DOCSLIB.ORG

Double Helix of Phyllotaxis

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

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
Recommended publications
  • Linguistic Presentation of Objects

    Linguistic Presentation of Objects

    Zygmunt Ryznar dr emeritus Cracow Poland [email protected] Linguistic presentation of objects (types,structures,relations) Abstract Paper presents phrases for object specification in terms of structure, relations and dynamics. An applied notation provides a very concise (less words more contents) description of object. Keywords object relations, role, object type, object dynamics, geometric presentation Introduction This paper is based on OSL (Object Specification Language) [3] dedicated to present the various structures of objects, their relations and dynamics (events, actions and processes). Jay W.Forrester author of fundamental work "Industrial Dynamics”[4] considers the importance of relations: “the structure of interconnections and the interactions are often far more important than the parts of system”. Notation <!...> comment < > container of (phrase,name...) ≡> link to something external (outside of definition)) <def > </def> start-end of definition <spec> </spec> start-end of specification <beg> <end> start-end of section [..] or {..} 1 list of assigned keywords :[ or :{ structure ^<name> optional item (..) list of items xxxx(..) name of list = value assignment @ mark of special attribute,feature,property @dark unknown, to obtain, to discover :: belongs to : equivalent name (e.g.shortname) # number of |name| executive/operational object ppppXxxx name of item Xxxx with prefix ‘pppp ‘ XXXX basic object UUUU.xxxx xxxx object belonged to UUUU object class & / conjunctions ‘and’ ‘or’ 1 If using Latex editor we suggest [..] brackets
  • 6 Phyllotaxis in Higher Plants Didier Reinhardt and Cris Kuhlemeier

    6 Phyllotaxis in Higher Plants Didier Reinhardt and Cris Kuhlemeier

    6 Phyllotaxis in higher plants Didier Reinhardt and Cris Kuhlemeier 6.1 Introduction In plants, the arrangement of leaves and flowers around the stem is highly regular, resulting in opposite, alternate or spiral arrangements.The pattern of the lateral organs is called phyllotaxis, the Greek word for ‘leaf arrangement’. The most widespread phyllotactic arrangements are spiral and distichous (alternate) if one organ is formed per node, or decussate (opposite) if two organs are formed per node. In flowers, the organs are frequently arranged in whorls of 3-5 organs per node. Interestingly, phyllotaxis can change during the course of develop- ment of a plant. Usually, such changes involve the transition from decussate to spiral phyllotaxis, where they are often associated with the transition from the vegetative to reproductive phase. Since the first descriptions of phyllotaxis, the apparent regularity, especially of spiral phyllotaxis, has attracted the attention of scientists in various dis- ciplines. Philosophers and natural scientists were among the first to consider phyllotaxis and to propose models for its regulation. Goethe (1 830), for instance, postulated the existence of a general ‘spiral tendency in plant vegetation’. Mathematicians have described the regularity of phyllotaxis (Jean, 1994), and developed computer models that can recreate phyllotactic patterns (Meinhardt, 1994; Green, 1996). It was recognized early that phyllotactic patterns are laid down in the shoot apical meristem, the site of organ formation. Since scientists started to pos- tulate mechanisms for the regulation of phyllotaxis, two main concepts have dominated the field. The first principle holds that the geometry of the apex, and biophysical forces in the meristem determine phyllotaxis (van Iterson, 1907; Schiiepp, 1938; Snow and Snow, 1962; Green, 1992, 1996).
  • Multidimensional Generalization of Phyllotaxis

