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This Amazingly 

Symmetrical World 

L. Tarasov 

Mir Publishers 

This Amazingly 



JL TapacoB 






L. Tarasov 

This Amazingly 



Translated from Russian by 
Alexander Repyev 

Typeset in XgKTgX by 
Damitr Mazanov 

Mir Publishers 

Translated from the Russian by Ram S. Wadhwa 
First published 1986 
Revised from the 1982 Russian edition 

This electronic version typeset using XgKTpX by Damitr Mazanov. 
Released on the web by http : //mirtitles . org in 2018. 


Preface 13 

A Conversation 15 

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 37 

6 Translational Symmetry 37 

7 Bad Neighbours 39 

8 Glide Plane (Axis) of Symmetry 41 


3 Borders and Patterns 43 

9 Borders 43 

10 Decorative Patterns 45 

11 Pattern Construction 49 

12 The ‘Lizards’Design 51 

4 Regular Poly he dr a 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 

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 




26 Polymorphism 76 

27 The Crystal Lattice and the External Appearance of a Crystal 

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 

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 Munchhausen 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 Lundamental 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 

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 


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 
Symmetry in Creative Arts 222 

Literature 231 


Nature! Out of the simplest matter it 
creates most diverse things, without the 
slightest effort, with the greatest 
perfection, and on everything it casts sort 
of fine veil. Each of its creations has its 
own essence, each phenomenon has 
separate concept, but everything is a 
single whole. 

- Gothe 


Symmetry is encountered everywhere - in nature, engineering, arts, and science. Note, for example, 
the symmetry of a butterfly and maple leaf, the symmetry of a car and plane, the symmetry of a verse 
and tune, the symmetry of patterns and borders, the symmetry of the atomic structure of molecules 
and crystals. 

The notion of symmetry can be traced down through the entire history of human creative endeav¬ 
ours. It has its beginnings in the well-springs of human knowledge and it has widely been used by all 
the modern sciences. So principles of symmetry dominate in physics and mathematics, chemistry and 
biology, engineering and architecture, painting and sculpture, poetry and music. The laws of nature, 
which govern the infinite variety of phenomena, in turn obey the principles of symmetry. 

Now then, what is symmetry? What profound idea under lies the concept? Why does symmetry 
literally permeate the entire world around us? Answers to these questions can be found in this book. 

The book is arranged in two parts. The first part considers the symmetry of positions, forms, and 
structures. This is the sort of symmetry that can be seen directly. And so it can be thought of as 
geometric symmetry. The second part is concerned with the symmetry of physical phenomena and 
law of nature. This symmetry lies at the very foundation of the naturalistic picture of the world, and 
so it can be called physical symmetry. 

The book is written in simple nontechnical language and is designed for the reader with but an 
elementary understanding of physics and mathematics, On the other hand the book is not intended to 
be an easy, entertaining read. The reader will have to be patient at times to grasp some relatively dif¬ 
ficult points, especially in the first chapters, which treat the notions of mirror rotational, translational, 
and other kinds of symmetry. No special background is required here, just a measure of patience to 
be rewarded, as I hope, by the satisfaction derived from reading the subsequent chapters. 

I would like to attract the reader’s attention to the very idea of symmetry and help him to discern 
a wide variety of the manifestations of symmetry in the surrounding world, and above all. to demon- 


strate the most important role played by symmetry in the scientific comprehension of the world and 
in human creative effort. 

The idea of writing the book was prompted by I. Gurevich, who was also very helpful in select¬ 
ing the material and format o f the book, for which I am grateful to him. Thanks are also due to A. 
Tarasova for her help with the manuscript. 

L. Tarasov 

Standing at the blackboard and drawing some figures on it with chalk I was suddenly 
struck by the idea: why is symmetry so pleasing to the eye? What is symmetry? It is an 
innate feeling, I answered myself. But what is it based on? 

Lev Tolstoy 

A Conversation Between 
the Author and the Reader 
About What Symmetry Is 

: There is an old parable about the ‘Buridan’s ass’. The 
ass starved to death because he could not decide on which heap 
of food to start with (Figure 1). The allegory of the ass is, of 
course, a joke. But take a look at the two balanced pans in the 
figure. Do the pans not remind us in some way of the parable? 

Figure 1: The ass starved to 
death as he could not decide 
which heap to start with. 

Reader : Really. In both cases the left and the right are so identi- 


cal that neither can be given preference. 

: In other words, in both cases we have a symmetry, 
which manifests itself in the total equity and balance of the left 
and right. And now tell me what you see in Figure 2. 

Figure 2: The pyramids are 
symmetric structures. 

Reader : The foreground shows a pyramid. Such pyramids were 
erected in Ancient Egypt. In the background are hills. 

