Contents

Preface 13

A Conversation 15

Part I Symmetry Around Us 23

1 Mirror Symmetry 27

1 An Object and Its Mirror Twin 27
2 Mirror Symmetry 30
3 Enantiomorphs 31

2 Other Kinds of Symmetry 35

4 Rotational Symmetry 35
5 Mirror-Rotational Symmetry
6 Translational Symmetry
7 Bad Neighbours 39
8 Glide Plane (Axis) of Symmetry

3 Borders and Patterns 43

9 Borders 43
10 Decorative Patterns 45
11 Pattern Construction 49
12 The ‘Lizards’ Design 51

4 Regular Polyhedra 53

13 The Five Platonic Solids 53
14 The Symmetry of the Regular Polyhedra 54
15 The Uses of the Platonic Solids to Explain Some Fundamental Problems 55
16 On the Role of Symmetry in the Cognition of Nature 57

5 Symmetry In Nature 59

17 From the Concept of Symmetry to the Real Picture of a Symmetrical World 59
18 Symmetry in Inanimate Nature. Crystals 61
19 Symmetry in the World of Plants 63
20 Symmetry in the World of Animals 66
21 Inhabitants of Other Worlds 69

6 Order in the World of Atoms 71

22 Molecules 71
23 The Puzzle of the Benzene Ring 72
24 The Crystal Lattice 73
25 The Face-Centred Cubic Lattice 74
26 Polymorphism 76
27 The Crystal Lattice and the External Appearance of a Crystal 77
28 The Experimental Study of Crystal Structures 78
29 The Mysteries of Water 79
30 Magnetic Structures 80
31 Order and Disorder 82

7 Spirality In Nature 85

32 The Symmetry and Asymmetry of the Helix 85
33 Helices in Nature 87
34 The DNA Molecule 89
35 The Rotation of the Plane of Light Polarization 91
36 Left and Right Molecules. Stereoisomerism 92
37 The Left-Right Asymmetry of Molecules and Life 93

Part II Symmetry Around Us 97

8 Symmetry and The Relativity of Motion 101

38 The Relativity Principle 101
39 The Relativity of Simultaneous Events 102
40 The Lorentz Transformations 103
41 The Relativity of Time Periods 105
42 The Speed in Various Frames 106

9 The Symmetry of Physical Laws 109

43 Symmetry Under Spatial Translations 109
44 Rotational Symmetry 111
45 Symmetry in Time 112
46 The Symmetry Under Mirror Reflection 113
47 An Example of Asymmetry of Physical Laws 114

10 Conservation Laws 117

48 An Unusual Adventure of Baron Münchhausen 117
49 The Problem of Billiard Balls 118
50 On the Law of Conservation of Momentum 121
51 The Vector Product of Two Vectors 122
52 Kepler’s Second Law 123
53 Conservation of the Intrinsic Angular Momentum of a Rotating Body 126

11 Symmetry and Conservation Laws 129

54 The Relationship of Space and Time Symmetry to Conservation Laws 129
55 The Universal and Fundamental Nature of Conservation Laws 130
56 The Practical Value of Conservation Laws 133
57 The Example of the Compton Effect 134
58 Conservation Laws as Prohibiting Rules 136

12 The World of Elementary Particles 139

59 Some Features of Particles 139
60 The Zoo of Elementary Particles 141
61 Particles and Antiparticles 142
62 Particles, Antiparticles and Symmetry 146
63 Neutrino and Antineutrino 148
64 The Instability of Particles 149
65 Inter-conversions of Particles 152

13 Conservation Laws and Particles 157

66 Conservation of Energy and Momentum in Particle Reactions 157
67 The Conservation of Electric Charge and Stability of the Electron 159
68 The Three Conservation Laws and Neutrino 160
69 Experimental Determination of Electron Antineutrino 162
70 Electron and Muon Numbers. Electron and Muon Neutrinos 163
71 The Baryon Number and Stability of the Proton 165
72 Discrete Symmetries. CPT-Invariance 167

14 The Ozma Problem 171

73 What Is the Ozma Problem? 171
74 The Ozma Problem Before 1956 173
75 The Mirror Asymmetry of Beta-Decay Processes 174
76 The Mirror Asymmetry in Decay Processes and the Ozma Problem 175
77 The Fall of Charge-Conjugation Symmetry 176
78 Combined Parity 177
79 Combined Parity and the Ozma Problem 179
80 The Solution to the Ozma Problem 180

15 Fermions and Bosons 183

81 The Periodic Table and the Pauli Principle 183
82 Commutative Symmetry. Fermions and Bosons 184
83 Symmetrical and Antisymmetrical Wave Functions 185
84 The Superfluidity of Liquid Helium. Superconductivity 187
85 Induced Light Generation and Lasers 188

16 The Symmetry of Various Interactions 189

86 The Principal Types of Interactions 189
87 Isotopic Invariance of Strong Interactions. The Isotopic Spin (Isospin) 191
88 Strangeness Conservation in Strong and Electromagnetic Interactions 194
89 Interactions and Conservations 197
90 A Curious Formula 198
91 The Unitary Symmetry of Strong Interactions 199

17 Quark-Lepton Symmetry 203

92 Quarks 203
93 The Charmed World 207
94 Quark-Lepton Symmetry 208
95 A New Discovery 210

A Conversation Between the Author and the Reader About the Role of Symmetry 213

The Ubiquitous Symmetry 213
The Development of the Concept of Symmetry 215
Symmetry-Asymmetry 217
On the Role of Symmetry in the Scientific Quest for Knowledge 219
Symmetry in Creative Arts 222

Literature 231