BIOMATHEMATICS Mathematics of Biostructures and Biodynamics This Page Intentionally Left Blank B IO M A T H E MAT I CS Mathematics of Biostructures and Biodynamics Sten Andersson Sandviks Forskningsinstitut, S-38o 74 LiJttorp, Sweden K~re Larsson KL Chern AB, S-237 3/4 Bj~irred and Carnurus Lipid Research, S-223 70 Lund, Sweden Marcus Larsson Lund University, Department of Clinical Physiology, S-214 ol MalmiJ, Sweden Michael Jacob Department of Inorganic Chemistry, Arrhenius Labatory, University of Stockholm, S-lo6 91 Stockholm, Sweden 1999 Elsevier Amsterdam - Lausanne- New Y o r k - O x f o r d - S h a n n o n - S i n g a p o r e - Tokyo ELSEVIER SCIENCE B.V Sara Burgerhartstraat 25 P.O Box 211, 1000 AE Amsterdam, The Netherlands 1999 Elsevier Science B.V All rights reserved This work and the individual contributions contained in it are protected under copyright by Elsevier Science, and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use Permissions may be sought directly from Elsevier Science Rights & Permissions Department, PO Box 800, Oxford OX5 I DX, UK; phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: permissions@elsevier.co.uk You may also contact Rights & Permissions directly through Elsevier's home page (http://www.eisevier.nl), selecting first 'Customer Support', then 'General Information', then 'Permissions Query Form' In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (978) 7508400, fax: (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London W IP 0LP, UK; phone: (+44) 171 631 5555; fax: (+44) 171 631 5500 Other countries may have a local reprographic rights agency for payments Derivative Works Tables of contents may be reproduced for internal circulation, but permission of Elsevier Science is required for external resale or distribution of such material Permission of the Publisher is required for all other derivative works, including compilations and translations Electronic Storage or Usage Permission of the Publisher is required to store or use electronically any material contained in this work, including any chapter or part of a chapter Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publisher Address permissions requests to: Elsevier Science Rights & Permissions Department, at the mail, fax and e-mail addresses noted above Notice No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made First edition 1999 Library of Congress Cataloging in Publication Data A catalog record from the Library of Congres has been applied for ISBN: 444 50273 OThe paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper) Printed in The Netherlands Contents Contents Chapter 1 Introduction References Chapter 2 Counting, Algebra and Periodicity - the Roots of Mathematics are the Roots of Life 2.1 Counting and Sine 2.2 Three Dimensions; Planes and Surfaces, and Surface Growth 2.3 The Growth of Nodal Surfaces - Molecules and Cubosomes References 7 16 26 Chapter 3 Nodal Surfaces 3.1 Non Cubic 3.2 Tetragonal 3.3 Hexagonal References of Tetragonal and Hexagonal Symmetry, and Rods Surfaces Nodal Surfaces and their Rod Structures Nodal Surfaces and their Rod Structures 27 27 27 36 45 Chapter 4 Nodal Surfaces, Planes, Rods and Transformations 4.1 Cubic Nodal Surfaces 4.2 Nodal Surfaces andPlanes 4.3 Cubic Nodal Surfaces and Parallel Rods 4.4 Transformations of Nodal Surfaces References 47 47 50 56 68 72 Chapter 5 Motion in Biology 5.1 Background and Essential Functions 5.2 The Control of Shape - the Natural Exponential or cosh in 3D 5.3 The Gauss Distribution (GD) Function and Simple Motion 5.4 More Motion in 3D References 73 73 76 81 93 102 Chapter 6 Periodicity in Biology - Periodic Motion 6.1 The Hermite Function 6.2 Flagella- Snake and Screw Motion 105 105 111 Contents 6.3 Periodic Motion with Particles in 2D or 3D 6.4 Periodic Motion with Rotation of Particles in 2D References 116 127 130 Chapter 7 Finite Periodicity and the Cubosomes 7.1 Periodicity and the Hermite Function 7.2 Cubosomes and the Circular Functions 7.3 Cubosomes and the GD-Function - Finite Periodicity and Symmetry P 7.4 Cubosomes and the GD-Function - Symmetry G 7.5 Cubosomes and the GD Function - Symmetry D 7.6 Cubosomes and the Handmade Function References 131 131 133 139 143 147 152 162 Chapter 8 Cubic Cell Membrane Systems/Cell Organelles and Periodically Curved Single Membranes 8.0 Introduction 8.1 Cubic Membranes 8.2 The Endoplasmatic Reticulum 8.3 Protein Crystallisation in Cubic Lipid Bilayer Phases and Cubosomes - Colloidal Dispersions of Cubic Phases 8.4 From a Minimal Surface Description to a Standing Wave Dynamic Model of Cubic Membranes 8.5 Periodical Curvature in Single Membranes References 163 163 163 169 175 177 183 190 Chapter 9 Cells and their Division - Motion in Muscles and in DNA 9.1 The Roots and Simple Cell Division 9.2 Cell Division with Double Membranes 9.3 Motion in Muscle Cells 9.4 RNA and DNA Modelling References 193 193 201 206 213 220 Chapter 10 10 Concentration Gradients, Filaments, Motor Proteins and again- Flagella 10.1 Background and Essential Functions 10.2 Filaments 10.3 Microtubulus and Axonemes 10.4 Motor Proteins and the Power Stroke 10.5 Algebraic Roots Give Curvature to Flagella References 10 223 223 227 235 244 247 255 Contents Chapter 11 11 Transportation 11.1 Background - Examples of Docking and Budding with Single Plane Layers, and Other Simple Examples 11.2 Docking and Budding with Curved Single Layers 11.3 Transport Through Double Layers References 11 257 257 265 273 284 Chapter 12 12 Icosahedral Symmetry, Clathrin Structures, Spikes, Axons, the Tree, and Solitary Waves 12.1 The icosahedral symmetry 12.2 Hyperbolic Polyhedra, Long Cones, Cylinders and