: And maybe in the foreground you see a hill as well? 

Reader : This is no hill. This looks like an artificial structure. A 
hill normally does not possess that regular, symmetric shape. 

: Quite so. Perhaps you could provide quite a few exam¬ 
ples of regular configuration (symmetry) of objects or structures 
created by man? 

Reader : A legion of them. To begin with, architectural structures. 
For example, the building of the Bolshoi Theatre in Moscow. 
Essentially all vehicles, from a cart to a jet liner. Household 
utensils (furniture, plates, etc.). Some musical instruments: a 
guitar, a violin, a drum, . . . 

: Man-made things are often symmetrical in shape. Al¬ 
though this is not always the case, just remember a piano or a 
harp. But why do you think symmetry is so often present in 
human products? 


Reader : The symmetry of shape of a given object may be deter¬ 
mined by purpose. Nobody wants a lop-sided ship or a plane 
with wings of different lengths. Besides, symmetrical objects are 

: That reminds me of the words of the German math¬ 
ematician Hermann Weyl (1885-1955), who said that through 
symmetry man always tried to ‘perceive and create order, beauty 
and perfection’. 

Reader : The idea of symmetry, it seems, is inherent in all spheres 
of human endeavour. 

: Exactly. It would be erroneous however, to think that 
symmetry is mostly present in human creations whereas nature 
prefers to appear in non-symmetrical (or asymmetrical) forms. 
Symmetry occurs in nature no rarer than in man-made things. 
Since the earliest times nature taught man to understand symme¬ 
try and then use it. Who has not been moved by the symmetry 
of snow-flakes, crystals, leaves, and flowers? Symmetrical are 
animals and plants. Symmetrical is the human body. 

Reader : Really, symmetrical objects seem to surround us. 

: Not just objects. Symmetry can also be seen in the regu¬ 
larity of the alternation of day and night, of seasons. It is present 
in the rhythm of a poem. Essentially we deal with symmetry ev¬ 
erywhere where we have a measure of order. Symmetry, viewed 
in the widest sense, is the opposite of chaos and disorder. 

Reader : It turns out that symmetry is balance, order, beauty, 
perfection, and even purpose. But such a definition appears to 
be a bit too general and blurry, doesn’t it? What then is meant 
by the term ‘symmetry’ more specifically? What features signal 
the presence, or absence for that matter, of symmetry in a given 

: The term ‘symmetry’ (avfinerpva) is the Greek for 
‘proportionality, similarity in arrangement of parts’. 

Reader : But this definition is again not concrete. 


: Right. A mathematically rigorous concept of symmetry 
took shape only recently, in the 19th century. The simplest ren¬ 
dering (according to Weyl) of the modern definition of symmetry 
is as follows: an object is called symmetrical if it can be changed 
somehow to obtain the same object. 

Reader : If I understand you correctly, the modern conception of 
symmetry assumes the unchangeability of an object subject to 
some transformations. 

: Exactly. 

Reader : Will you please clarify this by an 

: Take, for example, the symmetry of the letters U, H and 


Reader : But N does not appear to be symmetrical at all. 

: Let’s start with U. If one half of the letter were reflected 
in a plane mirror, as is shown in (Figure 3 a), the reflection would 
coincide exactly with the other half. This is an example of so- 
called mirror symmetry, or rather a symmetry in relation to 
mirror reflection. We have already encountered this form of 
symmetry: it then appeared as an equilibrium of the left and 
right sides (the left and the right in (Figure 1) can be viewed as 
mirror reflections of each other). 

Letter H is symmetrical to an even higher degree than U. It can 
be reflected in two plane mirrors (Figure 3 b). As for the letter 
N, it has no mirror symmetry, but it has the so-called rotational 
symmetry instead. If N were turned through 180° about the axis, 

o • 

Figure 3: Looking at sym¬ 
metries of the letters of 


that is normal to the letter’s plane and passes through its centre 
(Figure 3 c), the letter would coincide with itself. Put another 
way, the letter N is symmetrical in relation to a turn through 180°. 
Note that rotational symmetry is also inherent in H, but not in U. 

Reader : An Egyptian pyramid too possesses a rotational symme¬ 
try. It would coincide with itself if mentally turned through 90° 
about the vertical axis passing through the pyramid’s summit. 

: Right. In addition, a pyramid possesses a mirror symme¬ 
try. It would coincide with itself if mentally reflected in any of 
the four imaginary planes shown in (Figure 4) 

Reader : But how is one to understand the above-mentioned 
symmetry in the alternation of seasons? 

