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4 For Emile Boulanger CONTENTS Introduction 1 Arrays 2 Rotations and Reflections 4 Geometric SelfSimilarity 6 Radial 8 Sections and Skeletons 10 Spherical 12 Symmetries in 3D 14 Stacking and Packing 16 The Crystalline World 18 Basic Stuff 20 Dorsiventrality 22 Enantiomorphy 24 Curvature and Flow 26 Spirals and Helices 28 Fabulous Fibonacci 30 Branching Systems 32 Fascinating Fractals 34 Penrose Tiling and Quasicrystals 36 Asymmetry 38 Selforganising Symmetries 40 Symmetries in Chaos 42 Symmetry in Physics 44 Symmetry in Art 46 A Passion for Pattern 48 Symmetria 50 Formalism 52 Experiential Symmetries 54 Appendix 56 Glossary 58 Let proportions be found not only in numbers and measures, but also in sounds, weights, times and positions, and whatever force there is Leonardo da Vinci. Above Da Vincis conjecture that the total crosssectional area of a tree remains the same at all branching levels a balance illustrates the hidden symmetry of force equal to mass x distance. Overleaf a sample of natures infinite symmetrical variety Ernst Haekels drawings of various species of diatoms. All pictures by the author, except for Japanese pinebark pattern on page 39, reproduced from Japanese Patterns, by Jeanne Allen, with kind permission of Chronicle Books and the Portrait of Emmy Noether on page 47 by Jesse Wade. For further reading, try Symmetry and the Beautiful Universe, by Leon Lederman and Christopher Hill, Mario Livios The Equation that couldnt be Solved, or Symmetry, a Unifying Concept by Istvan and Magdolna Hargittai.
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