Symmetry in Plants

The following sections are included:

• FOREWORD

• PROLOGUE by a Mathematician: THE ROLE OF MATHEMATICS IN PHYLLOTAXIS

• PROLOGUE by a Botanist: PHYLLOTACTIC SYMMETRY IN PLANT GROWTH

• PROLOGUE by a Molecular Geneticist: THE ROLE OF HOMEOTIC GENES AND OF MOLECULAR MIMICRY IN THE DETERMINATION OF PLANT AND ANIMAL SYMMETRIES

• PROLOGUE by a Crystallographer: PHYLLOTAXIS

• CONTENTS

• LIST OF CONTRIBUTORS

The following sections are included:

• Introduction
  • Phyllotaxis
  • Genus Rhododendron
  • This Work

• Materials and Methods

• Results

• Discussion

• Conclusions

• Acknowledgments

• Appendix

• References

The following sections are included:

• Introduction

• Ontogenetic Changes

• E. bridgesiana, E. nicholii and E. acaciiformis

• Ironbarks

• Species with Heath-Like Seedlings

• Phyllotactic Complexity and the Rate of Leaf Production

• Phyllotaxis in the Conferruminatae

• Chirality of Shoot Architecture in Other Species of Conferruminatae

• The Sub-Decussate Arrangement

• Phyllotaxis in Eucalypts is Basically Bijugate

• Phase Change, Genetics and Phyllotaxis

• Acknowledgments

• References

The following sections are included:

• Introduction — Historical Context

• Pendulum Symmetry as a Suite of Features in Ulmus

• Permutations and Combinations Involving Distichous Pendulum Symmetry

• Further Complications of Pendulum Symmetry and Leaf Asymmetry

• Further Cases of Pendulum Symmetry without Leaf Asymmetry

• Conclusions

• Acknowledgements

• References

The following sections are included:

• Introduction

• Mature Shoot Architecture
  • Spiral Leaf Pattern, Terminal Inflorescences, Tendrils Absent (Leea guineensis, Cyphostemma juttae)
    • Leea guineensis
    • Cyphostemma juttae
  • Continuous Tendril-Inflorescences and Axillary Buds (Ampelopsis brevipedunculata, Tetrastigma voinerianum, Rhoicissus rhomboidea, R. digitata)
    • Ampelopsis brevipedunculata
    • Tetrastigma voinerianum
    • Rhoicissus rhomboidea
    • Rhoicissus digitata
  • Shoots with Tendril-Inflorescences at Two of Three Nodes (Interrupted)
    • Continuous Axillary Buds (Vitis riparia)
    • Interrupted Axillary Buds, at Two of Three Nodes (Parthenocissus inserta)
    • Interrupted Axillary Buds, at One of Three Nodes (Cissus antarctica, Parthenocissus tricuspidata)

• Pattern Transition: Juvenile to Adult

• Stipules

• Discussion
  • Interpretation of the Vitaceous Shoot
  • Dorsiventrality
  • Phyllotactic Theories

• References

The following sections are included:

• Primordium Arrangement in Flowers

• The Divergence Angle in Flowers

• Changes in the Plastochron Ratio and Primordium Area at Initiation During the Transition to Flowering
  • Primordium Area at Initiation in the Flower

• Origin of Primordium Arrangement in the Flower: Position Versus Sequence of Primordia

• Meristic Variation

• Phyllotaxis During Flower Reversion

• Phyllotaxis in Aberrant Silene Flowers

• Conclusions

• References

The following sections are included:

• Introduction

• Phyllotactic Parameters

• Linum Seedling Transition
  • Decussate Characteristics
  • Characteristics of the Spiral Transition

• Epilobium Reproductive Transition

• Discussion
  • Change in Divergence Angle versus Vertical Spacing
  • The Phyllotaxic Triangular Unit
  • Changes in Shoot Apical Meristem Initials
  • Changes in Shoot Apical Meristem Shape
  • Changes in van Iterson's Parameter

• Acknowledgements

• References

The following sections are included:

• Introduction

• Formation of the Receptacle
  • Origin
  • Maintenance of Symmetry in the Generative Area

• Floret Production

• Fibonacci Prevalence in Floret Row Numbers

• Wound-Induced Florets

• Initiation of Floret and Floret Bract Primordia
  • Polarity Change in the Generative Area
  • Surface Buckling as an Alternative Physical Theory
  • Chemical Inhibition of Floret Production

• Cytokinin Promotion of Floret Production
  • Response to Cytokinins
  • Response of the Dome to Cytokinin