: As the unchangeability of a certain set of phenomena 
(including weather, blooming of plants, the coming of snow, etc.) 
in relation to shift in time over 12 months 

Reader : The beauty of symmetrical objects and phenomena 
seems to come from their harmony and regularity. 

: The question of the beauty related to symmetry is not 
that obvious. 

Reader : Why not? I can imagine that looking at harmonious, bal¬ 
anced, recurrent parts of a symmetrical object must give one the 
feeling of peace, stability, and order. As a result, the object will 
be perceived as beautiful. And even more so, if in its symmetry 
we also see some purpose. On the contrary, any accidental viola¬ 
tion of symmetry (the collapsed corner of a building, the broken 
piece of a neon letter, an early snow) must be received negatively 
- as a threat to our trust in the stability and orderliness of the 
surrounding world. 

: That is all very well but it is known that symmetry 
may also produce negative emotions. Just look at some modern 
residential areas consisting of identical symmetrical houses 
(often fairly convenient and rational), do they not bore you to 
death? On the other hand, some deviations from symmetry used 
widely in painting and sculpture create the mood of freedom and 

Figure 4: The planes of 
symmetry for a pyramid. 


nonchalance, and impart an inimitable individuality to a work 
of art. Hardly anyone would be immune to the spell of a spring 
meadow in full bloom with an absolutely asymmetrical array 
of colours. Do you really think that a neatly mown lawn or a 
trimmed tree look prettier than a clearing in a forest or an oak 
growing in a field? 

Reader : Beauty is thus not always related to symmetry. 

: The truth is that when considering symmetry, you have 
to take into account not only symmetry as such, but also devi¬ 
ations from it. Symmetry and asymmetry must be approached 

Reader : Perhaps as is the case in nature? 

: Of course. But one important point here: one should 
not merely look at given violations of symmetry in a specific 
flower or organism of an animal. The issue of symmetry-asymmetry 
is far deeper. Symmetry may be said to express something gen¬ 
eral, characteristic of different objects and phenomena, it is 
associated with the structure and lies at the very foundation of 

Asymmetry, on the other hand, is related to realizations of the 
structure of a given specific object or phenomenon. 

Reader : Symmetry is general, asymmetry is specific? 

: If you like, you may think of it this way. In a specific 
object we find elements of symmetry, which link it to other sim¬ 
ilar objects. But the individual ‘face’ of a given object invariably 
shows up through the presence of some element of asymmetry. 
Spruces have much in common: a vertical stem, characteristic 
branches arranged in a certain rotational symmetry around the 
stem, a definite alternation of branches along the stem, and lastly 
the structure of needles. And still you take your time selecting 
the right spruce at a Christmas tree bazar. Among the many 
trees available you look for individual traits that you like most. 

Reader : It turns out then that the mathematical idea of symmetry 
in each case is embodied in real, not-very-symmetrical objects 


and phenomena. 

: Let’s try and visualize a world, arranged in a totally 
symmetrical manner. Such a world would have to be able to be 
reproduced in any rotation about any axis, in the reflection of 
any mirror, in any translation, and so on. This would be some¬ 
thing absolutely homogeneous, uniformly spread out over the 
entire space. 

Stated another way, in that sort of world you would observe 
nothing, neither objects nor phenomena. And so such a world is 

The world exists owing to the marriage of symmetry and asym¬ 
metry, which can in a way be treated as the unity of the general 
and the specific. 

Reader : Frankly, I never thought of symmetry in such a broad 

: Let’s draw some conclusions. Symmetry occurs widely, 
both in nature and in human life. Therefore, even a layman gen¬ 
erally has no difficulty in discerning symmetry in its relatively 
simple manifestations. 

Our world, all the things and phenomena in it, must be ap¬ 
proached as a manifestation of symmetry and asymmetry. In 
that case, symmetry is not just abundant, it is ubiquitous, in the 
broadest sense. 

Symmetry is utterly diverse. Objects may remain unchanged 
under turns, reflections, translations, interchanges, and so forth. 

Symmetry has many causes. It may be related to orderliness 
and equilibrium, proportionality and harmony of parts (and 
sometimes monotony), purposefulness and usefulness. 

Part I 

Symmetry Around Us 

Tyger! Tyger! burning bright 
In the forests of the night. 

What immortal hand or eye 
Dare frame thy fearful symmetry? 

- William Blake 

1 Mirror Symmetry 

1 An Object and Its Mirror Twin 

Figure 5 shows a simple example of an object and its ‘looking-glass 
twin - triangle ABC and triangle AiBiCi (here MN is the intersection 
of the mirror plane with the plane of the drawing). For each point on 
the triangle there is a corresponding point on the mirror twin. These 
points lie on the same perpendicular to straight line MN, on either side 
and at the same distance from it. For simplicity, the object in the figure 
is taken to be two-dimensional. In the general case, the object (and 
hence its mirror twin) are three-dimensional. 