Catenoids 12.3 Cylinder Division and Cylinder Fusion - Cylinder Growth 12.4 Solitary Waves, Solitons and Finite Periodicity References 12 285 285 294 299 305 311 Chapter 13 13 Axon Membranes and Synapses - A Role of Lipid Bilayer Structure in Nerve Signals 13.1 The Nerve Impulse 13.2 At the Action Potential Region of the Membrane there is a Phase Transition in the Lipid Bilayer 13.3 A Model of a Phase-Transition/Electric Signal Coupling at Depolarisation and its Physiological Significance 13.4 Transmission of the Nerve Signal at the Terminal Membrane of the Neurons - Synaptic Transmission 13.5 Synchronisation of Muscle Cell Activation 13.6 The General Anaesthetic Effect 13.7 Physiological Significance of Involvement of a Lipid Bilayer Phase Transition in Nerve Signal Conduction References 13 313 313 315 317 327 333 335 337 338 Chapter 14 14 The Lung Surface Structure and Respiration 14.1 The Alveolar Surface 14.2 Lung Surfactant 14.3 Structure of Tubular Myelin - A Bilayer arranged as the Classical CLP-Surface 14.4 The Existence of a Coherent Surface Phase Lining the Alveoli 14.5 Respiration 14.6 Physiological Significance of the Existence of an Organised Surface Phase at the Alveolar Surface References 14 341 341 342 344 349 357 359 361 Contents Chapter 15 15 Epilogue Acknowledgement References 15 363 372 372 Appendix The Plane, the Cylinder and the Sphere 375 Appendix Periodicity 385 Appendix The Exponential Scale, the GD function, Cylinder and Sphere Fusion 399 Appendix The Exponential Scale, the Planes and the Natural Function, Addition and Subtraction 409 Appendix Multiplication of Planes, Saddles and Spirals 419 Appendix Symmetry 431 Appendix The Complex Exponential, the Natural Exponential and the GDExponential - General Examples and Finite Periodicity 447 Appendix Classical Differential Geometry and the Exponential Scale 463 Appendix Mathematica (Contains the Mathematica scripts used for calculating the equations for the figures in this book.) 477 Introduction 1 Introduction There is no permanent place in the world for ugly mathematics [Hardy,1 ] This book deals with the shape of cells and cell organelles in plants and animals, and changes of shape associated with various life processes The cell membranes and cytoskeleton proteins build these shapes based on physical forces A mathematical/geometrical description of cellular and molecular shapes is presented in this book, and the biological relevance is discussed in the epilogue We demonstrate here new mathematics for cellular and molecular structures and dynamic processes Life began in water, and every single function of life takes place in an aqueous environment A profound way of classification in chemistry is the relation and interaction between molecules, or groups within molecules, and water Molecules (or parts of molecules) can attract water in which case they are called hydrophilic As the opposite extreme they can strive to avoid water; these molecules or molecular parts are termed hydrophobic Most biomolecules possess both these properties; they are amphiphilic This is a fundamental principle which determines the organisation of biomolecules- from the folding of peptide chains into native structures of proteins, to self-assembly of lipid and protein molecules into membranes One consequence of the existence of these two media is that the interface between them define surfaces that tend to be closed The lipid bilayer of membranes, for example, always form closed surfaces; the hydrocarbon chain core is never exposed to water The curvature of these surfaces is an important concept in order to understand structural features above the molecular level Surface and colloid science deals with forces involved in formation of such organisations The behaviour of the colloidal state of matter involves van der Waals interaction, electrostatic forces, so-called hydration forces and hydrophobic forces The colloidal level of structure extended towards curvature of surfaces and finite periodicity is a main theme in our book These concepts are seldom considered in molecular biology Our present understanding of the cell membrane dates back to Luzzati's classical work from 1960 [2], where the liquid character of the hydrocarbon chains in liquid-crystalline lipid-water phases with the combination of longrange order with short-range disorder first were revealed Another important aspect was introduced by Helfrich [3]; the curvature elastic energy Long time ago, two of us [4] proposed the idea that a bilayer Appendix 512 Sin[ Pi (x Cos[8 Pi/9] + y Sin[8Pi/9] )]-.5) +E^(.l(x^2+2y^2))==3.2,{x,-5,4},{y,-4,4}, PlotPoints-> 100] Fig 10.4.1: ContourPlot3D[ (z Cos[ Pi x]-y Sin[ Pi x]),{x,4.3,-1.5},{y,l.1,-1.3},{z,l.2,-1.1}, MaxRecursion->2,PlotPoints-> { {3,5 }, {3,5 }, {3,5 } },B oxed->False,Axes->True] Fig 10.4.2: ContourPlot3D[(E^-((.2(x)^2+y^2+z^2)) (z Cos[ Pi x]-y Sin[ Pi x])-.3), {x, 1.4,- 1.4 }, {y, 1.2,- 1.2 }, {z, 1.2,- 1.2 },MaxRecursion->2,PlotPoints-> {{6,4 }, {5,4 }, {5,4 } }, Boxed->False,Axes->True] Fig 10.4.3: ContourPlot3D[ E^-((x-3.5)^2+(y+.7)^2+z^2)+E^-((x-2)^2+y^2+z^2)-.75, {x,4.3,.5 }, {y, 8,- 1.3 }, {z, 1,- 1},MaxRecursion->2,PlotPoints-> { {5,3 }, {5,3 }, {5,3 } }, Boxed->False,Axes->True] Fig 10.4.4: ContourPlot3D[E^(E^-((.2(x)^2+y^2+z^2)) (z Cos[ Pi x]-y Sin[ Pi x])-.3)+ E^-((x-3.5)^2+(y)^2+z^2)+ E^-((x-2)^2+y^2+z^2)- 1, {x,4.3,- 1.5 },{y, 1.1,- 1.3 }, {z, 1.2,- 1.1 },MaxRecursion->2,PlotPoints-> { {7,4 }, {5,4 }, {5,4 } },Boxed->False,Axes->True] Show [%,ViewPoint-> {0.000,-0.000,3.384 }] Fig 10.4.5: ContourPlot3D[E^(E^-((.2(x)^2+y^2+z^2)) (z Cos[ Pi x]-y Sin[ Pi x])-.3)+ E^-((x-3.5)^2+(y+.35)^2+z^2)+.4 E^-((x-2)^2+y^2+z^2)-l,{x,4.3,-1.5},{y,l.1,-1.3 }, {z, 1.2,- 1.1 },MaxRecursion->2,PlotPoints-> { {7,4 }, {5,4 }, {5,4 }},Boxed->False,Axes->True] Show [%,ViewPoint-> {0.000,-0.000,3.384 }] Fig 10.4.6: ContourPlot3D[EA(E^-((.2(x)A2+yA2+z^2)) (z Cos[ Pi x]y Sin[ Pi x])-.3)+ E^-((x-3.5)A2+(y+.7)A2+z^2)+.4 E^-((x-2)^2+y^2+z^2)-l,{x,4.3,-1.5}, {y, 1.1 ,- 1.4 }, {z, 1.2,- 1.1 },MaxRecursion->2,PlotPoints-> { {7,4 }, {5,4 }, {5,4 } },Boxed->False, Axes->True] Sho w[%, V iewPo int-> {0.000,-0.000,3.384 }] Fig 10.5.1: ImplicitPlot[y-x+.5 0, {x,4,-4 }, {y,4,0 },PlotPoints-> 100] Fig 10.5.2: ImplicitPlot[y+E^-( 15xA2)(x-.5) -0, {x,7,-7 }, {y,2,-2},PlotPoints-> 100] Fig 10.5.3: ImplicitPlot[(-EA-(.15(x)A2)(x-.5)-y)A2-.01==0,{x,8,-8},{y,2,-2}, PlotPoints-> 100] Fig 10.5.4:ImplicitPlot[(EA-(.15(x)A2)(x-.5)(x-1)-2y)A2-.03 0,{x,8,-8},{y,4,-1 }, PlotPoints->200] Fig 10.5.5: ImplicitPlot[EA((E^-( 15(x)^2)(x-.5)+y)^2-.01)-EA-((x+5)^2+(y - 1)^2)-1+ EA-(x+ 10)+EA(x - 10)==0, {x, 8,-8 }, {y,2,-2 },PlotPoints-> 100] Fig 10.5.6: ImplicitPlot[E^((EA-(.15(x)^2)(x-.5)(x-1)-2y)^2-.02)-EA-((x+6)^2+(y)^2)-l+ EA-(x+ 10)+EA(x - 10)=-0, {x,8,- 10 }, {y,3,-2 },PlotPoints->200] Fig 10.5.7: ImplicitPlot[E^((-E^-( 15(x)^2)Cos[.25Pi x]-y)^2-.01)-E^-((x+4)^2+ (y- 1)^2)- 1+EA-(x+ 10)+E^(x - 10) 0, {x,6,- 12 }, {y,2,-2 },PlotPoints-> 100] Fig 10.5.8: ImplicitPlot[E^((-EA-(.15(x)A2)Cos[.5Pi x]-y)A2-.01)-EA-((x+4)A2+(y-1)^2)-l+ EA-(x+ 10)+EA(x - 10)==0, {x, 8,-8 }, {y,2,-2 },PlotPoints-> 150] Fig 10.5.9: ImplicitPlot[E^((-E^-(.15(x)^2)Cos[.75Pi x]-y)A2-.01)-EA-((x+4)^2+(y-1)^2)-l+ EA-(x+ 10)+EA(x - 10) =0, {x,7,-7 }, {y,2,-2 },PlotPoints-> 150] Mathematica 513 Fig 10.5.10: ImplicitPlot[E^((-E^-( 15(x)^2)Cos[.25Pi xl-y)^2-.01)-E^-((x+4)^2+(y-1)^2)1+E^-(x+ 10)+E^(x - 10)==0, {x,6,- 12 }, {y,2,-2 },PlotPoints-> 100] Fig 10.5.12:In[37]:=ImplicitPlot[E^-(x^2+E^(y-2)+E^-(y))+.2 E^-((x+y-3)^2+E^(x)+ E^-(x+5))+.2 E^-((-x+y-3)^2+E^(x-5)+E^-(x)) - 0, {x,6,-6}, {y,8,-2},PlotPoints-> 100] Fig 10.5.13: ImplicitPlot[E^((E^-(.15(x)^2)Cos [.25 Pi x]+y)^2-.01)-E^-((x+4)^2+ (y- 1)^2)- 1+E^-(x+ 10)+E^(x - 10) 0, {x, 8,-8 }, {y,2,-2 },PlotPoints-> 100] Fig 10.5.14:ImplicitPlot[E^((E^-(.15(y)^2)Cos[.25 Pi y]+x)^2-.01)-E^-((y+4)^2+ (x- 1)^2)- 1+E^-(y+ 10)+E^(y - 10) 0, {x,2,-2 }, {y, 8,-8 },PlotPoints-> 100] Fig 10.5.15: ImplicitPlot[E^-(E^((E^-( 15(y)^2)Cos[.5 Pi y]+x)^2-.01)-E^-((y+4)^2+ (x- 1)^2)- I+E^-(y+3)+E^(y-8))+E^-(E^((E^-( 15(x)^2)Cos [.5 Pi x]+y)^2-.01)-E^-((x+4)^2+ (y- 1)^2)- 1+E^-(x+3)+E^(x-8))- 97 0, {x, 10,-5 }, {y, 10,-5 },PlotPoints-> 100] Chapter 11 Fig 11.1.1: ContourPlot3D[E^-(x^2+y^2+(z-l.7)^2)+ E^-z-1 ,{x,2,-2},{y,2,-2},{z,2.2,-.1}, MaxRecursion->2,PlotPoints-> { {5,3 }, {5,3 }, {5,4 } },Boxed->False,Axes->True] Fig 11.1.2: ContourPlot3D[E^-(x^2+y^2+(z)^2)^2+ E^z-1 ,{x,2,-2},{y,2,-2},{z,.5,-3}, MaxRecursion->2,PlotPoints-> { {5,3 }, {5,3 }, {5,4 } },Boxed->False,Axes->True] Fig 11.1.3:ImplicitPlot[10^-((x-2) ^2+ 10^(y-8)+ 10^-(y+9))+ 10^-((x-7)^2+ 10^(y-9)+ 10^-(y+8))+ 10^-((x- 11)^2+ 10^(y-7)+ 10^-(y+9))+ 10^-((x- 17)^2+ 10^(y-9)+ 10^-(y+7))+ 10^-((x-3.5)^2+y^2)+ 10^-((x)^2+(y-2)^2)+ 10^-((x-5)^2+(y+3)^2)+ 10^-((x-8)^2+y^2)+ 10^-((x-9)^2+(y-2)^2)+ 10^-((x - 12)^2+(y+3)^2)+ 10^-((x- 13.5)^2+y^2)+ 10^-((x - 16)^2+ (y-2)^2)+ 10^-((x - 15)^2+(y+3)^2)+ 10^-((x-23.5)^2+y^2)+ 10^-((x - 19)^2+(y-2)^2)+ 10^-((x-20)^2+(y+3)^2)==.5,{x,-3,22},{y,10,-10},PlotPoints->200] Fig 11.1.4:ContourPlot3D[EA-(.5(xA2+y^2+30 (z)A2))+E^-(20((x+.5)^2+y^2+(z-.7)^2))+ E^-(20((x - 1)^2+y^2+(z-.4)^2))+E^-(20((x - 1)^2+(y+ 5)^2+(z+ 7)^2))- 2, {x,2,-2 }, {y,2,-2 }, {z, 1,-.9 },MaxRecursion->2,PlotPoints-> { {3,5 }, {3,5 }, {3,5 } },B oxed->False,Axes->True] Fig 11.1.5 b: ContourPlot3D[EA-((x)A2+(y)A2+(z)^2)+EA-((x-2)A2+(y)A2+(z)^2)+ EA-((x-4)AZ+(y)AZ+(z)^Z)+EA-((x-6)^2+(y)^z+(z)A2)+EA-((x-8)A2+(y)A2+(z)^2)+ EA-((x- 10)^2+(y)A2+(z)A2)+EA-((x - 12)^2+(y)AZ+(z)^Z)+E^-((x - 14.3)^2+(y)AZ+(z)^2)- 5, {x, 15.5,- 1}, {y, 1.7,- 1.7}, {z, 1.7,- 1.7},MaxRecursion->2,PlotPoints->{ {5,4}, {5,3 },{5,3 } }, Boxed->False,Axes->True] Fig 11.1.6: ImplicitPlot[.5EA-((x)A2+(y-l.6)A2-.5)A2+.5EA-((x-l.5)A2+(y)^2)+ EA-((x) ^2+ (y+l.6)AZ-.5)AZ .45,{x,3,-Z},{y,2.7,-2.7},PlotPoints->100] Fig 11.2.1: ContourPlot3D [EA-((x)A2+(z)A2+(y)A2-20)+EA-((x-6)A2+(z)A2+(y)^2+2)- 1, {x,7,-5 }, {y, 5,-5 }, {z, 5,-5 },MaxRecursion->2,PlotPoints-> { {5,4 }, {5,4 }, {5,4 } }, Boxed->False,Axes->True] Fig 11.2.2: ContourPlot3D [E^((x)^2+(z)^2+(y)^2-20)+E^-((x+3)^2+(z)^2+(y)^2+2)- 1, {x,6,-5 }, {y, 5,-5 }, {z, 5,-5 },MaxRecursion->2,PlotPoints-> { {5,4 }, {5,4 }, {5,4 } },Boxed-> False,Axes->True] 514 Appendix Fig 11.2.3: ContourPlot3D[(x)+(z)+(y),{x,2,-2},{y,2,-2},{z,2,-2},MaxRecursion->2, PlotPoints-> { {5,3 }, {5,3 }, {5,3 } },Boxed->False,Axes->Yrue] Fig 11.2.4 b: ContourP1ot3D[(x)A3+(z)A3+(y)A3,{X,2,-2},{y,2,-2},{Z,2,-2}, MaxRecursion->2,PlotPoints-> { {5,3 }, {5,3 }, {5,3 } },Boxed->False,Axes->True] Show[%,ViewPoint->{ 1.957,1.958,1.945}] Fig 11.2.5 b: ContourP1ot3D[(x)A7+(z)A7+(y)A7,{X,2,-2},{y,2,-2},{Z,2,-2}, MaxRecursion->2,PlotPoints-> { {5,3 }, {5,3 }, {5,3 } },Boxed->False,Axes->True] Show[%,ViewPoint->{ 1.957,1.958,1.945}] Fig 11.2.6 a: ContourPlot3D[(x)-(z)+(y),{x,2,-2},{y,2,-2},{z,2,-2},MaxRecursion->2, PlotPoints-> { {5,3 }, {5,3 }, {5,3 } },Boxed->False,Axes->Yrue] Fig 11.2.6 b: ContourPlot3D[(x)A3-(z)A3+(y)A3,{x,2,-2},{y,2,-2},{z,2,-2},MaxRecursion->2, PlotPoints-> { {5,3 }, {5,3 }, {5,3 } },Boxed->False,Axes->Yrue] Fig 11.2.7: ContourPlot3D[z (z-l) (z+l)+ x (x-l) (x+l)+ y (y-I) (y+l) ,{x,-2,2.5},{y,-2,2.5}, {z,-2,2.5 },MaxRecursion->2,PlotPoints-> { {5,3 }, {5,3 }, {5,3 } },Boxed->False,Axes->True] Fig 11.2.8: ContourPlot3D[EA-(z (z-I) (z+l)+ x (x-I) (x+l)+ y (y-I) (y+l))-I ,{x,-2,2.5}, {y,-2,2.5 }, {z,-2,2.5 },MaxRecursion->2,PlotPoints-> { {5,3 }, {5,3 }, {5,3 } },Boxed->False, Axes->True] Fig 11.2.9: ContourPlot3D [EA'(xA2+yA2+(Z- 1.4)A2)+EA-(xA2+yA2+(Z'2.4)A2)+EA'((X'3)A2+ y^2+(Z- 1.4)A2)+E^-((X-3)^2+y^2+(Z-2 )^2)+E^-((X-2)A2+(y-2.5)^2+(Z - 1.4)A2)+EA-((X-2)^2+ (y-2.5)A2+(Z-2.6)A2)+EA-((X-5)A2+(y-3.5)A2+(Z - 1.4)A2)+EA'((X-5)^2+(y'3.5)A2+(Z'2.5)A2)+ EA-((X-5.5)A2+(y" 1)A2+(Z" 1.4)^2)+EA-((X-5.5)A2+(y - 1)A2+(Z'2.3)A2)+EA'z" 1, {X,7,'2 }, {y,5,- }, {Z,3.3,0 },MaxRecursion->2,PlotPoints-> { {6,4 }, {6,4 }, {6,4 } }, Boxed->False,Axes->True] Fig 11.2.10: ContourPlot3D[EA-(z (z-I) (z+l)+ x (x-I) (x+l)+ y (y-I) (y+l))+ EA-(((X - 1.7)A2)+((y - 1.7)A2)+((Z - 1.7)A2) - 1)+EA-(((X-3)A2)+((y-3)A2)+((Z-3 )^2)- 1)- 1, {X,-2,4.5 }, {y,-2,4.5 }, {Z,-2,4.5 },MaxRecursion->2,PlotPoints-> { {7,4 }, {7,4 }, {7,4 } }, Boxed->False,Axes->True] Show [%,ViewPoint-> {0.359,-0.858,3.253 }] Fig 11.3.1: ContourP1ot3D[EA-((x)A2+(z)A2+(y)A2-20)A2+EA-((X-2.5)A2+(z)A2+(y)A2+ 1)-.2, {X,5.5,0 }, {y,2.5,-2.5 }, {Z, 5,-5.5 },MaxRecursion->2,PlotPoints-> { {5,4 }, {5,4 }, {5,4 } }, Boxed->False,Axes->True] Fig 11.3.2: ContourPlot3D[Cos[ Pi x]+Cos[.25 Pi y]+Cos[.25 Pi z] + (xA2)-2.8, {X,2,-2 }, {y, 10,- 10}, {Z, 10,- 10},MaxRecursion->2,PlotPoints-> { {3,5 }, {4,5 }, {4,5 } },B oxed->False,Axes->True] Fig 11.3.3: ContourPlot3D [EA-(xA2+(y+2)A2+zA2)+EA-(xA2+(y-3)A2+zA2)+EAzA2-1 95, {X,3 ,-3 }, {y, 5,-4 }, {Z, 75,- 75 },MaxRecursion->2,PlotPoints-> { {5,3 }, {5,3 }, {5,3 } }, Boxed->False,Axes->True] Fig 11.3.4: ImplicitPlot[EA((x)A2+ (y)A2-8)A2+EA-((X-2.83)A2+(y)A2)+EA-((X-2)A2+(y-2)A2) +EA-((x)AZ+(y-2.83)AZ)+EA-((x-Z)AZ+(y+2)^2) 2, {X,3,-4 }, {y,3,-4 },PlotPoints-> 100] Fig 11.3.5: ContourPlot3D [EA((x)^2+(y)A2-8+zA2)^2+EA-((X-2.8)^2+(y)A2+zA2)- 1.9, {x,l,3},{y,-2,2},{z,-2,2},MaxRecursion->2,PlotPoints->{ {5,4},{5,4},{5,4} }, Boxed->False,Axes->True] Mathematica 515 Fig 11.3.6: ImplicitPlot[E^-(((x-2.82)^2+(y-2.82)^2))+E^-(((x-4)^2+(y)^2)) Fig 11.3.7: ImplicitPlot[E^-(((x-2.82)^2+(y-2.82)^2))+E^-(((x-4)^2+(y)^2))+E^-(((x)^2+ +E^-(((x)^Z+(y-4)^Z))+E^-(((x-2.82)^2+(y+2.82)^2))+E^-(((x)^Z+(y+4)^2))+ E^-(((x+2.82)^2+(y+2.82)^2))+E^-(((x+4)^Z+(y)AZ))+E^-(((x+Z.SZ)^Z+(y-2.82)^2))+ E^-(((x-4.Z3)^Z+(y-4.Z3)^Z))+E^-(((x-6)^Z+(y)^Z))+E^-(((x)^Z+(y-6)^2))+ E^-(((x-4.23)^2+(y+4.23)^2))+E^-(((x)^Z+(y+6)^Z))+E^-(((x+4.23)^Z+(y+4.23)^2))+ E^-(((x+6)^Z+(y)^Z))+E^-(((x+4.23)^2+(y-4.23)^2))~ 18, {x,-8,8 }, {y,-8,8 }, PlotPoints-> 100] (y-4)^Z))+E^-(((x-2.82)^2+(y+Z.SZ)^Z))+E^-(((x)^Z+(y+4)^Z))+E^-(((x+2.82)^2+ (y+2.82)^2))+E^-(((x+4)^Z+(y)^Z))+E^-(((x+Z.gZ)^Z+(y-2.82)^2))+E^-(((x-4.23)^2+ (y-4.•3)^•))+E^-(((x-6)^•+(y)^•))+E^-(((x)^•+(y-6)^•))+E^-(((x-4.•3)^2+(y+4.•3)^•))+ E^-(((x)^Z+(y+6)^Z))+E^-(((x+4.Z3)^Z+(y+4.Z3)^Z))+E^-(((x+6)^Z+(y)^2))+ E^-(((x+4.Z3)^Z+(y-4.23)^2))+.5 E^-(((x-5.11)^2+(y-2.1)^2))+.5 E^-(((x-2.1)^2+ (y-5.1)^2))+.5 E^-(((x+2.11)^2+(y-5.1)^2))+.5 E^-(((x+5.1)^Z+(y-2.1)^2))+ E^-(((x+5.11)^Z+(y+2.1)^2))+.5 E^-(((x+Z.ll)^2+(y+5.1)^2))+.5 E^-(((x-5.11)^2+ (y+2.1)^2))+.5 E^-(((x-2.11)^2+(y+5.1)^2)) .35, {x,-8,8 }, {y,-8,S},PlotPoints-> 100] Fig 11.3.8 a: ContourPlot3D[E^-(((x-2.82)^2+(y-2.82)^2+(z)^2))+E^-(((x-4)^2+ (y)^Z+(z)^Z))+E^-(((x)^Z+(y-4)^Z+(z)^Z))+E^-(((x-2.82)^2+(y+2.82)^2+(z)^2))+ E^-(((x)^Z+(y+4)^Z+(z)^Z))+E^-(((x+2.82)^2+(y+2.82)^2+(z)^Z))+E^-(((x+4)^2+ (y)^Z+(z)^2)) +E^-(((x+Z.82)^Z+(y-Z.82)^Z+(z)^2)) +E^-(((x-a.z3)^Z+(y-4.23)^2+ (z)^Z))+E^-(((x-6) ^2+(y)^z+(z)^z))+E^-(((x)^z+(y-6)^z+(z) ^2))+E^-(((x-4.23)^2+ (y+4.•3)^•+(z)^•))+E^-(((x)^•+(y+6)^•+(z)^•))+E^-(((x+4.•3)^•+(y+4.•3)^•+(z)^•))+ E^-(((x+6)^Z+(y)^Z+(z)^2)) +E^-(((x+4.Z3)^Z+(y-4.Z3)^Z+(z)^Z))+.75(E^-(((x-5.11)^2+ (y-2.1)^2+(z- 1.8)^2))+ E^-(((x-2.1)^2+(y-5.1)^2+(z- 1.8)^2))+E^-(((x+2.11)^2+(y-5.1)^2+ (z-1.8)^2))+ E^-(((x+5.1)^Z+(y-Z.1)^Z+(z-l.g)^z))+E^-(((x+5.11)^2+(y+Z.1)^2+(z-l.8)^2))+ E^-(((x+2.1)^2+(y+5.1)^2+(z-1.8)^2))+E^-(((x-5.11)^2+(y+2.1)^2+(z-1.8)^2))+ E^-(((x-Z.1)^Z+(y+5.1)^Z+(z-l.8)^2)))+.75( E^-(((x-5.11)^2+(y-Z.1)^Z+(z+l.8)^2))+ E^-(((x-2.1)^2+(y-5.1)^2+(z+ 1.8)^2))+E^-(((x+2.11)^2+(y-5.1 )^2+(z+ 1.8)^2))+ E^-(((x+ 5.1)^2+(y-2.1)^2+(z+ 1.8)^2))+E^-(((x+ 5.11)^2+(y+2.1)^2+(z+ 1.8)^2))+ E^-(((x+2.1)^2+(y+5.1)^2+(z+ 1.8)^2))+E^-(((x-5.11)^2+(y+2.1)^2+(z+ 1.8)^2))+ E^-(((x-2.1 )^2+(y+5.1 )^2+(z+ 1.8)^2)))- 17 , {x,8,-8 }, {y,8,-8 }, {z,3.25,-3.25 }, MaxRecursion->2,PlotPoints-> {{7,4 }, {7,4 }, {7,4 }},Boxed->False,Axes->True] Fig 11.3.8 b: Fig 11.3.9: Show[%,ViewPoint->{0.000,-0.000,3.384}] ContourPlot3D [E^-(((x-2.82)^2+(y-2.82)^2+(z)^2))+E^-(((x-4)A2+(y)^2+(z)^2)) +EA-(((x)^Z+(y-4)AZ+(z)AZ))+EA-(((x-2.82)AZ+(y+Z.82)A2+(z)^2))+EA-(((x)^2+ (y+4)AZ+(z)^Z))+EA-(((x+Z.82)AZ+(y+Z.SZ)AZ+(z)AZ))+EA-(((x+4)^Z+(y)^Z+(z)^2))+ EA-(((x+2.82)^2+(y-2.82)^2+(z)^2)) +EA-(((x-4.23)^2+(y-4.Z3)AZ+(z)^2))+ EA-(((x-6)^Z+(y)^Z+(z)^2)) +EA-(((x)AZ+(y-6)AZ+(z)AZ))+EA-(((x-4.Z3)AZ+(y+4.23)^2+ (z)AZ))+EA-(((x)AZ+(y+6)^Z+(z)AZ))+EA-(((x+4.Z3)AZ+(y+4.Z3)^Z+(z)^2))+ EA-(((x+6)AZ+(y)AZ+(z)^2)) +EA-(((x+4.Z3)AZ+(y-4.Z3)AZ+(z)AZ))+.75(EA-(((x-5.11)^2+ (y-2.1)^2+(z- 1.5)^2))+ E^-(((x-2.1)^2+(y-5.1)^2+(z-1.5)^2))+E^-(((x+2.11)^2+(y-5.1)^2+ (z-1.5)^2))+ E^-(((x+5.1)^2+(y-2.1)^2+(z - 1.5)^2))+E^-(((x+5.11)^2+(y+2.1)^2+(z-1.5)^2))+ E^-(((x+2.1)^2+(y+5.1)^2+(z- 1.5)^2))+E^-(((x-5.11)^2+(y+2.1 )^2+(z- 1.5)^2))+ E^-(((x-2.1)^2+(y+5.1)^2+(z-1.5)^2)))+.75( E^-(((x-5.11)^2+(y-2.1 )^2+(z+ 1.5)^2))+ E^-(((x-2.1)^2+(y-5.1)^2+(z+ 1.5)^2))+E^-(((x+2.11)^2+(y-5.1 )^2+(z+ 1.5)^2))+ E^-(((x+5.1)^2+(y-2.1)^2+(z+ 1.5)^2))+E^-(((x+5.11)^2+(y+2.1)^2+(z+ 1.5)^2))+ E^-(((x+2.1)^2+(y+5.1)^2+(z+ 1.5)^2))+E^-(((x-5.11 )^2+(y+2.1)^2+(z+ 1.5)^2))+ E^-(((x-2.1)^2+(y+ 5.1)^2+(z+ 1.5)^2))) - 12 , {x,8,-8 }, {y,8,-8 }, {z,3.3,-3.3 }, MaxRecursion->2,PlotPoints-> {{7,4 }, {7,4 }, {7,4 }},Boxed->False,Axes->True] 