Phyllotaxis: Some progress, but a story far from over

•We review a variety of models for phyllotaxis.•We derive a model for phyllotaxis using biochemical and biomechanical mechanisms.•Simulation and analysis of our model reveals novel self-similar behavior.•Fibonacci progressions are a natural consequence of these mechanisms.•Common transitions between phyllotactic patterns occur during simulation.

This is a review article with a point of view. We summarize the long history of the subject and recent advances and suggest that almost all features of the architecture of shoot apical meristems can be captured by pattern-forming systems which model the biochemistry and biophysics of those regions on plants.