We all know that a mirror twin can be seen readily. Suffice it to 
place an illuminated object in front of a plane mirror and look into it. 

It is generally believed that the twin in the mirror is an exact replica 
of the object itself. But this is actually not so. A mirror does not just 
duplicate the object but transposes the front and back (in relation to 
the mirror) parts of the object. In comparison with the object itself, its 
mirror twin seems to be turned inside out in the direction perpendic¬ 
ular to the plane of the mirror. This is clearly seen in Figure 6 a and 
essentially imperceptible in Figure 6 b. When examining the cones in 
the figure the reader might disagree with our statement that the mirror 
reflection is not an exact copy of the object. After all, the object and its 
reflection in Figure 6 a only differ in their orientation: they are so ar¬ 
ranged that they face each other. This is all the more so for Figure 6 b. 
In this connection let us turn to a more interesting example. 

Suppose that a cone is rotated about its axis (Figure 7), the direction 

What can be more like my 
hand or my ear than their 
reflections in a mirror! And still 
the hand I see in the mirror 
cannot substitute for my real 

I. Kant 


A A, 

C C, 


Figure 5: Looking at sym¬ 
metry in a mirror reflection 
of a triangle, its looking 
glass twin. 


Figure 6: Looking at sym¬ 
metry of a cone in the 
mirror reflection. 

Figure 7: Looking at rota¬ 
tion of a cone in the mirror 

Figure 8: Looking at spiral- 
ity of a cone in the mirror 

of rotation being indicated by the arrow. If the rotation axis is normal 
to the mirror’s plane, the cone retains its direction of rotation when 
reflected in the mirror (Figure 7 a). If the rotation axis is parallel to the 
mirror, the direction of rotation is reversed in the reflection (Figure 7 b). 
No shifts or turns can now make the object coincide with its reflection. 
In other words, the rotating cone and its reflection are essentially 
different objects. To obtain a reflection without actually reflecting the 
cone in the mirror, we will have to reverse the direction of the rotation 


of the cone. 

We can do this without any rotation, though. If we make a thread 
on the cone (Figure 8), then the object’s thread and the reflection’s 
thread will have different directions: for the object’s thread to be 
driven into wood we will have to rotate its head clockwise, and the 
reflection’s one counterclockwise. The first thread is called right, and 
the second leftt. We generally use right screws. 

We have thus seen that, whatever the amount of similarity, an 
object and its reflection may be different, incompatible. Sometimes the 
difference is not very conspicuous; for instance, you generally attach 
no importance to the fact that a birth-mark on your right cheek in your 
‘looking-glass’ counterpart appears on the left cheek. In other cases 
the difference becomes so striking that we can only be surprised that 
it has been overlooked earlier. It is sufficient to compare a text with its 
reflection (Figure 9). Hold a book up to a mirror and try and read the 
reflection of the text in it. Or, worse, try and write just one line looking 
not at the paper but at its mirror reflection. 

Figure 9: Looking at reflec¬ 
tion of a text. 


2 Mirror Symmetry 

Suppose now that one half of an object is a mirror reflection of the 
other about a plane. Such an object is said to be mirror-symmetrical, 
and the plane is said to be the plane of symmetry. For a two-dimensional 
(flat) object instead of a plane of symmetry we have an axis of symme¬ 
try - the line of intersection of the plane of symmetry with the plane 
of the object. For a unidimensional (linear) object we have the centre 
of symmetry - the point of the intersection of the object?s straight line 
with the plane of symmetry. 

Figure 10 gives examples of mirror-symmetrical objects: (a) unidi¬ 
mensional object (O is the centre of symmetry), (b) two-dimensional 
object (MN is the axis of symmetry), (c) three-dimensional object (S is 
the plane of symmetry). 

A unidimensional object has no more than one centre of symmetry. 

A two-dimensional object may have several axes of symmetry, and 
a three-dimensional object, several planes of symmetry. So a regular 
hexagon has six axes of symmetry (the red lines in Figure 11). In Fig. 4 
were shown four planes of symmetry of a regular pyramid. The circle 
has an infinite number of axes of symmetry, just as the sphere, the 
circular cylinder, the circular cone, and the spheroid. 

Let us print on a sheet of paper in capital letters the words COOK 
and PAN. We will now get a mirror and place it vertically so that the 
line of the intersection of the mirror’s plane with the sheet’s plane 
divides these words in two along the horizontal line. Perhaps some 
will be surprised at finding that the mirror has not changed the word 
COOK, whereas it has changed the word PAN beyond recognition 
( Figure 12). This ‘trick’ can be explained very simply. To be sure, 
the mirror reflects the lower part of both words in a similar fashion. 
Unlike PAN, however, the word COOK possesses a horizontal axis of 
symmetry, that is why it is not distorted by the reflection. 