516 Appendix Fig 11.3.10: ContourPlot3D[E^-(E^-(x^2+(y)^2+z^2-5)+ E^(z^2)-l.5)+ E^-(((x^2+y^2)^.5-1.2)^2+(z)^2+.6)+ E^-(((x^2+y^2)^.5-2)^2+(z+2)^2+.5) + E^-(((x^2+y^2)^.5-2)^2+(z-2)^2+.5)-1 ,{x,n,-n},{y,4,0},{z,4,4}, MaxRecursion->2,PlotPoints-> {{3,5 }, {3,5}, {3,5 }},Boxed->False,Axes->True] Fig 11.3.11 a: ContourPlot3D[EA-(E^-(x^2+(y)^2+z^2-5)+E^(z^2)-l.5)+(E^-(x^2+ y^2+(z+3)^2-1))+(E^-( x^2+ y^2+(z+ 1)^2-1))+2 E^-(((x^2+y^2)^.5-3)^2+(z+2.5)^2)+ E^-(((x^2+y^2)^ 5-3 )^2+(z-2.5)^2)- 1, {x,4,-4 }, {y,4,-4 }, {z,-5,4 },MaxRecursion->2, PlotPoints->{ {3,5}, {3,5 }, {3,5} },Boxed->False,Axes->True] Fig 11.3.11 b:Show[%,ViewPoint->{1.359,-2.509,-1.819}] Chapter 12 Fig 12.1.1:ContourPlot3D[(EA((x+2.618 y)^4)+EA((-x+2.618 y)A4)+EA((y+2.618 z)^4)+ EA((-y+2.618z)A4)+E^((-2.618x+z)^4)+E^((2.618x+z)A4))+E^(1.618 (x+y+z))^4+ E^(1.618 (x-y-z))A4+E^(1.618 (-x-y+z))^4+ E^(1.618 (-x+y-z))^4-10^8, {x, 8,- }, {y, 8,- }, {z, 8,- },MaxRecursion->2,PlotPoints-> {{4,5 }, {4,5 }, {4,5 } }, Boxed->False,Axes->True] Fig 12.12: ContourPlot3D[E^(3.618(xA2+yA2+zA2))-E^((1.618x+y)A2)-E^((- 1.618x+y)A2)E^((1.618y+z)^Z)-E^(( - 1.618y+z)^Z)-E^((-x+ 1.618z)AZ)-EA((x+1.618z)^2), {x, 1.5,- 1.5 }, {y, 1.5,- 1.5 }, {z, 1.5,- 1.5 },MaxRecursion->2,PlotPoints-> {{4,5 }, {4,5 }, {4,5 }},Boxed-> False,Axes->Yrue] Fig 12.1.3: ContourPlot3D[E^(7.854(x^2+y^2+z^2))-(E^((- 1.618x+ 1.618y+ 1.618z)^2) +E^((1.618x+ 1.618y- 1.618z)A2)+E^((1.618x- 1.618y+ 1.618z)^2)+ E^((1.618x+ 1.618y+ 1.618z)A2)+E^((x+2.618y)A2)+EA((-x+2.618y)A2)+E^((2.618x+z)^2)+ E^((-2.618x+z)A2)+EA((-y+2.618z)^2)+EA((y+2.618z)^2)), {x, 1.6,-1.6}, {y, 1.6,- 1.6}, {z, 1.6,- 1.6 },MaxRecursion->2,PlotPoints-> {{5,5 }, {5,5 }, {5,5 }},B oxed->False,Axes->True] Fig 12.1.4: Plot[x Sin [ Pi (x)],{x,-8,8},PlotPoints->200,Axes->True] Fig 12.1.5: ContourPlot3D[z Sin[ Pi z]+ x Sin[ Pi x]+y Sin[ Pi y] -1,{x,1.2,-1.2}, {y, 1.2,- 1.2 }, {z, 1.2,- 1.2 },MaxRecursion->2,PlotPoints-> {{5,3 }, {5,3 }, {5,3 }}, Boxed->False,Axes->True] Fig 12.1.6:ContourPlot3D[Sin[Pi((x+2.618 y))]((x+2.618 y))+ Sin[Pi((-x+2.618 y))]((-x+2.618 y))+Sin[Pi((y+2.618 z))]((y+2.618 z))+ Sin[Pi((-y+2.618 z))]((-y+2.618z))+Sin[Pi((z-2.618 x))]((-2.618x+z))+ Sin[Pi((z+2.618 x))]((2.618x+z))+Sin[Pi(1.618(x+y+z))](1.618 (x+y+z))+ Sin[Pi(1.618(x-y-z))](1.618 (x-y-z))+Sin[Pi(1.618(-x-y+z))](1.618 (-x-y+z))+ Sin[Pi(1.618(-x+y-z))](1.618 (-x+y-z))2, {x,.82,-.82}, {y,.82,-.82}, {z,.82,-.82 }, MaxRecursion->2,PlotPoints-> {{3,5 }, {3,5 }, {3,5 }},B oxed->False,Axes->True] Fig 12.1.7: ContourPlot3D[Sin[Pi((1.618x+y))] ( (1.618x+y))+Sin[Pi((-1.618x+y))] ((-1.618x+y))+Sin[Pi((1.618y+z))] ( (1.618y+z))+Sin[Pi((-1.618y+z))] ( (-1.618y+z))+ Sin[Pi((x-l.618z))] ( (x-l.618z))§ ((x+l.618z))+2.7,{x,l.15,-1.15}, {y, 1.15,- 1.15 }, {z, 1.15,- 1.15 },MaxRecursion->2,PlotPoints-> {{5,4 }, {5,4 }, {5,4 }}, Boxed->False,Axes->True] Fig 12.1.8: ImplicitPlot[Cos[ Pi x] Cos[Pi (.5 x+.866 y)] Cos[Pi (-.5 x+.866 y)]==-.1, {x,2,2 }, {y,-2,2 },PlotPoints-> 100] Mathematica 517 Fig 12.1.9: ContourPlot3D[Sin[Pi((1.618x+y))] ( (1.618x+y))+Sin[Pi((-1.618x+y))] ((-1.618x+y))+Sin[Pi((1.618y+z))] ( (1.618y+z))+Sin[Pi((-1.618y+z))] ( (-1.618y+z))+ Sin[Pi((-x+l.618z))] ( (-x+l.618z))+Sin[Pi((x+l.618z))] ((x+l.618z))+3.2,{x,l.15,-1.15}, {y, 1.15,- 1.15 }, {z, 1.15,- 1.15 },MaxRecursion->2,PlotPoints->{ {5,4 }, {5,4}, {5,4} }, Boxed->False,Axes->True] Fig 12.1.10:ContourPlot3D[Cos[Pi((1.618x+y))] E^( (1.618x+y)^2)+ Cos[Pi((- 1.618x+y))] E^(( - 1.618x+y)^2)+Cos[Pi((1.618y+z))] E^( (1.618y+z)^2)+ Cos[Pi((-1.618y+z))] E^( (-1.618y+z)^2)+Cos[Pi((-x+l.618z))] E^( (-x+l.618z)^2)+ Cos[Pi((x+ 1.618z))] E^((x+ 1.618z)^2)+5.3, {x,.9,-.9}, {y,.9,-.9}, {z,.9,-.9}, MaxRecursion->2,PlotPoints-> { {5,4 }, {5,4 }, {5,4 } },Boxed->False,Axes->True] Fig 12.1.11 a: ContourPlot3D[+E^-((x-2)^2+y^2+z^2)+E^-(x^2+y^2+z^2)+ E^-((x-2)^2+(y-2)^2+z^2)+E^-(x^2+(y-2)^2+z^2)+E^-((x-4)^2+(y-2)^2+z^2)+ E^-((x-4)^2+y^2+z^2)+E^-((x-2)^2+(y-4)^2+z^2)+E^-(x^2+(y-4)^2+z^2)+ E^-((x-n)^2+(y-4)^2+z^2)- 8, {x,4.7,-.7 }, {y,4.7,- }, {z,.7,-.7 }, MaxRecursion->2,PlotPoints->{ {3,5}, {3,5 }, {5,3 }},Boxed->False,Axes->True] Fig 12.1.11 b: ContourPlot3D[EA-((x-2)A2+y^2+10 zA2)+E^-(xA2+y^2+10 z^2) E^-((x-2)^2+(y-2)^2+ 10 z^2)+E^-(x^2+(y-2)^2+ 10 z^2)+E^-((x-4)^2+(y,2)^2+ 10 z^2)+ E^-((x-4)^2+y^2+ I z^2)+E^-((x-2)^2+(y-4)^2+ 10 z^2)+E^-(x^2+(y-4)^2+l z^2)+ E^-((x-n)^2+(y-4)^2+ 10 z^2)-.56, {x,5,-1 }, {y,5,-1 }, {z,.35,-.35}, MaxRecursion->2,PlotPoints-> { {3,5 }, {3,5 }, {5,3 } },B oxed->False,Axes->True] Show[%,ViewPoint-> {0.864,-2.828,- 1.645 }] Fig 12.2.1: ContourPlot3D[EA-((x)^2)+E^-((z)^2)+E^-((y)^2)-2, {x,3,-3}, {y,3,-3 }, {z,3,-3 }, MaxRecursion->2,PlotPoints-> { {5,5 }, {5,5 }, {5,5 } },B oxed->False,Axes->True] Fig 12.2.2: ContourPlot3D[E^-((x)A2)+EA-((y)^2)-l.9,{x,.35,-.35},{y,.35,-.35},{z,1,-1}, MaxRecursion->2,PlotPoints-> { {5,3 }, {5,3 }, {5,3 } },B oxed->False,Axes->True] Fig 12.2.3: ContourPlot3D[EA-((x)A2)+E^-((y)A2)+E^-((z)A2)-l.9,{x,4,-4},{y,4,-4},{z,4,-4}, MaxRecursion->2,PlotPoints-> { {5,4 }, {5,4 }, {5,4 } },B oxed->False,Axes->True] Fig 12.2.4: ContourPlot3D[(EA-(x)^2)+(EA-(y)A2)+(EA-(z)^2)+(EA-(x-2.5)^2)+ (EA-(y-2.5)^2)+(EA-(z-2.5)^2)- 1.98, {x,-4,6 }, {y,-4,6 }, {z,-4,6 },MaxRecursion->2, PlotPoints-> { {4,5 }, {4,5 }, {4,5 } },Boxed->False,Axes->True] Fig 12.2.5: ContourPlot3D[(E^-(x)A2)+(E^-(y)^2)+(EA-(z)A2)+(E^-(x-8)^2)+ (E A-(y-8)^2 )+(EA-(z-8)^2)- 1.98, {X,-4,12 }, {y,-4,12 }, {Z,-4,12 },MaxRecursion->2, PlotPoints-> { {5,5 }, {5,5 }, {5,5 } },Boxed->False,Axes->True] Fig 12.2.6: ContourPlot3D[(Cos[Pi x])A8+(Cos[Pi y])A8+(Cos[Pi z])^8-1.98, {x, 1.5,-.5 }, {y, 1.5,- }, {z, 1.5,- },MaxRecursion->2,PlotPoints-> { {5,4 }, {5,4 }, {5,4 } },B oxed->False,Axes->Yrue] Fig 