• Non-Cytokinin Growth Regulators

• Mutant with Irregular Floret Patterning

• Conclusions

• References

The following sections are included:

• Introduction and Summary

• Material and Methods

• Four Parameters of a Two-Dimensional Analysis of Shoot Tips with Unijugate Phyllotactic Patterns
  • Analysis of Fibonacci Systems, as well as Spirodistichous and Distichous Systems
  • Phyllotaxis Analysis of Accessory Spiral Systems (See Classification in Table 1)

• Phyllotaxis Analysis of “Multiple Systems” = Patterns with Two or More Leaves per Node (see Classification in Table 1)
  • Whorled = k-Merous Systems
  • k-Jugate Spiral Patterns Related to the Fibonacci Systems: Bijugacy and Trijugacy

• Empirical Survey of the Plastochrone Ratios (R) and Leaf Arcs (i) Correlated with Various Phyllotactic Patterns

• Rising Phyllotaxis: Decrease of Plastochrone Ratio Values and Phyllotactic Transitions from the Beginning to Full Establishment of a Vegetative Shoot

• Decrease of Plastochrone Ratio Values and Acceleration of Leaf Initiation Rate During the Transition from Vegetative to Reproductive Development

• Plastochrone Ratios, Relative Size of Apical Meristems, Richards' Area Ratio (A)

• Outlook and Critical Comments

• Acknowledgements

• References

The following sections are included:

• Introduction

• Research Sites and Methods

• Methodological Notes

• Results
  • Changes in the Structure of Twig Phyllotactic Patterns
  • Changes in the Structure of Cone Phyllotactic Patterns
  • Changes of Phyllotactic Pattern Structure Throughout the Years
  • Changes of Phyllotactic Pattern Structure Observed in Different Tree Species

• Temperature Impact on Twig Phyllotactic Pattern Structure

• Conclusion

• Acknowledgements

• References

The following sections are included:

• Introduction
  • Allomery
  • Biastrepsis

• Material and Methods
  • Characteristics of Stems in Dipsacus sylvestris
  • Provenance and Treatment of the Study Material
  • Data Gathering

• Results
  • Data on the Persistence of the Two Major Aberrations
  • Morphological Aspects of Major Aberrations

• Discussion

• Acknowledgements

• References

The following sections are included:

• The Problem

• Choice of Surface

• The Normalized Cylinder

• Development of the Cylinder

• Elementary Properties of the Point-Lattice

• Parastichies

• Conspicuous Parastichies

• The Opposed Parastichy Triangle

• Contraction and Extension

• Kinds of Explanation Sought

• How a Model is Constructed

• The Assumptions

• Plotting Minimum Distance as a Function of d

• If r Is Small Enough, and the Minimum Distance Between Leaves is Maximized, then the Leaves Nearest 0 are Equidistant from It

• The Addition Rule

• More and More Maxima

• When Maximization Begins Early

• A Computer Program for the Model

• Weakening the Assumptions

• A Functional Interpretation

• Predictions Made by the Model

• Acknowledgments

• Appendix A

• Appendix B

• Appendix C

• Appendix D

• Appendix E. Computer Program for the Cylindrical Representation of a Disc Model

• References

The following sections are included:

• Introduction
  • Symmetry and Organ Numbers in Plants
  • Previous Work on the Relationship between Spiral Phyllotaxis and Numerical Canalization
    • Hirmer's Work and the Collision Model
    • Confirmation of the Collision Model
    • Discrepancies between Flowerhead Development and the Model
  • Objectives of the Current Work

• The Modified Collision Model
  • Placement of Primordia with Adjustments in the Divergence Angle
  • Growth of the Receptacle

• Evaluation of the Modified Collision Model
  • Spatial Arrangement of Florets on the Receptacle
    • Methods for Flowerhead Observations
    • Methods for Simulations
    • Methods for Comparing Observed and Modeled Flowerheads
    • Comparison Results
  • Numbers of Inner Phyllaries and Numerical Canalization
    • Observations
    • Simulations
    • Comparison Results
  • Ontogenetic Order of Inner Phyllary Positions
    • Observations
    • Simulations
    • Comparison Results

• Discussion
  • New Insights in Meristic Variation
    • Spatial Arrangement of Florets on the Receptacle
    • Canalization of Inner Phyllary Numbers
    • Ontogenetic Order of Inner Phyllaries
  • Suggestions for Future Work
  • Concluding Remarks