Figure 10: Examples of 
mirror-symmetrical objects. 

Figure 11: Symmetries of a 


Figure 12: Looking at 
reflection of a text. 

3 Enantiomorphs 

Suppose that an object is characterized by a single plane (axis) of sym¬ 
metry. Let us cut it along the plane (axis) of symmetry into two halves. 
These two halves are, clearly, mirror reflections of each other. It is es¬ 
sential that each half in itself is mirror-asymmetrical. The halves are 
said to be enantiomorphs. 

Thus, enantiomorphs are pairs of mirror-asymmetrical objects (figures) 
that are mirror reflections of each other. In other words, enantiomorphs 
are an object and its mirror reflection provided that the object itself is 
mirror-asymmetrical. Enantiomorphs may be individual objects and 
halves of an object cut appropriately. To distinguish between enan¬ 
tiomorphs of a given pair they are referred to as left and right. It is 
immaterial which is called left (right), it is only a matter of convention, 
tradition, or habit. 

Examples of three-dimensional enantiomorphs are given in Fig¬ 
ure 13: 

(a) left and right screws, (b) left and right dice, (c) left and right 
knots, (d) left and right gloves, (e) left and right reference frames, 

(f) left and right halves of a chair cut along the symmetry plane. Nor¬ 
mally used in practice are right screws, left dice, and right reference 
frames. Left and right gloves and knots occur with equal frequency. 

Given in Figure 14 are examples of two-dimensional enantiomorphs: 

(a) left and right spirals, (b) left and right traffic signs, (c) left and 


right reference frames, (d) left and right halves of an oak leave cut 
along the axis of symmetry. 

Figure 13: Examples of 
three-dimensional enan- 

Left Right 

spiral spiral 

Left Right 

system system 

Figure 14: Examples of 
two-dimensional enan- 

Two-dimensional enantiomorphs cannot be superposed on each 
other by any translations or turns in the space of these enantiomorphs, 
that is, in a plane. To make them coincide a turn in the three-dimensional 
space is required, that is, a turn as shown in Figure 15. As for three- 


dimensional enantiomorphs, for them to coincide exactly, turning in a 
fantastic non-existent four-dimensional space would be required. It is 
clear that to effect the turn or even to visualize it is impossible. There¬ 
fore, for three-dimensional enantiomorphs the following statement is 
valid: no translocations or turns can convert a left enantiomorph into its 
right one, and vice versa. So a left enantiomorph will always be left and 
a right one will always be right. Turn your left shoe as you might, it 
will never fit your right foot. Cast a left die as often as you might, it 
will never turn into a right die. 

Curiously, to prove the existence of an ‘other-worldly’ four-dimensional 
world they contrived tricky demonstrations involving conversions of 
left enantiomorphs into right ones, and vice versa. Such demonstra¬ 
tions were performed at so-called spiritualistic sessions, which were 
rather fashionable at the turn of the century in some decadent aristo¬ 
cratic circles. It goes without saying that those demonstrations were 
just adroit tricks based on sleight of hand. For example, the spiritualist 
would ask a participant in the session to give him his left glove, and 
after some distracting manipulations, the spiritualist would present to 
the spectators sitting in a half-dark room an identical right glove. This 
was claimed to be a proof of the short-term residence of the glove in 
the other world, where it had allegedly turned into the right glove. 

Figure 15: Coinciding 
two-dimensional enan¬ 
tiomorphs via turning in 
three-dimensional space. 

2 Other Kinds of Symmetry 

4 Rotational Symmetry 

Let there be an object that coincides with itself exactly when turned 
about some axis through an angle of 360 °/n (or its multiple), where 
n = 2, 3, 4,... In that case, we speak about rotational symmetry, and 
the above-mentioned axis is referred to as the n-fold rotation symmetry- 
axis, or n-fold symmetry, or axis of n-fold symmetry. In the earlier 
examples of letters H and N we dealt with a two-fold axis, and in the 
example with an Egyptian pyramid, with a 4-fold axis. Figure 16 gives 
examples of simple objects with rotation axes of different orders-from 
two-fold to five-fold. 

A three-dimensional object may have several rotation axes. For ex¬ 
ample, the first object in Figure 16 has three two-fold axes, the second 
object, in addition to a three-fold axis, has three two-fold axes, and 
the third object, in addition to a four-fold axis, has four two-fold axes 
(additional axes are shown in the figure by dash lines). 

