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Articles 1 - 5 of 5
Full-Text Articles in Physical Sciences and Mathematics
Phyllotaxis As Geometric Canalization During Plant Development, Christophe Godin, Christophe Golé, Stéphane Douady
Phyllotaxis As Geometric Canalization During Plant Development, Christophe Godin, Christophe Golé, Stéphane Douady
Mathematics Sciences: Faculty Publications
Why living forms develop in a relatively robust manner, despite various sources of internal or external variability, is a fundamental question in developmental biology. Part of the answer relies on the notion of developmental constraints: at any stage of ontogenesis, morphogenetic processes are constrained to operate within the context of the current organism being built. One such universal constraint is the shape of the organism itself, which progressively channels the development of the organism toward its final shape. Here, we illustrate this notion with plants, where strikingly symmetric patterns (phyllotaxis) are formed by lateral organs. This Hypothesis article aims first …
Convergence In A Disk Stacking Model On The Cylinder, Christophe Golé, Stéphane Douady
Convergence In A Disk Stacking Model On The Cylinder, Christophe Golé, Stéphane Douady
Mathematics Sciences: Faculty Publications
We study an iterative process modeling growth of phyllotactic patterns, wherein disks are added one by one on the surface of a cylinder, on top of an existing set of disks, as low as possible and without overlap. Numerical simulations show that the steady states of the system are spatially periodic, lattices-like structures called rhombic tilings. We present a rigorous analysis of the dynamics of all configurations starting with closed chains of 3 tangent, non-overlapping disks encircling the cylinder. We show that all these configurations indeed converge to rhombic tilings. Surprisingly, we show that convergence can occur in either …
Fibonacci Or Quasi-Symmetric Phyllotaxis. Part Ii: Botanical Observations, Stéphane Douady, Christophe Golé
Fibonacci Or Quasi-Symmetric Phyllotaxis. Part Ii: Botanical Observations, Stéphane Douady, Christophe Golé
Mathematics Sciences: Faculty Publications
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 num- bers reduced the complexity of the study and created a proliferation of generaliza- tions, among others the simple geometric model of lattices. Unfortunately, these simple descriptive method runs into difficulties when dealing with patterns pre- senting 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, …
The Geometric And Dynamic Essence Of Phyllotaxis, Pau Atela
The Geometric And Dynamic Essence Of Phyllotaxis, Pau Atela
Mathematics Sciences: Faculty Publications
We present a dynamic geometric model of phyllotaxis based on two postulates, primordia formation and meristem expansion. We find that Fibonacci, Lucas, bijugate and multijugate are all variations of the same unifying phenomenon and that the difference lies on small changes in the position of initial primordia. We explore the set of all initial positions and color-code its points depending on the phyllotactic type of the pattern that arises.
A Dynamical System For Plant Pattern Formation: A Rigorous Analysis, Pau Atela, Christophe Golé, S. Hotton
A Dynamical System For Plant Pattern Formation: A Rigorous Analysis, Pau Atela, Christophe Golé, S. Hotton
Mathematics Sciences: Faculty Publications
We present a rigorous mathematical analysis of a discrete dynamical system modeling plant pattern formation. In this model, based on the work of physicists Douady and Couder, fixed points are the spiral or helical lattices often occurring in plants. The frequent occurrence of the Fibonacci sequence in the number of visible spirals is explained by the stability of the fixed points in this system, as well as by the structure of their bifurcation diagram. We provide a detailed study of this diagram.