Fibonacci or quasi-symmetric phyllotaxis. Part II: botanical observations

Abstract

Historically, the study of phyllotaxis was greatly helped by the simple description of botanical patterns by only two integer numbers, namely the number of helices (parastichies) in each direction tiling the plant stem. The use of parastichy numbers reduced the complexity of the study and created a proliferation of generalizations, among others the simple geometric model of lattices. Unfortunately, these simple descriptive method runs into difficulties when dealing with patterns presenting transitions or irregularities. Here, we propose several ways of addressing the imperfections of botanical reality. Using ontogenetic analysis, which follows the step-by-step genesis of the pattern, and crystallographic analysis, which reveal irregularity in its details, we show how to derive more information from a real botanical sample, in particular, about its irregularities and transitions. We present several examples, from the first explicit visualization of a normal Fibonacci parastichy number increase, to more exotic ones, including the quasi-symmetric patterns detected in simulations. We compare these observations qualitatively with the result of the disk-packing model, presenting evidence for the relevance of the model.

Fibonacci or quasi-symmetric phyllotaxis. Part I: why?

“…Reliance on the convergence of the divergence angle imposes a limitation on the speed of decrease of the radius and length of transition not often observed in plants. And indeed these types of oscillating patterns are seen in plants [1,23,27,39], and seem far from exceptional. The fluctuations in vertical spacing, also a sign of the order switching, are clearly visible in Fig.…”

“…5. Part II [23] provides more botanical examples of these patterns and evidence of the route to quasi-symmetry as explained with the model here.…”

“…In [6] and Part II of this work [23], evidence is presented that rhombic tilings are actually a better model of constant parastichy phyllotaxis than the more restrictive lattices, as they fit plant data better, accounting for instance for permutations of order in the pattern observed by botanists [37]. Nonetheless, the set of rhombic lattices and its van Iterson diagram forms some kind of skeleton, and serves as reference for the much more complicated set of rhombic tilings.…”

“…See section "Heuristic arguments for MGRS and QSS" for a heuristic argument for its prevalence. Part II [23] of this work will provide clear botanical examples of such Fibonacci transitions, both increasing and decreasing. ■ How does this argument compare to previous ones in the literature?…”

“…Increasing b induces cascades of pentagon transitions, usually affecting the parastichy number corresponding to the most numerous front vectors, as they are the most likely to contribute to the side of the notch with two vectors. See [23] for botanical examples.…”