Consider the cube. We can see immediately that it has three four¬ 
fold rotation axes (Figure 17 a). Further examination reveals six two¬ 
fold axes passing through the centres of opposite parallel edges (Fig¬ 
ure 17 b), and four three-fold axes, which coincide with the inner 
diagonals of the cube (Figure 17 c). The cube thus has 13 rotation axes 
in all, among them two-, three-, and four-fold ones. 

If we venture beyond our 
habitual notion of symmetry 
as a property that is in all 
ways related to our external 
appearances, we can find 
quite a few figures that are 
symmetrical in one or another 
sense. - A.S. Kompaneets 

Figure 16: Symmetries 
axes of various three- 
dimensional solids. 


Also interesting is the rotational symmetry of the circular cylinder. 
It has an infinite number of two-fold axes and one infinite-fold axis 
(Figure 18). 

To describe the symmetry of a concrete object requires specifying 
all the axes and their orders, and also all the planes of symmetry. Let 
us take, for example, a geometrical body composed of two identical 
regular pyramids (Fig. 19). It has one four-fold rotation axis (AB), 
four two-fold axes ( CE , DF, MP , and NQ), five planes of symmetry 

Figure 17: The 13 axes of 
symmetry in a cube. 

Figure 18: The infinite 
two-fold axes of symmetry, 
and one three-fold axes of 
symmetry for a cylinder. 
Figure 19: The axes of sym¬ 
metry of a body composed 
of two identical regular 


5 Mirror-Rotational Symmetry 

Let us cut a square out of thick paper and inscribe into it obliquely 
another square (Figure 20). Now we will bend the corners along the pe¬ 
riphery of the inner square as shown in Figure 21. The resultant object 
will have a two-fold symmetry axis (AB) and no planes of symmetry. 
We will view this object first from above and then from below (from 
the other side of the sheet). We will find that the top and bottom views 
look the same. This suggests that the two-fold rotational symmetry 
does not exhaust all the symmetry of a given object. 

The additional symmetry of this object is the so-called mirror- 
rotational symmetry, in which the object coincides with itself when 
turned through 90° about axis AB and then reflected from plane CDEF. 
Axis AB is called the four-fold mirror-rotational axis. We thus have a 
symmetry relative to two consecutive operations - a turn by 90° and a 
reflection in a plane normal to the rotation axis. 


Figure 20: Combining 
two squares to create a 


6 Translational Symmetry 

Consider the plane figure in Figure 22 a. A translation along line AB 
by a distance a (or its multiple) makes the figure seem unchanged. In 
that case we speak about translational symmetry. AB is called the axis 
of translation, and a is called the fundamental translation, or period. 
Strictly speaking, a body having a translational symmetry must be 
infinite in the direction of the translation. But the concept of transla¬ 
tional symmetry in a number of cases is also applied to bodies of finite 
sizes when partial coincidence is observed. It is seen in Figure 22 b 
that when a finite figure is displaced by distance a along line AB, part 1 
coincides with part 2. 

Figure 21: Axes of sym¬ 
metry for the composite 
3-d object created from 
inscribed square. 

Associated with translational symmetry is the important concept of 
a two-dimensional periodic structure - the plane lattice. A plane lattice 
can be formed by crossing two sets of parallel equi-spaced straight 
lines (Figure 23), their intersections being called sites. To specify a 
lattice, it is sufficient to specify its unit cell and then to displace it along 
AB by distances multiple of a, or along AC by distances multiple of b. 
Note that in a given lattice a unit cell may be chosen in a wide variety 


A i B 

A B 

Figure 22: A translation 
along line gives rise to 
translational symmetry. 

of ways. So we may select the red cell in Figure 23. Or we may also use 
any of the shaded ones. 

Figure 23: Different ways of 
choosing the unit cell in a 

The translational symmetry of a plane lattice is totally defined by 
a combination of two vectors (a and b in Figure 23). Five types of 
plane lattices (five types of translational symmetry in a plane) are 
distinguished. These are given in Figure 25: (a) a = b, y = 90° (square 
lattice), (b) a + b, y = 90° (rectangular lattice), (c) a = b, y — 60° 


(hexagonal lattice), (d) a = b, y j= 90°, y =£ 60° (rhombic lattice), and 
(e) a + b, y ± 90° (oblique lattice). 

Translational symmetry in three-dimensional space is associated 
with the notion of a three-dimensional periodic structure - the space 
lattice. This lattice can be thought of as a result of the crossing of 
three sets of parallel planes. The translational symmetry of a three- 
dimensional lattice is defined by the three vectors that specify the unit 
cell of the lattice. Figure 24 shows a unit cell described by a, (oblique 
lattice) b, and c. In the simplest case, all the edges of a cell are equal 
in length and the angles between them are 90°. In that case, we have 
a cubic lattice. All in all, there are 14 types of space lattices, differing 
in their type of space symmetry. In other words, there are 14 Bravais 
lattices (named after a French crystallographer of the 19th century). 

