MaterialPress DOUBLE HELIX OF PHYLLOTAXIS Copyrighted BrownWalker MaterialPress Copyrighted BrownWalker DOUBLE HELIX OF PHYLLOTAXIS ANALYSIS OF THE GEOMETRIC MODEL OF PLANT MORPHOGENESIS MaterialPress BORIS ROZIN Copyrighted BrownWalker BrownWalker Press Irvine • Boca Raton Double Helix of Phyllotaxis: Analysis of the geometric model of plant morphogenesis Copyright © 2020 Boris Rozin. All rights reserved. No part of this publication may be reproduced, distributed, or transmitted in any form or by any means, including photocopying, recording, or other electronic or mechanical methods, without the prior written permission of the publisher, except in the case of brief quotations embodied in critical reviews and certain other noncommercial uses permitted by copyright law. For permission to photocopy or use material electronically from this work, please access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC) at 978-750- 8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payments has been arranged. BrownWalker Press / Universal Publishers, Inc. Irvine • Boca RatonMaterial USA • 2020 Press www.BrownWalkerPress.com 978-1-62734-748-8 (pbk.) 978-1-59942-604-4 (ebk.) Typeset by Medlar Publishing Solutions Pvt Ltd, India Cover design by Ivan Popov US Library of Congress Cataloging-in-Publication Data is available at https://lccn.loc.gov Names: Rozin, Boris, 1964- author. Title: Double helix of phyllotaxis : analysis of the geometric model of plant morphogenesis / BorisCopyrighted Rozin. Description: Irvine.BrownWalker California : Brown Walker Press, [2020] | Includes bibliographical references. Identifiers: LCCN 2020019498 (print) | LCCN 2020019499 (ebook) | ISBN 9781627347488 (paperback) | ISBN 9781599426044 (ebook) Subjects: LCSH: Phyllotaxis--Morphogenesis. | Plant morphogenesis--Mathematical models. Classification: LCC QK649 .R855 2020 (print) | LCC QK649 (ebook) | DDC 581.4/8--dc23 LC record available at https://lccn.loc.gov/2020019498 LC ebook record available at https://lccn.loc.gov/2020019499 SYNOPSIS his book is devoted to anyone who is in search of beauty in mathematics, and mathematics in the beauty around us. Attempting to combine mathe- matical rigor and magnificence of the visual perception, the author is pre- Tsenting the mathematical study of phyllotaxis, the most beautiful phenomenon of the living nature. The distinctive feature of this book is an animation feature that explains the work of mathematical models and the transformation of 3D space. The analysis of the phyllotactic pattern as a system of discrete objects together with the mathematical tools of generalized sequencesMaterial made it Presspossible to find a uni- versal algorithm for calculating the divergence angle. In addition, it is serving as a new proof of the fundamental theorem of phyllotaxis and analytically confirming well-known formulas obtained intuitively earlier as well as casting some doubts on a few stereotypes existing in mathematical phyllotaxis. The presentation of phyllotaxis morphogenesis as a recursive process allowed the author to formulate the hydraulic model of phyllotaxis morphogenesis and pro- pose a method for its experimental verification. With the help of artificial intelli- gence, the author offered methodology for the digital measurement of phyllotaxis allowing a transition to a qualitatively new level in the study of plant morphogenesis. Due to the successful combination of mathematical constructions and their visual presentation, the materials of this study are comprehensible to readers with high school advanced mathematical levels. Copyrighted BrownWalker MaterialPress Copyrighted BrownWalker CONTENTS Acknowledgments xi I. INTRODUCTION 1 II. MATHEMATICAL FOUNDATIONS 3 II.1. The Golden Ratio 3 II.2. Recursion and recursive sequence 5 II.2.1. The ibonacciF sequence MaterialPress 5 II.2.2. Generalized recursive sequence 6 II.2.3. The sum of ecursiver sequences 8 II.3. Spirals 9 III. PHYLLOTAXIS 13 III.1. Phyllotaxis: concept and terminology 13 III.2. Phyllotactic patterns classification and recursive sequence hierarchy 16 III.3. Visual perception of phyllotactic patterns 18 III.4. Double morphogenetic phenomenon of phyllotaxis 19 IV. THE DOUBLE HELIX MODEL OF PHYLLOTAXIS 21 IV.1. Geometric interpretation of the model 21 CopyrightedIV.2. Bravais-Bravais theorem 24 IV.3. BrownWalkerThe fundamental theorem of phyllotaxis 25 V. THE PLANAR DH-MODEL ON ARCHIMEDEAN SPIRALS 27 V.1. The criterion of the minimum distance between the nodes of the phyllotactic lattice 28 V.2. Fibonacci phyllotaxis 29 V.2.1. Finding the divergence coefficient for Fibonacci phyllotaxis 29 V.2.2. The phyllotactic pattern with divergence coefficient 1/t2 31 viii Double Helix of Phyllotaxis V.3. The analysis of the minimum distances between the nodes in the Fibonacci phyllotactic lattice 32 V.4. The hreadst of the nodes of the phyllotactic lattice and their visual perception 33 V.5. Diapasons and nodes of “the phyllotaxis rises” 35 V.6. Diameter of an element of the phyllotactic lattice 39 V.7. The angle between tangent lines to the opposed parastichies 40 V.8. Is phyllotaxis a fractal or a recursive structure? 