Rhombic Tilings and Primordia Fronts of Phyllotaxis

We introduce and study properties of phyllotactic and rhombic tilings on the cylinder. These are discrete sets of points that generalize cylindrical lattices. Rhombic tilings appear as periodic orbits of a discrete dynamical system S that models plant pattern formation by stacking disks of equal radius on the cylinder. This system has the advantage of allowing several disks at the same level, and thus multi-jugate configurations. We provide partial results toward proving that the attractor for S is entirely composed of rhombic tilings and is a strongly normally attracting branched manifold and conjecture that this attractor persists topologically in nearby systems. A key tool in understanding the geometry of tilings and the dynamics of S is the concept of primordia front, which is a closed ring of tangent disks around the cylinder. We show how fronts determine the dynamics, including transitions of parastichy numbers, and might explain the Fibonacci number of petals often encountered in compositae.