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 {643} cones of the Norway spruce, Beal[578] 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[579]. 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 {644} 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[580].
Fig. 325.
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 fir-cone, regarded as cylindrical, and unwrapped to form a plane surface. A, a, at the two ends of the base-line, represent the same initial leaf or scale: O is 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 aO. 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, l, it can easily be shewn that the arrangement is composite, and that there are l fundamental, or genetic spirals, and l leaves (including A) which are situated exactly on the line Aa. 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 a) to the leaf which is situated nearest to the base-line Aa. The fundamental spiral will thus be right-handed (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 {645} 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, like 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
Fig. 326.