E-Book Content
Contents
Chapter 1 1 Introduction References 1 Publications on 'The Exponential Scale'
Chapter 2 2 The Roots of Mathematics - the Roots of Structure 2.1 Multiplication of Polynomials 2.2 Addition of Polynomials 2.3 Saddles Exercises 2 References 2
7 7 16 26 31 38
Chapter 3 3 The Natural Function and the Exponential Scale 3.1 Polygons and Planar Geometry 3.2 Polyhedra and Geometry 3.3 Curvature 3.4 The Fundamental Polyhedra- and Others 3.5 Optimal Organisation and Higher Exponentials Exercises 3 References 3
39 39 47 52 55 62 67 71
Chapter 4 4 Periodicity and the Complex Exponential 4.1 The Translation Vector 4.2 The Complex Exponential and Some Variants 4.3 Some Other Exponentials Exercises 4 References 4
73 73 80 88 94 98
Chapter 5 5 The Screw and the Finite Periodicity with the Circular Punctions 5.1 Chirality, the Screw and the Multi Spiral 5.2 The Bending of a Helix 5.3 Finite Periodicity- Molecules and the Larsson Cubosomes Exercises 5 References 5
99 99 109 112 119 122
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Contents
Chapter 6 6 Multiplication, Nets and Planar Groups 6.1 Lines and Saddles 6.2 Nets with Two Planes, and Variations 6.3 Nets with Three Planes, and Variations 6.4 Nets with Four Planes, and Variations 6.5 Structures in 3D from the Nets 6.6 Quasi 6.6.1 Four Planes and Quasi 6.6.2 Five Planes and Quasi Exercises 6 References 6
123 123 125 127 130 132 136 136 137 139 146
Chapter 7 7 The Gauss Distribution Function 7.1 The GD Function and Periodicity 7.1.1 Handmade Periodicity 7.2 The GD Function and Periodicity in 3D 7.3 The BCC and Diamond Symmetries 7.4 The Link to Cosine Exercises 7 References 7
147 147 155 156 161 176 186 190
Chapter 8 8 Handmade Structures and Periodicity 8.1 Prelude 8.2 Simplest of Periodic Structures 8.3 Contact of Spheres in Space- Structures and Surfaces 8.4 How Tetrahedra and Octahedra meet in Space Exercises 8 References 8
191 191 201 205 226 229 236
Chapter 9 9 The Rods in Space 9.1 Primitive Packing of Rods 9.2 Body Centred Packing of Rods 9.3 Tetragonal and Hexagonal Packing of Rods 9.4 Larsson Cubosomes of Rods 9.5 Packing of Rods, and their Related Surfaces Exercises 9 References 9
237 237 242 246 253 258 262 265
Contents
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Chapter 10 10 The Rings, Addition and Subtraction 10.1 Some Simple Examples of Subtraction and Addition in 3D 10.2 The Rings 10.3 More Ways to make Rings 10.4 More Subtraction- Hyperbolic Polyhedra Exercises 10 References 10
267 267 273 278 283 289 292
Chapter 11 11 Periodic Dilatation- Concentric Symmetry 11.1 Dilatation and Translation in 2D 11.2 Dilatation and Translation in 3D 11.3 Pure Dilatation Exercises 11 References 11
293 293 300 312 320 324
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
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 transformation