    Multidimensional Generalization of Phyllotaxis

    CYBERNETICS AND PHYSICS, VOL. 8, NO. 3, 2019, 153–160 MULTIDIMENSIONAL GENERALIZATION OF PHYLLOTAXIS Andrei Lodkin Dept. of Mathematics & Mechanics St. Petersburg State University Russia [email protected] Article history: Received 28.10.2019, Accepted 28.11.2019 Abstract around a stem, seeds on a pine cone or a sunflower head, The regular spiral arrangement of various parts of bi- florets, petals, scales, and other units usually shows a ological objects (leaves, florets, etc.), known as phyl- regular character, see the pictures below. lotaxis, could not find an explanation during several cen- turies. Some quantitative parameters of the phyllotaxis (the divergence angle being the principal one) show that the organization in question is, in a sense, the same in a large family of living objects, and the values of the divergence angle that are close to the golden number prevail. This was a mystery, and explanations of this phenomenon long remained “lyrical”. Later, similar pat- terns were discovered in inorganic objects. After a se- ries of computer models, it was only in the XXI cen- tury that the rigorous explanation of the appearance of the golden number in a simple mathematical model has been given. The resulting pattern is related to stable fixed points of some operator and depends on a real parame- ter. The variation of this parameter leads to an inter- esting bifurcation diagram where the limiting object is the SL(2; Z)-orbit of the golden number on the segment [0,1]. We present a survey of the problem and introduce a multidimensional analog of phyllotaxis patterns.
  • The Ordered Distribution of Natural Numbers on the Square Root Spiral

    The Ordered Distribution of Natural Numbers on the Square Root Spiral

    The Ordered Distribution of Natural Numbers on the Square Root Spiral - Harry K. Hahn - Ludwig-Erhard-Str. 10 D-76275 Et Germanytlingen, Germany ------------------------------ mathematical analysis by - Kay Schoenberger - Humboldt-University Berlin ----------------------------- 20. June 2007 Abstract : Natural numbers divisible by the same prime factor lie on defined spiral graphs which are running through the “Square Root Spiral“ ( also named as “Spiral of Theodorus” or “Wurzel Spirale“ or “Einstein Spiral” ). Prime Numbers also clearly accumulate on such spiral graphs. And the square numbers 4, 9, 16, 25, 36 … form a highly three-symmetrical system of three spiral graphs, which divide the square-root-spiral into three equal areas. A mathematical analysis shows that these spiral graphs are defined by quadratic polynomials. The Square Root Spiral is a geometrical structure which is based on the three basic constants: 1, sqrt2 and π (pi) , and the continuous application of the Pythagorean Theorem of the right angled triangle. Fibonacci number sequences also play a part in the structure of the Square Root Spiral. Fibonacci Numbers divide the Square Root Spiral into areas and angle sectors with constant proportions. These proportions are linked to the “golden mean” ( golden section ), which behaves as a self-avoiding-walk- constant in the lattice-like structure of the square root spiral. Contents of the general section Page 1 Introduction to the Square Root Spiral 2 2 Mathematical description of the Square Root Spiral 4 3 The distribution
  • Golden Ratio: a Subtle Regulator in Our Body and Cardiovascular System?

    Golden Ratio: a Subtle Regulator in Our Body and Cardiovascular System?

    See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/306051060 Golden Ratio: A subtle regulator in our body and cardiovascular system? Article in International journal of cardiology · August 2016 DOI: 10.1016/j.ijcard.2016.08.147 CITATIONS READS 8 266 3 authors, including: Selcuk Ozturk Ertan Yetkin Ankara University Istinye University, LIV Hospital 56 PUBLICATIONS 121 CITATIONS 227 PUBLICATIONS 3,259 CITATIONS SEE PROFILE SEE PROFILE Some of the authors of this publication are also working on these related projects: microbiology View project golden ratio View project All content following this page was uploaded by Ertan Yetkin on 23 August 2019. The user has requested enhancement of the downloaded file. International Journal of Cardiology 223 (2016) 143–145 Contents lists available at ScienceDirect International Journal of Cardiology journal homepage: www.elsevier.com/locate/ijcard Review Golden ratio: A subtle regulator in our body and cardiovascular system? Selcuk Ozturk a, Kenan Yalta b, Ertan Yetkin c,⁎ a Abant Izzet Baysal University, Faculty of Medicine, Department of Cardiology, Bolu, Turkey b Trakya University, Faculty of Medicine, Department of Cardiology, Edirne, Turkey c Yenisehir Hospital, Division of Cardiology, Mersin, Turkey article info abstract Article history: Golden ratio, which is an irrational number and also named as the Greek letter Phi (φ), is defined as the ratio be- Received 13 July 2016 tween two lines of unequal length, where the ratio of the lengths of the shorter to the longer is the same as the Accepted 7 August 2016 ratio between the lengths of the longer and the sum of the lengths.
  • A Method of Constructing Phyllotaxically Arranged Modular Models by Partitioning the Interior of a Cylinder Or a Cone