7 Bad Neighbours 

Translational and rotational symmetries may live side by side with 
each other. So, the square lattice (Figure 25 a) has a four-fold rotational 
symmetry, and the hexagonal lattice (Figure 25 c) has a six-fold rota¬ 
tional symmetry. It is easily seen that a lattice has an infinite number 
of rotation axes. For example, in the case of the square lattice, the ro¬ 
tation axes (four-fold) pass through the centre of each square cell and 
through each lattice site. 

Figure 24: Vectors defining 
a three-dimensional unit 

But the translational and rotational symmetries are bad neighbours. 
In the presence of a translational symmetry only two-, three-, four-, and 
sixfold rotation axes are possible. Let’s prove this. 

Let points A and B in Figure 26 be sites of some plane lattice 
(|AB| = a). Suppose that through these sites ra-fold rotation axes 
pass perpendicular to the plane of the lattice. We will now turn the 
lattice about axis A through an angle a = 360°, and will designate by C 
the new position of site B. If the lattice were turned through the angle 
a around axis B in the opposite direction, site A would come to D. 


The presence of translational symmetry requires that points C and 
D coincide with lattice sites. Hence, 

Figure 25: Five types of 
plane lattices (five types of 
translational symmetry in a 

|CD| = m |AB| = ma 

where m is an integer. The trapezoid ABDC (see the figure) being 
equilateral, we have |CD| = a + 2a cos a. Thus, 

a(l ± 2 cos a) = ma. 

that is, cos a = ±(m - l)/2. Since | cos a\ < 1, 

—2 < (m — 1) < 2, 

Figure 26: Vectors defining 
a three-dimensional unit 

It follows that only the following five cases are possible: 

(1) m = — 1, cosa = —1, a = 180°, n = 2 (two-fold rotational symmetry); 

(2) m = 0, cos a = —1/2, a = 120°, n = 3 (three-fold rotational symmetry); 

(3) m = 1, cos a = 0, a = 90°, n = 4 (four-fold rotational symmetry); 

(4) m = 2, cos a = 1/2, a = 60°, n = 6 (six-fold rotational symmetry); 

(5) m = 3, cos a = 1, a = 0. 


Accordingly, with translational symmetry, five-fold rotation axes are 
impossible in principle, as well as axes of an order higher than six. 

8 Clide Plane (Axis) of Symmetry 


Figure 27: An example of 
glide symmetry with period 


It was shown earlier that successive turns and reflections may give 
rise to a new type of symmetry - the mirror-reflection symmetry. Com¬ 
bining turns or reflections with translations can also produce new- 
types of symmetry. By way of example, consider a symmetry involving 
the so-called glide plane of symmetry (or rather glide axis of symmetry, 
since the figure in question is a plane). Figure 27 depicts a design ex¬ 
hibiting a translational symmetry along axis AB with period 2a. It is 
easily seen that another symmetry can be revealed here - the symme¬ 
try relative to the displacement along AB with period a followed by the 
reflection about axis AB. Axis AB is termed a glide axis of symmetry 
with period a. 

3 Borders and Patterns 

9 Borders 

A periodically recurring pattern on a tape is called a border. Borders 
come in various types. These may be frescoes, decorating walls, gal¬ 
leries, stairs, and also iron casting used as fencing for parks, bridges, 
embankments. These also may be gypsum plaster-reliefs or earthen¬ 

A mathematician, just like 
an artist or a poet, produces 
designs. - C. Hardy 

Figure 28 (a) presents 14 borders in seven pairs. Each pair consists 
of borders similar in symmetry type. Altogether, there are seven types of 
symmetry of borders. Any border has a translational symmetry along 
its axis (translation axis). In the simplest case, the border has only 
translational symmetry (Figure 28 (a) a). Figure 28 (b) a is a schematic 
representation of that sort of border, the triangle stands for a recurring 
asymmetric element. 

The borders shown in Figure 28 (a) b, apart from a translational 
symmetry, also have a mirror symmetry: they are mirror-symmetrical 
relative to the straight line dividing the border tape in half longitudi¬ 
nally. This sort of border is shown schematically in Fig. Figure 28 (b) b, 
where the translation axis doubles as an axis of symmetry. 

In the borders depicted inFigure 28 (a) c and Figure 28 (b) c the 
translation axis is a glide axis. The borders in Figure 28 (a) d have 
transverse axes of symmetry. They are given in Figure 28 (b) d as 
segments of straight lines perpendicular to the translation axis. 