44 V.9. The planar DH-Model for “accessory” sequences 45 V.10. The planar DH-model for “lateral” sequences 49 V.11. Converse of the fundamental theorem of phyllotaxis 57 V.12. The planar DH-model for “multijugate” sequences 59 V.13. The universal algorithm for calculating the divergence coefficient 66 V.14. Interim results 67 VI. FIBONACCI LATTICES ON THE PLANAR NON-ARCHIMEDEANMaterial SPIRALS 69 VI.1. Fibonacci lattices on the planar power genetic spiralsPress 70 VI.2. Fibonacci lattices on the planar logarithmic genetic spirals 74 VI.3. Interim results 77 VII. THE CYLINDRICAL DH-MODEL 79 VII.1. The cylindrical Fibonacci phyllotaxis 82 VII.2. The cylindrical phyllotaxis generated by a non-multiple recurrent sequence 90 VII.3. The cylindrical phyllotaxis generated by the generalized recurrent sequence 92 VIII. PHYLLOTACTIC LATTICES WITH RATIONAL DIVERGENCE COEFFICIENT 97 VIII.1. Phyllotactic lattices with the divergence coefficientβ = 0 97 CopyrightedVIII.2. Phyllotactic lattices with the divergence coefficientβ = 0.5 98 VIII.3. Two-pair phyllotactic lattices with the divergence coefficientBrownWalker β = 0.5 99 VIII.4. Multi-pair phyllotactic lattices with the divergence coefficient β = 0.5 100 VIII.5. Phyllotactic lattices with the divergence coefficientβ = 0.25 101 VIII.6. The two-pair phyllotactic lattice with the divergence coefficient β = 0.25 102 VIII.7. Interim results 103 Contents ix IX. DISPUTE OF THE STATICS DH-MODEL 105 IX.1. Novelty and validity of the DH-Model 105 IX.2. Edge function and divergence coefficient 106 X. SOME ASPECTS OF THE DYNAMIC DH-MODEL 109 X.1. Phyllotaxis under the microscope or where is the inflorescence forming from? 109 X.2. Invisible proprimordium and visible primordium 110 X.3. The phenomenon of cutting”“ 112 X.4. The ubet model of the periodic phyllotaxis 114 X.5. The mathematical genesis of the Fibonacci numbers or why Turing had failed in his research in phyllotaxis 115 X.6. The onceptc of the dynamic DH-Model 118 X.7. The ydraulich aspect of the dynamic DH-Model 118 X.8. Experimental verification of the hydraulic model 121 XI. WHAT WE OBJECTIVELY KNOW AND DON’TMaterial KNOW ABOUT PHYLLOTAXIS Press125 XII. HOW PHYLLOTAXIS SHOULD BE MEASURED 127 XII.1. Digital technology and Artificial Intelligence in the measurement of phyllotaxis 127 XII.2. The network project “Open Phyllotaxis” 128 XIII. CONCLUSION OR FIVE CRITICAL QUESTIONS 131 APPENDIX A. THE GOLDEN SECTION AND MORPHOGENESIS 135 A.1. The Golden Ratio in the works of man 135 CopyrightedA.2. The Golden Section in nature 136 A.3. The Golden Section in technique 140 A.4. BrownWalkerWhy does nature need the Golden Section? 140 B. EXTENSIONS TO THE FIBONACCI SEQUENCE AND THE GOLDEN SECTION 143 B.1. p-numbers Fibonacci 143 B.2. Metallic Means 144 B.3. Continuous Fibonacci functions 145 x Double Helix of Phyllotaxis B.4. Bodnar’s “golden” hyperbolic functions 150 B.5. The Golden sine 151 C. THE FIBONACCI NUMBERS IN CYBERNETICS 155 D. THE ANGLE OF THE TANGENT TO THE SPIRAL 157 E. SOLVE A LINEAR DIOPHANTINE EQUATION 159 F. TABLE OF THE DEPENDENCE OF THE DIVERGENCE COEFFICIENT AND ENCYCLIC NUMBERS ON THE INITIAL TERMS OF THE GENERATING RECURRENT SEQUENCE 161 References 165 Index 169 About Author MaterialPress 173 Copyrighted BrownWalker ACKNOWLEDGMENTS he author is grateful to all researchers of the phenomenon of phyllotaxis. Biologists who have made thousands of observations of phyllotaxis patterns, accumulated and systematized a big array of information. To the math- Tematicians who tried to explain the numerical relationships in biological objects. Special thanks to Professor Roger V. Jean, for trying to systematize and summarize this knowledge. The author thanks Intel and Microsoft, all the mathematicians and engineers who created modern computers and SW. WithoutMaterial using a computerPress with MS Visual Studio, MS WORD, and MS Excel, this study would have been impossible. The author is grateful to his wife Natali and daughter Miriam for their help and support; To Arkadiy Boshoer (Brooklyn, NY) for the criticisms and assistance in prepar- ing this book; To Dr. Alla Kolunuk (Kamianets-Podilskyi, Ukraine), to Valeria Boshoer (San Francisco, CA), to Robert Fuchs (München, Germany) for the help