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The Nature of Mathematics and the Mathematics of Nature PDF

355 Pages·1998·25.28 MB·English
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The Nature of Mathematics and the Mathematics of Nature TThhiiss PPaaggee IInntteennttiioonnaallllyy LLeefftt BBllaannkk The Nature of Math and the Mathematics of Nature by Michael Jacob Department of Inorganic Chemistry Arrhenius Laboratory University of Stockholm S- 10691 Stockholm, Sweden and Sten Andersson Sandviks Forskningsinstitut S. L~nggatan 27 S-38074 L6ttorp, Sweden 1998 ELSEVIER AMSTERDAM. LAUSANNE. NEWYORK. OXFORD. SHANNON. SINGAPORE. TOKYO ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands ISBN: 0-444-82994-6 Library oF Congress Cataloging-in-Publication Data Jacob, Nichae]. The nature of mathematics and the mathematics of nature / Nichae] Jacob, Sten Andersson. p. cm. Includes bibliographical references and index. ISBN 0-444-82994-6 (alk. paper) 1. Mathematics. I. Andersson, Sten. II. Tit]e. QA39.2.J313 1998 510--dc21 98-36663 CIP (cid:14)9 1998 Elsevier Science B.V. All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any from or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publishers, Elsevier Science B.V., Copyright & Permissions Department, P.O. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the U.S.A.- This publication has been registered with the Copyrigt Clearance Center Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the publisher. O The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper). Printed in The Netherlands. Contents Chapter 1 1 Introduction References 1 Publications on 'The Exponential Scale' Chapter 2 2 The Roots of Mathematics - the Roots of Structure 7 2.1 Multiplication of Polynomials 7 2.2 Addition of Polynomials 16 2.3 Saddles 26 Exercises 2 31 References 2 38 Chapter 3 3 The Natural Function and the Exponential Scale 39 3.1 Polygons and Planar Geometry 39 3.2 Polyhedra and Geometry 47 3.3 Curvature 52 3.4 The Fundamental Polyhedra- and Others 55 3.5 Optimal Organisation and Higher Exponentials 62 Exercises 3 67 References 3 71 Chapter 4 4 Periodicity and the Complex Exponential 73 4.1 The Translation Vector 73 4.2 The Complex Exponential and Some Variants 80 4.3 Some Other Exponentials 88 Exercises 4 94 References 4 98 Chapter 5 5 The Screw and the Finite Periodicity with the Circular Punctions 99 5.1 Chirality, the Screw and the Multi Spiral 99 5.2 The Bending of a Helix 109 5.3 Finite Periodicity- Molecules and the Larsson Cubosomes 112 Exercises 5 119 References 5 122 vi Contents Chapter 6 6 Multiplication, Nets and Planar Groups 123 6.1 Lines and Saddles 123 6.2 Nets with Two Planes, and Variations 125 6.3 Nets with Three Planes, and Variations 127 6.4 Nets with Four Planes, and Variations 130 6.5 Structures in 3D from the Nets 132 6.6 Quasi 136 6.6.1 Four Planes and Quasi 136 6.6.2 Five Planes and Quasi 137 Exercises 6 139 References 6 146 Chapter 7 7 The Gauss Distribution Function 147 7.1 The GD Function and Periodicity 147 7.1.1 Handmade Periodicity 155 7.2 The GD Function and Periodicity in 3D 156 7.3 The BCC and Diamond Symmetries 161 7.4 The Link to Cosine 176 Exercises 7 186 References 7 190 Chapter 8 8 Handmade Structures and Periodicity 191 8.1 Prelude 191 8.2 Simplest of Periodic Structures 201 8.3 Contact of Spheres in Space- Structures and Surfaces 205 8.4 How Tetrahedra and Octahedra meet in Space 226 Exercises 8 229 References 8 236 Chapter 9 9 The Rods in Space 237 9.1 Primitive Packing of Rods 237 9.2 Body Centred Packing of Rods 242 9.3 Tetragonal and Hexagonal Packing of Rods 246 9.4 Larsson Cubosomes of Rods 253 9.5 Packing of Rods, and their Related Surfaces 258 Exercises 9 262 References 9 265 Contents vii Chapter 10 10 The Rings, Addition and Subtraction 267 10.1 Some Simple Examples of Subtraction and Addition in 3D 267 10.2 The Rings 273 10.3 More Ways to make Rings 278 10.4 More Subtraction- Hyperbolic Polyhedra 283 Exercises 10 289 References 10 292 Chapter 11 11 Periodic Dilatation- Concentric Symmetry 293 11.1 Dilatation and Translation in 2D 293 11.2 Dilatation and Translation in 3D 300 11.3 Pure Dilatation 312 Exercises 11 320 References 11 324 Appendix 1 Mathematica 325 Appendix 2 Curvature and Differential Geometry 327 Appendix3 Formal Way to Derive the Shapes of Polyhedra 330 Appendix 4 More Curvature 333 Appendix 5 Raison d'etre 335 Subject Index 339 TThhiiss PPaaggee IInntteennttiioonnaallllyy LLeefftt BBllaannkk 1 Introduction 'every chapter is an &troduction' (from D'Arcy Thompson) We are chemists, and as chemists we find it necessary to build models for the understanding and description of structures in science. This book concerns the tool we found in order to build and describe structures with the use of mathematics. Chemistry, as well as the rest of natural science, is awfully complicated - because it is Nature. Mathematics is man-made and therefore not as complicated. We found good use of it from group theory for crystal structure determination and description [1], and we used the intrinsic curvature to explain reactions and structures in inorganic solid state chemistry. We dealt with minimal surfaces, isometric transformations and applications in natural science [2]. Together with mathematicians and bio- scientists THE LANGUAGE OF SHAPE [3] was born. We found the daily use of the mathematics involved somewhat heavy, for instance the differential geometry, Riemann surfaces or Bonnet transformations. As chemists we are, we tried new routes - other branches - of mathematics. We introduced the Exponential Scale a few years ago, and the articles published are collected below. The field is unusually rich, and instead of writing more articles we decided to write this book, and search as deep as possible into the mathematics. We found that the 3D representations of the hyperbolic functions are the concave adding of planes, and the convex subtraction of planes. These give polyhedra in the first case, and saddles in the second. We also found that the multiplication of planes give the general saddle equations and the multispirals. And that the simplest complex exponential in 3D (also composed of planes) is a fundamental nodal surface, within 0.5 % the same as the famous Schwartz minimal surface as found by Schwartz himself. This surface is in a way identical to a classical chemical structure. We found that the functions we do can be dissecting into planes or lines, which may be the roots that build the fundamental theorem of algebra, and our finite periodicity.

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Chemistry, physics and biology are by their nature genuinely difficult. Mathematics, however, is man-made, and therefore not as complicated. Two ideas form the basis for this book: 1) to use ordinary mathematics to describe the simplicity in the structure of mathematics and 2) to develop new branche
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