Fibonacci numbers in phyllotaxis : a simple model

A simple model is presented which explains the occurrence of high order Fibonacci number paras-tichies in asteracae flowers by two distinct steps. First low order parastichies result from the fact that a new floret, at its appearance is repelled by two former ones, then, in order to accommodate for the increase of the radius, parastichies numbers have to evolve and can do it only by applying the Fibonacci recurrence formula. The beautiful spirals observed on sunflowers or daisies have puzzled scientists since it was realized that the numbers of these spirals belonged to the Fibonacci sequence. The basic principle of phyllotaxis has been formulated as early as 1868 by Hofmeister [1] and states that each new structure (leaf, floret, seed,etc.) appears " opposite " to the previous one-or ones. For instance a new leaf often appears opposite to the previous one, which leads to an alternate pattern. When however the new leaf feels the presence, with a smaller amplitude, of the next previous one then the resulting pattern is starry as depicted in fig 1. If leaves appear in pairs, opposite to each other (de-FIG. 1: The new structure N is repelled by its predecessor N-1 and somewhat less by the former one N-2. It appear at an angle α from its predecessor with α between 135 • and 144 •. At steady state, a starry pattern is formed where each structure is close to its 5th and 8th predecessors and successors. cussate pattern), each pair is perpendicular to the previous one. It is worth noting that alternate and spiral patterns can coexist in the same plant and that a decus-sate pattern can turn into a spiral one on a stem: the pattern is sensitive to local influences or noise. In many observed cases and in simulations with a wide range of parameters the angle of the starry pattern lies between 135 • = 360 • × 3/8 and 144 • = 360 • × 2/5 which means that the " structure " of order n will be close to those of order n ± 8 and n ± 5.If the growth is axial this leads to the commonly observed pattern of crossing helices of order 5 and 8. A pineapple is a typical example. If the growth is radial, spirals (in place of helices) of higher order appear. These orders are not random but belong to the Fibonacci sequence …