    A Method of Constructing Phyllotaxically Arranged Modular Models by Partitioning the Interior of a Cylinder Or a Cone

    A method of constructing phyllotaxically arranged modular models by partitioning the interior of a cylinder or a cone Cezary St¸epie´n Institute of Computer Science, Warsaw University of Technology, Poland [email protected] Abstract. The paper describes a method of partitioning a cylinder space into three-dimensional sub- spaces, congruent to each other, as well as partitioning a cone space into subspaces similar to each other. The way of partitioning is of such a nature that the intersection of any two subspaces is the empty set. Subspaces are arranged with regard to phyllotaxis. Phyllotaxis lets us distinguish privileged directions and observe parastichies trending these directions. The subspaces are created by sweeping a changing cross-section along a given path, which enables us to obtain not only simple shapes but also complicated ones. Having created these subspaces, we can put modules inside them, which do not need to be obligatorily congruent or similar. The method ensures that any module does not intersect another one. An example of plant model is given, consisting of modules phyllotaxically arranged inside a cylinder or a cone. Key words: computer graphics; modeling; modular model; phyllotaxis; cylinder partitioning; cone partitioning; genetic helix; parastichy. 1. Introduction Phyllotaxis is the manner of how leaves are arranged on a plant stem. The regularity of leaves arrangement, known for a long time, still absorbs the attention of researchers in the fields of botany, mathematics and computer graphics. Various methods have been used to describe phyllotaxis. A historical review of problems referring to phyllotaxis is given in [7]. Its connections with number sequences, e.g.
  • Some Curves and the Lengths of Their Arcs Amelia Carolina Sparavigna

    Some Curves and the Lengths of Their Arcs Amelia Carolina Sparavigna

    Some Curves and the Lengths of their Arcs Amelia Carolina Sparavigna To cite this version: Amelia Carolina Sparavigna. Some Curves and the Lengths of their Arcs. 2021. hal-03236909 HAL Id: hal-03236909 https://hal.archives-ouvertes.fr/hal-03236909 Preprint submitted on 26 May 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Some Curves and the Lengths of their Arcs Amelia Carolina Sparavigna Department of Applied Science and Technology Politecnico di Torino Here we consider some problems from the Finkel's solution book, concerning the length of curves. The curves are Cissoid of Diocles, Conchoid of Nicomedes, Lemniscate of Bernoulli, Versiera of Agnesi, Limaçon, Quadratrix, Spiral of Archimedes, Reciprocal or Hyperbolic spiral, the Lituus, Logarithmic spiral, Curve of Pursuit, a curve on the cone and the Loxodrome. The Versiera will be discussed in detail and the link of its name to the Versine function. Torino, 2 May 2021, DOI: 10.5281/zenodo.4732881 Here we consider some of the problems propose in the Finkel's solution book, having the full title: A mathematical solution book containing systematic solutions of many of the most difficult problems, Taken from the Leading Authors on Arithmetic and Algebra, Many Problems and Solutions from Geometry, Trigonometry and Calculus, Many Problems and Solutions from the Leading Mathematical Journals of the United States, and Many Original Problems and Solutions.
  • Phyllotaxis: a Remarkable Example of Developmental Canalization in Plants Christophe Godin, Christophe Golé, Stéphane Douady

