[Illustration: FIG. 29.] § 12. The third form is more complex, and we must take the pains to follow out what we left unobserved in last chapter respecting the way a triplicate plant gets out of its difficulties. Draw a circle as in Fig. 29, and two lines, AB, BC, touching it, equal to each other, and each divided accurately in half where they touch the circle, so that AP shall be equal to PB, BQ, and QC. And let the lines AB and BC be so placed that a dotted line AC, joining their extremities, would not be much longer than either of them. [Illustration: FIG. 30.] Continue to draw lines of the same length all round the circle. Lay five of them, AB, BC, CD, DE, EF. Then join the points AD, EB, and CF, and you have Fig. 30, which is a hexagon, with the following curious properties. It has one side largest, CD, two sides less, but equal to each other, AE and BF; and three sides less still, and equal to each other, AD, CF, and BE. [Illustration: FIG. 31.] [Illustration: FIG. 32.] Now put leaves into this hexagon, Fig. 31, and you will see how charmingly the rhododendron has got out of its difficulties. The next cycle will put a leaf in at the gap at the top, and begin a new hexagon. Observe, however, this geometrical figure is only to the rhododendron what the _a_ in Fig. 25 is to the oak, the icy or dead form. To get the living normal form we must introduce our law of succession. That is to say, the five lines A B, B C, &c., must continually diminish, as they proceed, and therefore continually approach the centre; roughly, as in Fig. 32. [Illustration: FIG. 33.] § 13. I dread entering into the finer properties of this construction, but the reader cannot now fail to feel their beautiful result either in the cluster in Fig. 26, or here in Fig. 33, which is a richer and more oblique one. The three leaves of the uppermost triad are perfectly seen, closing over the bud; and the general form is clear, though the lower triads are confused to the eye by unequal development, as in these complex arrangements is almost always the case. The more difficulties are to be encountered the more licence is given to the plant in dealing with them, and we shall hardly ever find a rhododendron shoot fulfilling its splendid spiral as an oak does its simple one. Here, for instance, is the actual order of ascending leaves in four rhododendron shoots which I gather at random. [Illustration: FIG. 34.] Of these, A is the only quite well-conducted one; B takes one short step, C, one step backwards, and D, two steps back and one, too short, forward. [Illustration: FIG. 35.]