• Acknowledgements

• Appendix A: Proof of Hirmer's Conjecture

• Appendix B: Minimizing the Sum of the Squared Distances between Corresponding Primordium Positions

• References

The following sections are included:

• Introduction

• Simple Continued Fractions: Definition and Notation

• The Principal Convergents

• The Klein Lattice

• Intermediate Convergents

• The Coxeter Lattice

• ‘Heads’ and ‘Tails‘ of a Continued Fraction

• Shape and Squashing

• Optimal Squashing

• Intermediate Convergents and Near Maximum Elongation

• Disc Phyllotaxis

• Conclusion

• Acknowledgement

• Appendix A: Zero Squash and Varied α

• Appendix B: Some Textbook Formulae

• Appendix C: Full Squash and Maximum Elongation

• References

The following sections are included:

• Introduction

• The Cylindrical Lattice
  • Description
  • Equations of (m, n) Pattern
  • The Set of Solutions
  • The Relation between l and r
  • The Whorled and Multijugate Cases

• Selection of the Normal Versus Abnormal Solutions
  • The Compactness Criterion
  • The Local Minimum Criterion
  • The Absolute Minimum Criterion

• Discussion and Conclusion

• Acknowledgement

• References

The following sections are included:

• Introduction
  • Models — General Aspects
  • Propagation — General Aspects
  • Classification
    • Whorled
    • Spiral

• Close-Packing Theories — Virtues and Drawbacks

• Monotonic Field Theories — Virtues

• Monotonic Fields — Limitations
  • Assumptions
  • Breadth of Applicability beyond Pattern Propagation
    • Explaining Stability of Propagation
    • The Distinction between Organ Pattern and Boundary Pattern
    • The Origin of Pattern

• Undulatory Fields — General
  • General Features
  • Chemical Fields (Reaction-Diffusion)
  • Physical Undulating Fields: Buckling

• Simulations
  • De novo Pattern Formation
  • Propagation
    • A Propagation Model with Step-Function Parameters
    • Models with Continuous Function Parameters

• Discussion
  • Comparison with Other Theories
  • Is the Undulating Field in the Controls or the Response?
  • Applicability to Other Systems
  • Cause and Effect
    • Familiar Transductions, Viewed as Algebraic Conversions
    • A Transduction as an Integration

• Appendix. Propagation of a Fibonacci Spiral Pattern

• References

The following sections are included:

• Introduction

• The Crocograph and its Latest Improvements

• Crocograms and Results

• Conclusions

• References

The following sections are included:

• Introduction

• Fibonacci Numbers and the Golden Mean

• Models of Phyllotaxis
  • Coxeter Model
  • van Iterson Model
  • Rivier Model

• Spacing Properties

• Farey Series
  • Definition and Properties of Farey Series
  • Properties of Farey Series
  • Farey Series and Continued Fractions
  • Representation of Irrational Numbers by Continued Fractions
  • Noble Numbers and Anomalous Phyllotaxis
  • Number and Phyllotaxis

• Wythoff's Game

• A Fibonacci Number System

• The Gears of Life

• Quasi-crystals

• The Circle Map and Chaos

• Mode Locking

• Mode Locking and the Circle Map

• A Dynamical Systems Approach to Phyllotaxis

• Conclusion

• Appendix A: The Golden Mean and Optimal Spacing

• Appendix B
  • To Convert a Rational Number into a Continued Fraction
  • To Generate the Convergents of a Continued Fraction if you Know the Indices
  • Converting a Noble Number to a Continued Fraction

• Appendix C: Wythoff's Game

• References

The following sections are included:

• Phyllotaxis

• Geometrical Description of Phyllotaxis
  • The Basics of Continued Fractions
  • Van Iterson's Geometrical Description of Spiral Patterns
  • Model of Hard Disks

• Reaction-Diffusion Model of Phyllotaxis

• An Iteration Model of Phyllotaxis

• Conclusion: Phyllotaxis as a Geometrical and Dynamical System

• Appendix
  • Construction of the Lattice of Tangent Disks
  • Parastichy Transitions
  • Transition Rings in the Bénard Structure

• References

The following sections are included:

• Introduction

• Gnomonical Growth
  • The Snail
  • The Stem
  • Expansion Versus Stacking

• Phyllotaxis and Differentiation

• The Stack-and-Drag Model
  • Pattern Determining Parameters
  • Translation Parameters

• Discussion

• Appendix

• References

The following sections are included:

• Morphogenetic Aspects
  • Random Cell Division — Proliferation
  • Control of New Meristematic Centres — Mutual Influences
  • Persistence of Established Patterns — Continuity

