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Geometrical study of phyllotactic patterns by Bernoulli spiral lattices
Takamichi Sushida1, Yoshikazu Yamagishi2
1Research Institute for Electronic Science, Hokkaido University, Sapporo, 060-0812, Japan.
Development, Growth & Differentiation
|July 14, 2017
Summary
This study reviews two centric models for phyllotactic patterns: disk packings and Voronoi tilings of Bernoulli spiral lattices. The Voronoi tiling model
Area of Science:
- Mathematical Biology
- Geometry
- Crystallography
Background:
- Phyllotactic patterns are studied using centric and cylindrical models based on ideal lattices.
- Van Iterson's diagram, based on disk packings of Bernoulli spiral lattices, explains bifurcation processes.
- Previous work explored geometrical properties of disk packings.
Purpose of the Study:
- To review two centric models for ideal phyllotactic patterns: disk packings and Voronoi tilings.
- To present a mathematical framework for Voronoi tilings of Bernoulli spiral lattices.
- To compare the phase diagrams of both models.
Main Methods:
- Utilizing a mathematical framework for Voronoi tilings of Bernoulli spiral lattices.
- Graph-theoretical analysis to compare phase diagrams.
- Reviewing existing literature on disk packing models.
Main Results:
- The phase diagram of Voronoi tilings is graph-theoretically dual to Van Iterson's diagram.
- Established a mathematical framework for Voronoi tilings of Bernoulli spiral lattices.
- Provided a comparative review of disk packing and Voronoi tiling models.
Conclusions:
- Voronoi tilings offer an alternative centric model for phyllotactic patterns.
- The duality between Voronoi tiling and Van Iterson's diagrams provides new insights.
- This review consolidates understanding of geometric models in phyllotaxis.
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