The Nature of Mathematics and the Mathematics of Nature

355 Pages·1998·25.28 MB·English

by  M. Jacob

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
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(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.