• Initiation of New Primordia
  • The Basic Requirements
  • The Existing (Older) Primordia
  • The Mechanism of Control by Existing Primordia Over the Positioning of the Next New One
  • The Strength of the Inhibitory Effects Emanating from the Older Primordia
  • The Importance of the Vertical Distances between Primordia
  • Polar Diffusion of the Inhibitory Morphogen

• Examples of Divergence Control by Length of Incipient Internodes

• The Computer Model

• Discussion

• Acknowledgments

• References

The following sections are included:

• Introduction
  • Geometrical Approach
  • Dynamical Approach
  • Physiology of the Apical Meristem

• The Iterative Model
  • Principle
  • The Simulation
    • The Repulsive Interaction
    • The Geometry of the Apex
    • Definition of the Control Parameter
    • Simulations Set-Up
    • Running the Simulations: Steady Regimes and Transients

• Results on the Steady Regimes
  • Distichous and Fibonacci Spiral Modes
    • The T(Γ) and d(Γ) Diagrams
    • Interpretation
    • Relation with Richards' Plastochrone Ratio
    • Stability of the Spiral Patterns
  • The Whorled Modes
  • The Bijugate and Multijugate Spiral Modes
  • The Mean Periodicity of the Iteration, the Compactness of the Packing and the Selection of the Patterns
  • The Role of the Conicity and of the Interaction Stiffness and of the Growth of the Elements Size

• Results: The Transients Corresponding to Ontogeny
  • The Transient Regimes Creating the Vegetative Modes
    • Transitions to the Main Fibonacci Spiral
    • Transitions into the Range of the Decussate Mode
    • Transitions to Values of Γ below the Range of Decussation
  • The Formation of High Order Modes during the Floral Initiation
    • The Transient Leading to the Formation of High Order Spiral Modes
    • Transient Regimes Creating High Order Whorled Modes (m, m) or Spiral Modes of the Accessory Series (m, m +1)
    • Rare Modes

• Discussion and Conclusions

• References

The following sections are included:

• Introduction
  • Phyllotaxis: A Special Case of Two-Dimensional Botanical Patterns
  • Various Standpoints with Regard to Phyllotaxis
  • Aim of the Present Chapter

• General Properties of the Biological Model
  • The Fundamental Feature of the Model

• Convexity Properties of the Elementary Potentials and Numerical Values of the Divergence
  • The Statistical Character of the Golden Divergence
  • Distichous Phyllotaxis

• Rational Phyllotaxis Resulting from the Quasi-Concavity of the Elementary Potentials

• Irrational Phyllotaxis as a Consequence of the Irregularity of the Elementary Potentials

• Spectra of Divergences Depending on Two Characteristic Parameters

• Conclusion

• Acknowledgements

• References

The following sections are included:

• Introduction

• Origins of Phyllotaxis
  • Homologies
  • Phylogeny in Phyllotaxis

• Prominent Elementary Processes of Growth in Nature
  • Spirality
  • Branching
  • Gnomonic Growth
  • Allometry

• Branching in Phyllotaxis
  • Hierarchical Systems
  • Multijugacy: A Unifying Concept
  • Perturbed Systems
  • Rare Systems

• Allometry in Phyllotaxis
  • The Dynamical Model
  • Links between (m, n) and other Phyllotactic Parameters

• Redefining Phyllotaxis
  • Systemic Phyllotaxis
  • Main Challenges in Systemic Phyllotaxis

• Conclusion

• Acknowledgements

• References

The following sections are included:

• Introduction
  • The Regularity in Plants
  • Why Fibonacci Numbers?

• Mechanical Theory
  • Growth and Stress
  • Some Historical Remarks

• The Geometric Model
  • Cylindrical Lattices and Parastichies
  • The Space of Cylindrical Lattices

• The Energy Model
  • The Deformation History
  • Symmetries of the Energy

• Universality of Fibonacci Numbers
  • Plan of the Discussion
  • Modular Symmetry and Farey Triangles
  • Farey Partition and the Trajectories of Energy Minima

• Robustness of the Problem
  • Deformation vs. Anisotropic Growth
  • Disk and Cone Geometry

• Summary

• References

The following sections are included:

• Introduction

• The Establishment of Chemical Periodicity

• From Chemical to Biological Periodicity

• Periodicity in the Minerals

• The Ordered Addition of Components in Atoms, Molecules and Minerals

• Carbon Atoms Build Icosahedral Balls

• Silicon Atoms Maintain Radial Configurations as their Number Increases

• Water does not Need Genes to Produce a Constant Pattern and a Specific Variation

