The equations of phyllotaxis were discovered by Douady and Couder: these describe the inhibition "force" generated by an incipient primordium, preventing the initiation of other primordia in its vicinity: the method that was lacking was to take into account the distance of primordia to the meristem apex over the course of its development. With these parameters, our model integrates both biochemical "forces" (inhibition due to auxin) and mechanical forces (contact pressure). This powerful tool allows us to revisit many important notions of plant biology. For example, we model auxin concentrations at the plant apex, as well as the development of vegetative and reproductive meristems. We also explain why the whorls of monocots are trimerous and those of plants with quincuncial perianths are pentamerous. We design a geometric method for reconstructing inflorescences from their building blocks, i.e. floral meristems. We also show that phyllotaxic spirals are only the application of a general property of symmetry, the advantages of which have been exploited by natural selection.