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Geometrical study of phyllotactic patterns by B ernoulli spiral lattices
Author(s) -
Sushida Takamichi,
Yamagishi Yoshikazu
Publication year - 2017
Publication title -
development, growth and differentiation
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.864
H-Index - 66
eISSN - 1440-169X
pISSN - 0012-1592
DOI - 10.1111/dgd.12378
Subject(s) - voronoi diagram , phase diagram , spiral (railway) , ideal (ethics) , combinatorics , bernoulli's principle , mathematics , geometry , statistical physics , physics , phase (matter) , mathematical analysis , thermodynamics , quantum mechanics , philosophy , epistemology
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.