Applying the Dynamics of Circle Maps to Phyllotactic Patterns

Scott Hotton

Table of Contents


    i     Abstract

1.0     Introduction
1.1     Examples of Spiral Phylloatxis
1.2     The Golden Angle
1.3     The Golden Angle in Aonium
1.4     (5,8) Spiral Lattice
1.5     The Space of Spiral Lattices
1.6     Spiral Applet
1.7     Early Stages of Development
1.8     Parts of a Plant Shoot
1.9     Hofmeister's Hypothesis

2.0     Cylindrical Case
2.1     Geometric Specification
2.2     Configuration from Angles
2.3     Dynamical System
2.4     The Maximin Principal
2.5     Graph of D(p)
2.6     Effect on Configuration
2.7     Making use of Symmetry
2.8     Bifurcation Theorem
2.9     Bifurcation Diagram
2.10     Dynamical Model Applet
3.0     Circular Disk Case
3.1     Sunflower Capitulum
3.2     Geometric Specification
3.3     Dynamical System
3.4     The Maximin Principal
3.5     Making use of Symmetry
3.6     The Exponential map
3.7     Higher Order Phyllotaxis
3.8     Space of Helical Lattices
3.9     Space of Spiral Lattices
3.10     Bifurcation Diagram
3.11     Developing Magnolia Carpels
3.12     Tucker's Table
3.13     Period 8 Orbit
3.14     Model's Configuration

4.0     Elliptical Disk Case
4.1     Oval Shaped Flowers
4.2     Shifted Parastichy Numbers
4.3     Geometric Specification
4.4     Parametrizing the Ellipses
4.5     Normal Hyperbolicity
4.6     Configuration for a=1.00
4.7     Configuration for a=1.01
4.8     Configuration for a=1.02
4.9     Configuration for a=1.03
4.10     Configuration for a=1.04
4.11     Parastichies in Configuration
4.12     Diffeomorphism
4.13     Diffeomorphisms Residual
4.14     Diffeomorphisms Amplitude
4.15     The Conjugating Map
4.16     The Residual Map
4.17     Extending Conjugating Map
4.18     Undulating Parastichy

5.0     Avoiding Preconceptions
5.1     SEM of a Gerber Daisy
5.2     Change Center
5.3     Graph of Change Center
5.4     Example
5.5     More Examples
5.6     Recovering Divergence Angles

6.0     Postscript

7.0     References