    Phyllotaxis: a Remarkable Example of Developmental Canalization in Plants Christophe Godin, Christophe Golé, Stéphane Douady

    Phyllotaxis: a remarkable example of developmental canalization in plants Christophe Godin, Christophe Golé, Stéphane Douady To cite this version: Christophe Godin, Christophe Golé, Stéphane Douady. Phyllotaxis: a remarkable example of devel- opmental canalization in plants. 2019. hal-02370969 HAL Id: hal-02370969 https://hal.archives-ouvertes.fr/hal-02370969 Preprint submitted on 19 Nov 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Phyllotaxis: a remarkable example of developmental canalization in plants Christophe Godin, Christophe Gol´e,St´ephaneDouady September 2019 Abstract Why living forms develop in a relatively robust manner, despite various sources of internal or external variability, is a fundamental question in developmental biology. Part of the answer relies on the notion of developmental constraints: at any stage of ontogenenesis, morphogenetic processes are constrained to operate within the context of the current organism being built, which is thought to bias or to limit phenotype variability. One universal aspect of this context is the shape of the organism itself that progressively channels the development of the organism toward its final shape. Here, we illustrate this notion with plants, where conspicuous patterns are formed by the lateral organs produced by apical meristems.
  • Archimedean Spirals ∗

    Archimedean Spirals ∗

    Archimedean Spirals ∗ An Archimedean Spiral is a curve defined by a polar equation of the form r = θa, with special names being given for certain values of a. For example if a = 1, so r = θ, then it is called Archimedes’ Spiral. Archimede’s Spiral For a = −1, so r = 1/θ, we get the reciprocal (or hyperbolic) spiral. Reciprocal Spiral ∗This file is from the 3D-XploreMath project. You can find it on the web by searching the name. 1 √ The case a = 1/2, so r = θ, is called the Fermat (or hyperbolic) spiral. Fermat’s Spiral √ While a = −1/2, or r = 1/ θ, it is called the Lituus. Lituus In 3D-XplorMath, you can change the parameter a by going to the menu Settings → Set Parameters, and change the value of aa. You can see an animation of Archimedean spirals where the exponent a varies gradually, from the menu Animate → Morph. 2 The reason that the parabolic spiral and the hyperbolic spiral are so named is that their equations in polar coordinates, rθ = 1 and r2 = θ, respectively resembles the equations for a hyperbola (xy = 1) and parabola (x2 = y) in rectangular coordinates. The hyperbolic spiral is also called reciprocal spiral because it is the inverse curve of Archimedes’ spiral, with inversion center at the origin. The inversion curve of any Archimedean spirals with respect to a circle as center is another Archimedean spiral, scaled by the square of the radius of the circle. This is easily seen as follows. If a point P in the plane has polar coordinates (r, θ), then under inversion in the circle of radius b centered at the origin, it gets mapped to the point P 0 with polar coordinates (b2/r, θ), so that points having polar coordinates (ta, θ) are mapped to points having polar coordinates (b2t−a, θ).
  • Phyllotaxis: a Model

    Phyllotaxis: a Model

    Phyllotaxis: a Model F. W. Cummings* University of California Riverside (emeritus) Abstract A model is proposed to account for the positioning of leaf outgrowths from a plant stem. The specified interaction of two signaling pathways provides tripartite patterning. The known phyllotactic patterns are given by intersections of two ‘lines’ which are at the borders of two determined regions. The Fibonacci spirals, the decussate, distichous and whorl patterns are reproduced by the same simple model of three parameters. *present address: 136 Calumet Ave., San Anselmo, Ca. 94960; [email protected] 1 1. Introduction Phyllotaxis is the regular arrangement of leaves or flowers around a plant stem, or on a structure such as a pine cone or sunflower head. There have been many models of phyllotaxis advanced, too numerous to review here, but a recent review does an admirable job (Kuhlemeier, 2009). The lateral organs are positioned in distinct patterns around the cylindrical stem, and this alone is often referred to phyllotaxis. Also, often the patterns described by the intersections of two spirals describing the positions of florets such as (e.g.) on a sunflower head, are included as phyllotactic patterns. The focus at present will be on the patterns of lateral outgrowths on a stem. The patterns of intersecting spirals on more flattened geometries will be seen as transformations of the patterns on a stem. The most common patterns on a stem are the spiral, distichous, decussate and whorled. The spirals must include especially the common and well known Fibonacci patterns. Transition between patterns, for instance from decussate to spiral, is common as the plant grows.
  • Spiral Phyllotaxis: the Natural Way to Construct a 3D Radial Trajectory in MRI

