Symmetry and its transition in phyllotaxis.

Symmetry is an important component of geometric beauty and regularity in both natural and cultural scenes. Plants also display various geometric patterns with some kinds of symmetry, of which the most notable example is the arrangement of leaves around the stem, i.e., phyllotaxis. In phyllotaxis, reflection symmetry, rotation symmetry, translation symmetry, corkscrew symmetry, and/or glide reflection symmetry can be seen. These phyllotactic symmetries can be dealt with the group theory. In this review, we introduce classification of phyllotactic symmetries according to the group theory and enumerate all types of phyllotaxis, not only major ones such as spiral and decussate but also minor ones such as orixate and semi-decussate, with their symmetry groups. Next, based on the mathematical model studies of phyllotactic pattern formation, we discuss transitions between phyllotaxis types different in the symmetry class with a focus on the transition into one of the least symmetric phyllotaxis, orixate, as a representative of the symmetry-breaking process. By changes of parameters of the mathematical model, the phyllotactic pattern generated can suddenly switch its symmetry class, which is not constrained by the group-subgroup relationship of symmetry. The symmetry-breaking path to orixate phyllotaxis is also accompanied by dynamic changes of the symmetry class. The viewpoint of symmetry brings a better understanding of the variety of phyllotaxis and its transition.