it is found in the great majority of cases, in fir-cones and a host of other examples of phyllotaxis, that = 2. In other words, in the

great majority of cases, we have what corresponds to an arrangement next in order of simplicity to the simplest case of all: next, that is to say, to the arrangement which consists of opposite and alternate leaves. “These simple considerations,” as Tait says, “explain completely the so-called mysterious appearance of terms of the recurring series 1, 2, 3, 5, 8, 13, etc.* The other natural series, usually but misleadingly represented by convergents to an infinitely extended continuous fraction, are easily explained, as above, by thking t = 3, 4, 5, etc., etc.” Many examples of these latter series have been given by Dickson and other writers.

We have now learned, among other elementary facts, that wherever any one system of helical spirals is present, certain others invariably and of necessity accompany it, and are definitely related to it. In any diagram, such as Fig. 326, in which we represent our leaf-arrangement by means of uniform and regularly interspaced dots, we can draw one series of spirals after another, and one as easily as another. But in our fir-cone, for instance, one particular series, or rather two conjugate series, are always conspicuous, while the others are sought and found with comparative difficulty.