  We introduce and study properties of phyllotactic and rhombic tilings on the
cylin- der. 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 config- urations. 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
pri- mordia 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.
