Rhombic Tilings and Primordia Fronts of Phyllotaxis (Preprint, Oct 2007) (P. Atela & C. Golé)(pdf 5.2 MB)
The Snow Dynamical System for Plant Pattern Formation (DRAFT) (P. Atela & C. Golé) (pdf 2.2 MB)

These two papers present new results and concepts based on the "hard disk" Snow model, which can be thought as piling disks in the cylinder, at the lowest level possible, without overlapping existing disks. Primordia fronts are strings of tangent disks enclosing the cylinder, and represent the newest level of primordia. Primordia front are in a sense dual to parastichies and they contain the same information, but in localized form. In particular, the front parastichy numbers is an alternative to divergence angles for recording phyllotactic transtions. Rhombic tilings, that arise naturally as periodic orbits of the Snow dynamical system offer more flexibility than lattices in describing phyllotactic patterns. They include lattices of all jugacies, and point to the possible deformations between different lattices. . The second paper is an earlier, less technical draft of the first.

The possible and the actual in phyllotaxis: Bridging the gap between empirical observations and iterative models. Journal of Plant Growth Regulation 25: 313-323 (Hotton, S, V Johnson, J Wilbarger, K Zwieniecki, P Atela, C Gole, J Dumais (2006)) (pdf .7 MB)

This paper presents new methods for the geometrical analysis of phyllotactic patterns and their comparison with patterns produced by simple, discrete dynamical systems.   We introduce the concept of ontogenetic graph as a parsimonious and mechanistically relevant representation of a pattern.   The ontogenetic graph is extracted from the local geometry of the pattern and does not impose large-scale regularity on it as for the divergence angle and other classical descriptors.    We exemplify our approach by analyzing the phyllotaxis of two asteraceae in the light of a hard disk model.   The simulated patterns offer a very good match to the observed patterns for over 150 iterations of the model.

A Dynamical System for Plant Pattern Formation: Rigorous Analysis (P. Atela, C. Golé & S. Hotton, J. Nonlinear Sci. Vol. 12, Number 6 (2002)) (pdf 2.7 MB)

This paper presents our research on a discrete dynamical system inspired by a model of Douady and Couder which implements Hoffmeister's hypothesis of primordia formation. We prove that helical lattices appearing in plants can be seen as fixed points of our dynamical systems. Furthermore, we prove that these fixed points are stable. We then provide a detailed analysis of the fixed point bifurcation diagram in the space of planar lattices. Hyperbolic geometry is used to analyse the regions in parameter space traversed by the self similar bifurcation diagram. The main paths of the bifurcation diagram and the stability of fixed points explain the occurence of Fibonacci numbers of helixes, and the inverse of the golden mean as a limiting divergence angle. Results on periodic orbits are alluded to, and there is a comparaison with other scientists' work (mainly Adler, Douady and Couder, Kunz and Levitov).

Finding the Center of a Phyllotatic Pattern (S. Hotton, J. Theor. Biology , Vol. 225, Issue 1 (2003) pp. 15-32) (pdf 341 K)

While there are many well formed specimens of phyllotactic patterns that closely conform to a spiral lattice it is not at all uncommon for naturally occuring phyllotactic patterns to deviate slightly from the ideal. This has made phyllotaxis a difficult field to work in. This paper provides a robust definition of the center that works well in specimens that do not conform perfectly to a spiral lattice and helps lay the groundwork to objectively compare theory against experiment in a more general way than has been done before. Click here for sample programs which find the center of a phyllotactic pattern.


Symmetry of Plants (Scott Hotton's Ph. D. thesis, UC Santa Cruz, 1999) (pdf 932 K)

Contains many of the results in the dynamical system paper above, but with different proofs. A chapter is devoted to the group theoretic study of lattices in arbitrary dimensions, with classifications relevant to phyllotaxis. A later chapter shows that our "ideal" dynamical system presents robust phenomena: taking into account the repulsive effect of all primordia on the incoming one (instead of only the closest primordium) does not change the qualitative behavior of the system, even for arbitrary small values of the growth parameter (in a precise sense).


Talk for a general mathematical audience (pdf 980 K)
Talk for a dynamical systems audience (pdf 340 K)
"Applying the Dynamics of Circle Maps to Phyllotactic Patterns"   (HTML ~4.5M)

The first two talks present, in brief detail, the work in the first paper above. The first talk presents the material in the spiral, or equivalntly in the centric representation. The second talk uses helixes. The third talk is about extending the dynamical system to elliptical shaped apecies.