12.2.7: ContourPlot3D[ E^-(z)+EA-(x)^2+EA-(y)A2+EA-(x-4)A2+EA-(y-4)^2-1.95, {x,8,-4 }, {y,8,-4 }, {z,6,- 1},MaxRecursion->2,PlotPoints-> { {7,4},{ 7,4}, {7,4} }, Boxed->False,Axes->True] Fig 12.3.1 a: ContourPlot3D[E^-(10(y^2+(z)^2)+E^-(2 x)+E^(2 (x-ll)))-.9,{x,12,-1}, {y, 1,- 1}, {z, 1,- 1},MaxRecursion->2,PlotPoints-> { {5,4 }, {5,3 }, {5,3 } }, Boxed->False,Axes->True] 518 Appendix Fig 12.3.1 b: ContourP1ot3D[Eh-(10(yh2+(z)h2)+Eh-(2 x)+Eh(2 (x-ll)))-.9,{X,10.5,7}, {y, 15,- 15 }, {Z, 15,- 15},MaxRecursion->2,PlotPoints-> {{5,4 }, {5,3 }, {5,3 }},Boxed->False, Axes->True] Fig 12.3.2: ContourPlot3D[Eh-(10(yh2+zh2)+ Eh((x - 11))+ Eh-(X+2))+ Eh'((yh2+ (x)h2+zh2))-.96, {X,6,-2},{y,.45,-.45 }, {Z,.45,-.45},MaxRecursion->2, PlotPoints-> {{5,4 }, {5,3 }, {5,3 }},Boxed->False,Axes->True] Fig 12.3.3: ContourPlot3D[Eh(z Cos[Pi xh2]+ y Sin[Pi xh2])+Eh( (zh2+yh2))'l.95, {X,2,-2}, {y,.55,'.55 }, {Z,.55,'.55 },MaxRecursion->2,PlotPoints-> {{5,5 }, {5,4 }, {5,4 }}, Boxed->False,Axes->False] Fig 12.3.4 a: ImplicitP1ot[10h-((x)h2)+ 10h'((x)h2+10h(y'15)+10h'(y'10)) 0.1, {X,'8,8 }, {y,4,20 },PlotPoints-> 100] Fig 12.3.4 b: ImplicitP1ot[10h-((x)h2+10h(y-ll))+10h-((x+.l y-1)h2+10h(y-15)+ 10h-(y- 10))+ 10h-((-X+ y- 1)h2+l 0h(y- 15)+10h-(y- 10)) 0.1,{X,-8,8},{y,4,20}, PlotPoints-> 100] Fig 12.3.5:ImplicitP1ot[10h-((x)h2+10h(y-11))+.12 10h-(((x)h2+(y-15.7)h2))+ 12 10h-(((X+2)h2+(y-15.4)h2))+2 10h-((X+.2 y-2)h2+10h(y-15)+10h-(y-10))+ 10h-((-X+.2 y-2)h2+10h(y-15)+10h-(y-10))==0.1,{X,-8,8},{y,4,20},P1otPoints->200] Fig 12.3.6: ImplicitP1ot[10h-((x)h2+l 0h(y-6)+ 10h'(y+6)) +.2 10h'(('X'.9 y'5)h2+ 10h-(y+ 15)+ 10h(y+4))+.2 10h'((X'.4 y'2)h2+ 10h'(y+ 15)+ 10h(y+4))+.2 10h'(( " lx'y'4)h2+ 10h(x-10)+10h-(x))+.12 10h-((X-.2 y-7.5)h2+10h-(y+20)+10h(y+10))+.15 10h'((-.lx'y" 10)h2+ 10h'(x+ 12)+ 10h(x+2))+ 10h'((X+.4 y'2)h2+ 10h(y"19)+ 10h'(y'5))+ 10h-((-X+.4 y'2)h2+ 10h(y"17)+ 10h-(y'5))+ 10h'((X+2 y'25)h2+ 10h(y-21)+ 10h-(y" 15))+ 10h'(('X+ y'8)h2+ 10h(y-22)+ 10h-(y- 12))+ 10h-((X+1.5 y-27)h2+ 10h(y-21)+ 10h-(y- 17))+ 10h-((-X+ y- 16)h2+ 10h(y-24)+ 10h-(y- 13)) 0.1,{X,-18,16}, {y,-20,27},PlotPoints-> 100] Fig 12.4.1: ImplicitPlot[ Eh(- 2(X-6)h2)-y +Eh-(2 xh2) -0,{X,8,-3},{y,-1,1}, PlotPoints-> 100] Fig 12.4.2 a: ImplicitPlot[-12(3+4Cosh[2x-8 5]+Cosh[4x-64 5])/(3 Cosh[x-28 5]+ Cosh[3x-36 5])h2+y 0,{X,12,-6}, {y,- 1,10},PlotPoints->100] Fig 12.4.2 b: ImplicitPlot[-12(3+4Cosh[2x-8 2]+Cosh[4x-64 2])/(3 Cosh[x-28 2]+ Cosh[3x-36 2])h2+y 0,{X,12,-6}, {y,- 1,10},PlotPoints-> 100] Fig 12.4.3 a: ImplicitPlot[ (Sech[x])h2-y==0,{x,5,-5},{y,6,-1 },PlotPoints->100] Fig 12.4.3 b: ImplicitPlot[ (Sech[x-8])h2-y==0,{x,12,-5},{y,6,-1},PlotPoints->100] Fig 12.4.4 a: ImplicitPlot[4 (Sech[x])h2+8 (Sech[x-8])h2-y==0,{x,10,-5},{y,9,-1 }, PlotPoints-> 100] Fig 12.4.4 b: ImplicitPlot[4 (Sech[x])h2+8 (Sech[x-4])h2-y==0,{x,10,-5},{y,8,-1 }, PlotPoints-> 100] Fig 12.4.5: ImplicitPlot[{y- ( Sech[ x])h2-( Sech[ X-3])h2-( Sech[ X-6])h2-( Sech[ X-9])h2 0,Eh-xh2+Eh-(x-3)h2+Eh-(x-6)h2+Eh-(x-9)h2-y }, {X,-4,13}, {y,-2,2 },PlotPoints->200, Axes->False] Mathematica 519 Fig 124.6:ImplicitPlot[Cos[(2/100)^.5 x]^100-y -0,{x,-3,50},{y,-1,1},PlotPoints->200, Axes->False] Chapter 13 Fig 13.3.1:ContourPlot3D[E^(-8 x^2)-E^(-8 (x-1)^2)+E^(-8 (x-2)^2)-E^(-8 (x-3)^2) + (E^(-8 (x-4)^2)-E^(-8 (x-5)^2)+ E^(-8 (x-6)^2))-l(E^(-8 (x-7)^2)-E^(-8 (x-8)^2) + E^(-8 (x-9)^2)) +E^(-8 (x-10)^2)-E^(-8 (x-11)^2) +E^(-8 (x-12)^2) - E^(-8 (x-13)^2) + E^(y^2+z^2)-3.5, {x, 15,-2 }, {y,3 ,-3 }, {z,3,-3 },MaxRecursion->2, PlotPoints-> {{7,4 }, {5,4 }, {5,4 }},Boxed->False,Axes->True] Fig 13.3.2:ContourPlot3D[E^(-8 x^2)-E^(-8 (x-1)^2)+E^(-8 (x-2)A2)-E^(-8 (x-3)^2) + 91 (E^(-8 (x-4)^2)-E^(-8 (x-5)^2)+ E^(-8 (x-6)^2))-l(E^(-8 (x-7)^2)-E^(8 (x-8)^2)+ E^(-8 (x-9)^2)) +E^(-8 (x-10)^2)-E^(-8 (x-11)^2)+E^(-8 (x-12)^2)- E^(-8 (x-13)^2) + E^(y^2+z^2)-3.5, {x, 15,-2 }, {y,3,-3 }, {z,3,-3 },MaxRecursion->2, PlotPoints->{ {7,4},{5,4},{5,4} },Boxed->False,Axes->True] Fig 13.3.3:ContourPlot3D[E^(-8 xA2)-E^(-8 (x-1)^2)+E^(-8 (x-2)^2)-E^(-8 (x-3)^2) + (E^(-8 (x-4)^2)-E^(-8 (x-5)^2)+E^(-8 (x-6)^2))-.l(E^(-8 (x-7)^2)-E^(-8 (x-8)^2)+ E^(-8 (x-9)^2))+E^(-8 (x-10)^2)-E^(-8 (x-11)^2)+E^(-8 (x-12)^2) - E^(-8 (x-13)^2)+ E^(y^2+z^2)-3.5, {x, 15,-2 }, {y,3,-3 }, {z,3,-3 },MaxRecursion->2,PlotPoints-> {{7,4 }, {5,4 }, {5,4 }},B oxed->False,Axes->True] Fig 13.3.4:ContourPlot3D[E^(-8 xA2)-E^(-8 (x-1)A2)+E^(-8 (x-2)A2)-E^(-8 (x-3)^2) + (E^(-8 (x-4)^2)-E^(-8 (x-5)^2)+ E^(-8 (x-6)^2))-.l(E^(-8 (x-7)A2)-E^(-8 (x-8)^2)+ E^(-8 (x-9)^2))+E^(-8 (x-10)^2)-E^(-8 (x-11)^2)+E^(-8 (x-12)^2)- E^(-8 (x-13)^2)+ EA(yA2+z^2)-3.5, {x, 15,-2 }, {y,3,-3 }, {z,3,-3 },MaxRecursion->2,PlotPoints-> {{7,4 }, {5,4 }, {5,4 } },B oxed->False,Axes->True] Fig 13.3.5:ContourPlot3D[E^-(.5 EA(x)+.5(Cos[Pi x]+Cos[Pi y]+Cos[Pi z])+ E^(y^2+z^2)) - 05, {x,-8,1.5 }, {y,- 1.2,1.2 }, {z,- 1.2,1.2 },MaxRecursion->2,PlotPoims-> {{5,3 }, {5,3 }, {5,3 }},Boxed->False,Axes->True] Fig 13.3.6:ContourPlot3D[E^-(.5 EA(-x)+.5(Cos[Pi x]Sin[Pi y]+Cos[Pi z]Sin[Pi x]+ Sin[Pi z]Cos[Pi y])+EA(y^2+z^2))-.05,{x,8,-1.5},{y,-1.2,1.2},{z,-1.2,1.2},MaxRecursion->2, PlotPoints-> { {5,3 }, {5,3 }, {5,3 } },Boxed->False,Axes->True] Fig 13.3.7:ContourPlot3D[E^-(.5 EA-(x)+.5(Cos[Pi x]Sin[Pi y]+Cos[Pi z]Sin[Pi x]+ Sin[Pi z]Cos[Pi y])+EA(yA2+zA2))+E^-(.5 E^(x)+.5(Cos[Pi x]+Cos[Pi y]+Cos[Pi z])+ EA(yA2+z^2))- 05, {x,4,-3 }, {y,- 1.2,1.2 }, {z,- 1.2,1.2 },MaxRecursion->2,PlotPoints-> {{5,4 }, {5,3 }, {5,3 } },B oxed->False,Axes->True] Fig 13.3.8 a: ContourPlot3D[.5 Cos[Pi z]+(yA2+x^2)-12,{x,-4,4},{y,-4,4},{z,-4,4}, MaxRecursion->2,PlotPoints-> {{5,4 }, {5,4 }, {5,4 }},Boxed->False,Axes->True] Fig 13.3.8 b: ContourPlot3D[(Cos[Pi z]+Cos[Pi x]+Cos[Pi y])+(yA2+xA2)-12,{x,-4,4}, {y,-4,4 }, {z,-4,4 },MaxRecursion->2,PlotPoints-> {{5,4 }, {5,4 }, {5,4 }},Boxed->False,Axes>True] Fig 13.3.8 e: ContourPlot3D[.6 (Cos[Pi x] Sin[Pi z]+Cos[Pi y] Sin[Pi x]+ Cos[Pi z] Sin[Pi y])+(y^2+x^2)-12,{x,-4,4},{y,-4,4},{z,-4,4},MaxRecursion->2, PlotPoints-> {{5,4 }, {5,4 }, {5,4 } },Boxed->False,Axes->True] 520 Appendix Fig 13.4.3 a: ContourPlot3D[EA-( EA(2(X+2))+.2 (yA2+zA2))+EA-(EA(-2(X-2))+ (yA2+zA2))'EA('2((y" 1)A2+zA2+(X+3)A2-.5))-EA(-6((y-1)^2+(Z-2)A2+(X+3)A2-.5)) En(-6((y - 1)A2+(Z+2)A2+(X+3)A2-.5))'EA('2((y+ 1)A2+zA2+(X+3)A2-.5))-En(-6((y+ 1)A2+ (z-2)A2+(x+3)A2-.5))-EA(-6((y+•)A2+(z+2)A2+(x+3)A2-.5))-EA(-2((y-•)A2+zA2+(x+5)A2-.5))EA(-6((y - 1)A2+(Z'2)A2+(X+5)A2- 5))-EA(-6((y- 1)A2+(Z+2)A2+(X+5)A2- 5))-EA(-2((y+ 1)A2+ zA2+(X+ 5)A2- 5))-EA(-6((y+ 1)A2+(Z-2)A2+(X+5)A2- 5))-EA(-6((y+ 1)A2+(Z+2)A2+(X+5)A2- 5)).04,{X,-7,3 }, {y,- 1,4.1 }, {Z,-4.1,4.1 },MaxRecursion->2,PlotPoints->{ {6,4}, {6,4},{6,4} }, Boxed->False,Axes->True] Show[%, ViewPoint->{-1.344,-2.329, 2.054}] Fig 13.4.3 b: ContourPlot3D[EA-( E^(2(x+2))+.2 (yA2+zA2))+EA-(EA(-2(X-2))+ (yA2+zA2))-EA(-2((y-1)A2+zA2+(X+2.75)A2-.5))-EA(-6((y -1)A2+(z-2)A2+(x+2.75)A2-.5))EA(-6((y - 1)A2+(Z+2)A2+(X+2.75)A2- 5))-EA(-2((y+ 1)A2+zA2+(X+2.75)A2- 5)) EA(-6((y+ 1)A2+(Z'2)A2+(X+2.75)A2- 5))-EA(-6((y+ 1)A2+(Z+2)A2+(X+2.75)A2- 5))E^(-2((y - 1)A2+zA2+(X+4.5)A2- 5))-EA(-6((y-1 )A2+(Z-2)^2+(X+4.5)A2- 5))" En(-6((y - 1)A2+(Z+2)A2+(X+4.5)A2- 5))-EA(-2((y+ 1)A2+zA2+(X+4.5)A2- 5))EA(-6((y+ 1)A2+(Z-2)A2+(X+4.5)A2- 5))-EA(-6((y+ 1)A2+(Z+2)A2+(X+4.5)A2- 5))- 04, {X,-7,3 }, {y,- 1,4.1 }, {Z,-4.1,4.1 },MaxRecursion->2,PlotPoints-> { {3,5 }, {3,5 }, {3,5 } }, Boxed->False,Axes->True] Show[%, ViewPoint->{-1.344,-2.329, 2.054}] Chapter 14 Fig 14.3.3:ContourPlot3D[(Cos[.25 Pi ( x-y)] EA(.025 Cos[Pi z])-( Cos[.25 Pi(x+ y)])), {x,4.6,-4.6 }, {y,4.6,-4.6 }, {z,2,-2 },MaxRecursion->2,PlotPoints-> { {5,4 }, {5,4 }, {4,4 } }, Boxed->False,Axes->True] Fig 14.5.1:ContourPlot3D[En(Cos[.25Pi ( x-y)] EA(.05 Cos[Pi z])-( Cos[.25 Pi(x+ y)]))+ EA-((y-8)A2)+EA-((y+8)A2)- 1, {X,6,- 10 }, {y,8.3,-8.3 }, {Z,2,-2 },MaxRecursion->2,PlotPoints-> { {5,4}, {5,4}, {5,4} },Boxed->False,Axes->True] Subject Index 521 Subject Index A C actin 9206; 207; 208; 211; 223; 228; 229; 230; 244; 431 Alhambra 432 alveolar surface" 341; 342; 343; 349; 350; 351; 352; 353; 354; 356; 357; 358; 361; 420 alveolar surface phase 9354; 356; 357; 358 amalgamation 9163; 463 amoeba 9168 amphiphilic 91; 337; 358 anaesthetic agents 92; 183; 184; 317; 330; 332; 335; 336; 337 anaesthetic effects 9335 apoptosis 168 Archaebacteria 366 Archimedes 9105 axon membrane 3; 313; 314; 315; 316; 317; 320; 321; 326; 327; 336; 363 axoneme 9223; 236; 240; 242; 244; 255 catenoids 928; 34; 38; 39; 41; 52; 57; 58; 61; 62; 68; 73; 76; 97; 99; 141; 153; 181; 211; 223; 227; 229; 234; 270; 292; 299; 388; 390; 396; 399; 409; 415; 416; 472 cell division 4; 193; 194; 196; 198; 199; 200; 201; 204; 227; 232; 285; 299; 301 cell hybrid 264; 265 cell membrane systems 9163 cell membranes 91; 2; 3; 4; 41; 73; 131; 162; 163; 164; 167; 169; 180; 184; 363; 364; 417 cell organelles 91; 182; 257 chemical synapse 327; 328; 332 chirality 9187; 301 Chlamydomona 9252; 253; 254 chloroplasts 168; 364 cilia 223; 236; 252; 253; 255 clathrin 92; 189; 257; 285; 292 cnoidal waves 9311 colloidal particles 163; 164; 176; 353; 398 colloidal state complex exponential 447; 448; 449 complex number 447 compound-460 concentration gradient 2; 83; 105; 225; 227; 229; 313 connectivity 131; 153; 154; 155; 158; 159; 160; 161; 180; 301; 388; 390; 392; 393; 396; 398 connexons 9327; 332 cosh 974; 76 crawling of cells 9116 cruciform 9193; 217; 219 cubic lipid bilayers 9164; 177 cubic phases 2; 164; 180; 184 cubosomes" 4; 16; 131; 132; 133; 135; 147; 163; 164; 169; 179; 180; 182; B B 12HI22 9289 bacteriorhodopsin 175 bilateral symmetry 9431; 432 bilayer compressibility 181 bilayer motions 93; 187 biological motion 4; 76; 122; 129; 139; 212; 223; 247; 302 Bloch walls 9311 body centred structure 9391; 440 Bonnet transformation 47; 68; 463 breast stroke 223; 252 budding off" 188; 189 butter fly 9255 Subject Index 22 187; 310; 333; 353; 366; 398; 431; 452 curvature elasticity 9181 cyclic crawling 9116 cytoskeleton proteins" D D surface 22; 23; 24; 25; 48; 65; 154; 157; 395 D'Arcy Thompson 363; 366 de Moivre 448 depolarisation 9314; 316; 317; 318; 321; 326; 333 Diffusion equation 74 dilatation 111; 151; 223; 237; 242; 276; 287; 289; 445 Diophante 9268; 270; 271 DNA 3; 27; 100; 101; 102; 193; 213; 214; 217; 219; 220; 310; 366; 419; 426; 429; 431; 432 dodecahedron 138; 144; 285; 287; 289; 290; 291; 292; 293; 440; 441; 459 dynein 93; 244; 255 E earthquake 305 eccentric wheel 244 eigenvibrations 398 electrical synapses 327; 332 ELF structure 917 elliptic point-464 endocytosis 992; 116 endoplasmatic reticulum 92; 3; 4; 163; 164; 167; 169; 170; 171; 172; 174; 188; 260; 265; 273; 333 erythrocyte membranes 9187 Escher 9432 excitation 314; 315; 323; 327; 330; 334; 363 exocytosis 992; 116; 329; 331 exponential scale 3; 76; 180; 212; 237; 247; 399; 401; 402; 405; 407; 409; 411; 419; 425; 433; 447; 456; 460; 461; 463; 472 F Fermat 268 fetal rat lung 9345 Fibonacci 151; 152; 238 Filament construction 9235 finite periodicity 1; 73; 83; 107; 131; 141; 151; 152; 318; 319; 447; 450; 452 flagella 4; 105; 111; 113; 115; 116; 126; 223; 236; 244; 247; 248; 249; 251; 252; 255; 409 flagella of bacteria 9113 flagella of sperms 9111 fractal 4; 299 fundamental theorem 7; 8; 9; 16; 105; 106; 107; 385; 431 G gap junctions 327; 328; 332 garnet 9231 Gauss distribution 75; 81; 399 gaussian curvature 911; 166; 180; 181; 319; 349; 416; 463; 464; 466; 467 goke 236 golden mean 9285 Golgi 93; 4; 184; 188; 