    Spiral Phyllotaxis: the Natural Way to Construct a 3D Radial Trajectory in MRI

    Magnetic Resonance in Medicine 66:1049–1056 (2011) Spiral Phyllotaxis: The Natural Way to Construct a 3D Radial Trajectory in MRI Davide Piccini,1* Arne Littmann,2 Sonia Nielles-Vallespin,2 and Michael O. Zenge2 While radial 3D acquisition has been discussed in cardiac MRI methods feature properties that are particularly well suited for its excellent results with radial undersampling, the self- for cardiac MRI applications. navigating properties of the trajectory need yet to be exploited. In the field of cardiac MRI, it has to be considered that Hence, the radial trajectory has to be interleaved such that the anatomy of the heart is very complex. Furthermore, the first readout of every interleave starts at the top of the sphere, which represents the shell covering all readouts. If this in many cases, the location and orientation of the heart is done sub-optimally, the image quality might be degraded by differ individually within the thorax. Up to now, only eddy current effects, and advanced density compensation is experienced operators are able to perform a comprehen- needed. In this work, an innovative 3D radial trajectory based on sive cardiac MR examination. As a consequence, the option a natural spiral phyllotaxis pattern is introduced, which features to acquire a 3D volume that covers the whole heart with optimized interleaving properties: (1) overall uniform readout high and isotropic resolution is a much desirable goal (11). distribution is preserved, which facilitates simple density com- In this configuration, some of the preparatory steps of a pensation, and (2) if the number of interleaves is a Fibonacci cardiac examination are obsolete since the complexity of number, the interleaves self-arrange such that eddy current effects are significantly reduced.
  • Linguistic Presentation of Object Structure and Relations in OSL Language

    Linguistic Presentation of Object Structure and Relations in OSL Language

    Zygmunt Ryznar Linguistic presentation of object structure and relations in OSL language Abstract OSL© is a markup descriptive language for the description of any object in terms of structure and behavior. In this paper we present a geometric approach, the kernel and subsets for IT system, business and human-being as a proof of universality. This language was invented as a tool for a free structured modeling and design. Keywords free structured modeling and design, OSL, object specification language, human descriptive language, factual markup, geometric view, object specification, GSO -geometrically structured objects. I. INTRODUCTION Generally, a free structured modeling is a collection of methods, techniques and tools for modeling the variable structure as an opposite to the widely used “well-structured” approach focused on top-down hierarchical decomposition. It is assumed that system boundaries are not finally defined, because the system is never to be completed and components are are ready to be modified at any time. OSL is dedicated to present the various structures of objects, their relations and dynamics (events, actions and processes). This language may be counted among markup languages. The fundamentals of markup languages were created in SGML (Standard Generalized Markup Language) [6] descended from GML (name comes from first letters of surnames of authors: Goldfarb, Mosher, Lorie, later renamed as Generalized Markup Language) developed at IBM [15]. Markups are usually focused on: - documents (tags: title, abstract, keywords, menu, footnote etc.) e.g. latex [14], - objects (e.g. OSL), - internet pages (tags used in html, xml – based on SGML [8] rules), - data (e.g. structured data markup: microdata, microformat, RFDa [13],YAML [12]).