• Constancy and Variation in the Pattern of Water Crystals

• Nucleic Acids Maintain Ordered Configurations as their Number of Components Increases

• Association in Crystals Follows Order

• The Properties of Crystals Open the Way to Our Understanding of Biological Transformation

• The Relationship between the Mineral and the Flower Pattern
  • Two-, Three-, Four-, Six-Fold Symmetry
  • Five-Fold Symmetry
  • Crystal and Flower Symmetries

• Root and Mineral Organization

• Leaves and Fruits Follow the Atomic Order of Minerals

• The Compositae Exemplify the Rigid Order that Determines Flower Organization

• The Periodicity is also Put in Evidence by the Occurrence of the Same Solution in Different Organs

• The Distribution of the Arms in Living and Fossil Starfishes

• The Human Body is Built on the Atomic Plan of a Twin Crystal

• Summary
  • The Transfer of the Chemical Periodicity to Living Organisms
  • Molecular Mimicry
  • The Conservation of Homeotic Genes in Plants, Invertebrates and Vertebrates Including Humans
  • Molecular Mechanisms Leading to the Reappearance of Ancient Structures
  • Molecular Cascades Result in Coordinated Patterns
  • The Periodicity of Plant Symmetries
  • The Periodicity Found in Atoms, Minerals, Plants and Animals has a Common Material Basis

• References

The following sections are included:

• Introduction

• Mathematical Representation of Uniform Spacing Models
  • The Morphological Unit and the Interaction Function
  • Dynamic and Static Analyses
  • Informing Geometry, Forgetful Dynamics, and Morphogenesis

• Icosahedral Virus Capsids

• Coaxial Helices
  • Fourier Representation of Helical Symmetry: Subhelices
  • Optimally Interacting Coaxial Helices

• Simple Helices
  • The Unperturbed Helix
  • The Perturbed Helix
  • The Least Squares Fit and Gradient Dynamics
  • The Evolution of Σ²
  • The Evolution of Δφ

• Phyllotaxis
  • The Static Problem Via : Disk MUs
  • The Mixed Representation
  • as a Function of Δφ
  • The Numerical Evolution of the Mixed Representation Phyllotaxis Model

• Discussion

• Appendix
  • Fourier Coefficients for a Helically Symmetrical Field
  • Fourier Coefficients A_n for an Azimuthally Symmetrical MU
  • The Unperturbed Interaction
  • The Linear Perturbations

• Acknowledgements

• References

The following sections are included:

• Introduction

• Pattern Formation by Local Autocatalysis and Long Range Inhitition
  • Activator-Inhibitor Systems
  • Activator-Substrate Systems
  • The Dividing Cell as Pattern Forming System
  • Generation of Regular and Irregular Patterns in Large Fields
  • Generation of Stripes

• Cell States and Differentiation
  • Cell Determination Requires Autocatalytic (Autoregulatory) Genes
  • Mutual Activation and Stabilization of Differently Determined Cell Types
  • Formation of Filament-Like Branching Structures
  • How to Generate Structures at Opposite Positions of a Field
  • Switch from Shoot to Floral Meristem

• Phyllotaxis
  • Spacing of Leaves by an Activator-Inhibitor Mechanism
  • Local Exclusion and Long Range Help: The Shoot Meristem Enables Leaf Initiation in its Vicinity
  • Segmentation in Plants
  • The Role of a Long Term Memory in Phyllotaxis
  • Altered Phyllotaxis after Experimental Interference

• Conclusion

• References

The following sections are included:

• Introduction

• Materials and Methods

• Simulation Equations

• Results

• Discussion

• References

The following sections are included:

• Introduction

• On Symmetry

• On The Origin of Symmetry in Land Plants

• Symmetry and Process Morphology

• On Phyllotaxis

• On Branching

• Conclusions

• Acknowledgements

• References

The following sections are included:

• Introduction

• Quasicrystalline Structure for Phyllotaxis: Quasiperiodic Penrose Tiling Pattern

• Nonlinear Dynamics and Chaos in Iterative Processes

• Quantum Mechanics: Concepts and Limitations for Real World Systems
  • Quantum-like Mechanics in Fluid Flows: A Physically Consistent Theory
  • Quantum-like Mechanics and Phyllotaxis

• Discussions and Conclusions

• Acknowledgements

• References

The following sections are included:

• SUBJECT INDEX

• AUTHOR INDEX