257; 259; 260; 261; 262; 264; 265; 273; 333; 409 groves 9101 gyroid surface" 47; 48; 139; 156; 165; 170; 230; 301; 321; 393 H harmonic oscillator 106; 132; 247 Helfrich 5; 183; 185; 191 helicoid 15; 16; 114; 244; 245; 425; 428; 456; 466; 467 helicoidal tower surface 9428 Hermite function 83; 105; 106; 108; 109; 110; 111; 115; 131; 132 Hermite polynoms 9106 hexagon 9187; 209; 431 hexagonal 2; 27; 28; 36; 38; 41; 42; 58; 62; 131; 154; 175; 182; 184; 207; 208; 209; 327; 433; 434; 435; 436; 439; 443 Hilbert 398; 466; 472; 476 Holliday junction 9193; 214 Hyde 5; 31; 45; 72; 162; 165; 190; 221; 338; 339; 346; 362; 373; 464; 467; 472; 476 Subject Index 523 hydrophilic 91; 343; 354; 355 hydrophobic 91; 343 hyperbolic polyhedra 9285; 310 icosahedron 9285; 286; 289; 292; 293; 444; 459; 460 infinite product 8; 9; 18; 23; 25; 132; 248; 395 infinite products-9; 23; 25; 395 Invaginations 9271; 272 IPMS" 2; IWP surface" 24 K kick 255 kinesin L lamellar bodies 342; 353; 354; 355 Laplace equation 74 leech 371; 372 Lidin 95; 31; 42; 45; 103; 162; 190; 221; 256; 338; 339; 346; 362; 373; 417; 472; 476 lipid bilayers 3; 163; 164; 165; 177; 180; 188; 315; 323; 337; 345; 347; 360 lung surfactant 9342; 343; 361 lungs 341; 342; 356; 361 Luzzati 91; 372 M magnetic domains 9311 magnetite 394; 395; 442 mathematical mirror 444 Max Born 72 mean curvature 270; 456; 464 membrane lipids 92; 41; 164; 184; 364; 366; 368 microtubuli microtubulus 9235; 236; 239; 255 minimal surface structures 9164 mitochondria 9164; 168; 169; 193; 204; 260 mitochondrion 9193; 201; 204; 206; 409 monkey saddle- 22; 23; 25; 26; 36; 37; 58; 61; 267; 270; 397; 422; 423; 424; 426; 464; 466 motion 4; 73; 74; 76; 87; 96; 97; 101; 105; 111; 112; 113; 114; 115; 116; 117; 118; 119; 120; 122; 125; 126; 127; 129 ;139; 180; 181; 193; 194; 201; 207 ;208; 212; 223; 224; 227; 232; 236; 244; 247; 252; 255; 257; 302; 305; 314; 320; 331; 337; 363; 375; 385; 409 muscle cell 9131; 166; 193; 206; 244; 313; 327; 333 myelinated 9315; 318; 326 myosin 206; 207; 208; 211; 223; 244; 247 N natural exponential 76; 78; 409; 447; 449; 461 natural number 76 Neovius surface 25; 26 nerve signal conduction 9313; 363 nerve trunk 326 Nesper 45; 49; 72; 177; 178; 191 neuromuscular junction 9328 Newton 9105 nonagon 236 nuclear pore complex 94; 257; 276; 282; 283 O octahedron 918; 80; 107; 132; 149; 294; 295; 442; 457; 458; 459; 460 organelles 91; 163; 182; 223; 225; 257; 364 orthorhombic" 436;437; 439 p P surface 47; 48; 153 parabolic point 464 pentagonal dodecahedron 285 periodic motion 9111; 116; 118; 120; 193 photoreceptor membrane 9364 Subject lndex 524 pinocytosis 989; 189 polymerase 9101; 214 polynomial algebra 257 potassium channels 9315 power expansion 99; 16; 108 power stroke 9207; 223; 244; 246; 247 precipitation 9225; 227; 228 presynaptic membrane 329; 330; 331; 332; 333 pretzel 473; 474 prolamellar body 9168 proteins 1; 4; 92; 113; 129; 131; 160; 163; 165; 168; 175; 188; 211; 223; 227; 229; 234; 236; 244; 255; 257; 276; 282; 317; 327; 332; 333; 335; 342; 343; 361; 368; 431 pulmonary capillaries 341 Q quasi symmetry 237 R Ranvier node 9315 repolarisation 9314; 315; 318 respiration 341; 359; 360 rhombic dodecahedron 138; 144; 440; 441; 459 RNA- 213; 214; 217; 276; 367; 368 rod packing 27; 28; 230 ruffling 299 S sacromere 27 saddle point 9464; 465; 466 sarcoplasmatic reticulum 9333 Scherk 913; 369; 468; 469; 470; 471 Schr0dinger 9106; 107; 132; 306; 307; 310 sech 75; 83; 311 Sendai virus 264; 265 shoulders 301 Signal propagation- 315 sodium channels 314; 315; 318; 323; 326; 337 solitons 975; 305; 306; 307; 308; 309; 310; 311 solubility 4; 223; 225 spikes 285; 297; 299; 409 standing wave conformations 92; 180; 184 standing wave dynamics 9163 stella octangula 9442; 457; 458 Streptomyces hygroscopicus 9183 symmetry group 432 synaptic transmission 327; 329; 330; 332; 336 Synge 9105; 130 T tetragonal 27; 28; 29; 30; 31; 32; 34; 36; 47; 61; 62; 64; 65; 160; 161; 341; 349; 361; 410; 436; 439; 472 tetrahedron 80; 98; 99; 124; 126; 137; 456 The CLP surface 64 thylakoid membranes 9168 tidal breathing 347; 359 topology 2; 31; 126; 131; 160; 214; 217; 225; 253; 257; 266; 376; 416 torus- 99; 101; 102; 282; 472; 473 tower surfac 913; 15; 370; 420; 421; 423; 424; 426; 427; 428; 468; 469 tower surface 13; 15; 370; 420; 421; 423; 424; 426; 427; 428; 468; 469 transmitter molecules 313; 329; 330; 331; 332 travelling waves 75 tree 4; 285; 301; 305; 341; 361; 409 truncated octahedron 9460 tubular myelin 341; 342; 345; 350; 354; 360 Tubulin 235 twin operation 34; 223; 252; 253; 254; 255 U unmyelinated axons 9326 V vesicle 3; 85; 86; 87; 88; 89; 91; 92; 93; 96; 153; 163; 180; 184; 188; 189; 190; 225; 257; 258; 259; 265; 266; 270; 273; 274; 313; 329; 331; 332; 368; 388; 405 Subjec t lndex 525 vesicle transport 163; 189 virus 4; 264; 265; 285; 292; 361; 444 von Schnering 26; 45; 49; 72; 177; 178; 191 W Wave equation 74 wave motions 92; 177; 180; 182; 189; 190; 208; 338 wave packet 9109; 111 wavelet 9109 Weierstrass 931; 42; 346; 463; 472 w h e e l 111; 243; 244; 463; 475; 476 This Page Intentionally Left Blank ... or otherwise, without prior written permission of the Publisher Address permissions requests to: Elsevier Science Rights & Permissions Department, at the mail, fax and e-mail addresses noted above... K Larsson and S Andersson, Z Kristallogr 212 (1997) K Larsson, M Jacob and S Andersson, Z Kristallogr 211 (1996) 875 M Jacob and S Andersson; THE NATURE OF MATHEMATICS AND THE MATHEMATICS OF... animals, and changes of shape associated with various life processes The cell membranes and cytoskeleton proteins build these shapes based on physical forces A mathematical/geometrical description