Geometrical study of phyllotactic patterns by Bernoulli spiral lattices.

Abstract

Geometrical studies of phyllotactic patterns deal with the centric or cylindrical models produced by ideal lattices. van Iterson (Mathematische und mikroskopisch - anatomische Studien über Blattstellungen nebst Betrachtungen über den Schalenbau der Miliolinen, Verlag von Gustav Fischer, Jena, 1907) suggested a centric model representing ideal phyllotactic patterns as disk packings of Bernoulli spiral lattices and presented a phase diagram now called Van Iterson's diagram explaining the bifurcation processes of their combinatorial structures. Geometrical properties on disk packings were shown by Rothen & Koch (J. Phys France, 50(13), 1603-1621, 1989). In contrast, as another centric model, we organized a mathematical framework of Voronoi tilings of Bernoulli spiral lattices and showed mathematically that the phase diagram of a Voronoi tiling is graph-theoretically dual to Van Iterson's diagram. This paper gives a review of two centric models for disk packings and Voronoi tilings of Bernoulli spiral lattices.