## A Mathematical Model

We present the dynamical system model of phyllotaxis which has been the object of our research. It is a mathematical derivation of a model by the physicists Douady and Couder, which itself is based on observations by the 19th Century botanist Hofmeister. This model reproduces all spiral patterns exhibited by plants and explains why Fibonacci phyllotaxis is predominant. The applet Dynamical Model gives a simulation of this process.

##### Hofmeister's Rules

In 1868 the botanist Wilhelm Hofmeister published his microscopic study of meristems. He proposed that in spiral phyllotaxis, botanical units form one by one with the newest in the least crowded spot. Following Douady and Couder (1996, Part I), we make these rules a bit more precise:

Primordia initiate one at a time in the least crowded spot around a circular meristem.
• Primordia are radially displaced away from the center as they grow.

 Compare a scanning electron micrograph of a Norway spruce (Picea abies) shoot apical meristem with a computer generated arrangement of primordia. The computer program implemented the mathematical model based on Hofmeister's rules and converged naturally to this configuration. Primordia are numbered according to their age - the higher the number, the older the primordium. Notice the remarkable similarity between plant and model. Electron micrograph courtesy of Rolf Rutishauser, published in Rutishauser (1998).

##### Dynamical System Model

Our model is a discrete dynamical system. Generally, a discrete dynamical system consists of iterating a function (also called map) F on a set S (also called the state space). In this model, each element in S corresponds to a configuration of primordia, which are represented by points on concentric circles, with one point per circle. The innermost circle represents the periphery of the meristem. From circle to circle the radius increases by a constant factor G, that we call here expansion rate (Plastochrone ratio in the literature). Each configuration is encoded as a list of divergence angles between successive points. Hence, S can be seen as the set of all possible lists (d1 , d2 , . . . ,dN) of angles.

 Under the function F, points of a configuration move out radially to the next circle. The outermost point is eliminated and a new point (green) is generated on the innermost circle, where the distance to the closest point is maximum. The "distance-to-closest" function along the inner circle is represented as a graph (red). the "distance-to-closest" function D(z), with variable z representing a point on the edge of the meristem is implemented as D(z) = mink(dis(z, zk)) where zk is the point representing the center of primordium k. Clearly each zk is function of the variable d1 , d2 , . . . ,dN. The least crowded rule chooses the point z that maximizes D.