The borders of Figure 28 (a) e have two-fold rotation axes perpen- 


dicular to the border plane. The intersections of those axes with the 
border plane are marked in Figure 28 (b) e by lentil-shaped figures. 
Combining glide axes with two-fold rotation axes normal to the bor¬ 
der plane produces .the borders given in Figure 28 (a) f, which possess 
transverse axes of symmetry. The scheme of that kind of border is 
presented in Figure 28 (b) f. 



(a) Seven different types of borders showing 
translational symmetry. 


(b) Schematic illustration of the translational 
symmetry in the seven different types of 

Figure 28: The seven border 
types and their schemat¬ 
ics showing translational 


Lastly, Figure 28 (a) g and Figure 28 (b) g provide borders based 
on combinations of mirror reflections. Apart from a longitudinal axis, 
such borders also have transverse axes of symmetry. As a consequence, 

two-fold rotation axes emerge. Please note that in the original 

book, the border figures are 
separate figures 28 and 29. 

. Here we have put them with 

10 Decorative Patterns same figure number. Hence 

there is no fig 29. 

You have admired decorative patterns - those amazing designs that 
occur so widely in applied arts. In them you can find bizarre marriages 
of translational, mirror, and rotational symmetries. Examples abound. 
Just look at the design of the wall-paper in your room. Some patterns 
are shown in Figures 30-32. Two of them have been produced by the 
eminent Dutch artist Escher: ‘Flying Birds’ (Figure 30) and ‘Lizards’ 
(Figure 32). 

Figure 30: Flying Birds by 
Dutch artist M. C. Escher. 

Any pattern is based on one of the five plane lattices discussed in 
Chapter 2. The type of the plane lattice determines the character of the 
translational symmetry of a given pattern. So the ‘Flying Birds’ pattern 
is based on an oblique lattice, the characteristic Egyptian ornament in 
Figure 31 on a square lattice, and the ‘Lizards’ pattern on a hexagonal 

In the simplest case, a pattern is characterized by a translational 
symmetry alone. Such is, for example, the ‘Flying Birds’ pattern. To 
construct that pattern one must select an appropriate oblique lattice 


Figure 31: A characteris¬ 
tic Egyptian ornamental 

and ‘fill’ a unit cell of the lattice with some design, and then reproduce 
it repeatedly by displacing the cell without changing its orientation. 

Figure 32: Lizards by Dutch 
artist M. C. Escher. 


In Figure 33 a unit cell of the pattern is hatched. Note that the 
surface area of the cell is equal to the total area occupied by birds of 
different colours. 

Figure 33: Finding the unit 
cell in Flying Birds by Dutch 
artist M. C. Escher. 

The symmetry of the Egyptian design is analyzed in Figure 34. The 
translational symmetry of the pattern is given by a square lattice with 
a unit cell singled out in Figure 34 a. That cell has two-fold rotation 
axes, conventional and glide axes of symmetry. In Figure 34 b conven¬ 
tional axes of symmetry are indicated by solid lines, and glide axes by 
dash lines. 

Intersections of the two-fold rotation axes with the plane of the pat¬ 
tern are marked by lentil-shaped figures. Unlike the ’Flying Birds’, this 
design has a higher symmetry, as follows from the presence of rotation 
axes, and also of conventional and glide axes of mirror symmetry. 

The symmetry of the Egyptian pattern would be yet higher if we 
simplified the colouring - instead of the red and blue colours used 
one colour, for example red. In that case, we would also have four¬ 
fold rotation axes and more glide axes of symmetry. The symmetry 
of this design is indicated in Figure 34 c, where the black squares are 
intersections of four-fold rotation axes with the plane of the pattern. 

Figure 34 b and c contains all the information about the elements 
of symmetry of the corresponding patterns. If in the figures we re¬ 
moved the design and only retained the conventional and glide axes 




Ss> L 




Figure 34: Finding the unit 
cell in Egyptian ornamental 

Fig. 34 

and intersections of rotation axes with the plane of the pattern, we 
would obtain schematic representations of two different types of the 
symmetry patterns. In all there are 17 types of symmetry plane designs. 
They are given in Figure 35. Here solid straight lines are conventional 
axes of symmetry and dash lines are glide axes. The lentils are the 
intersections of two-fold axes with the plane of the pattern; the trian¬ 
gles, three-fold axes; squares, four-fold axes; and hexagons, six-fold 
axes. The pattern of Figure 34 b is represented in Fig. 35 by # 9, and the 
pattern of Figure 34 c, by # 12; the ‘Flying Birds’, by # 1. 


Figure 35: The 17 types of 
symmetry plane designs. 

11 Pattern Construction 

Any pattern can in principle be constructed along the lines of ‘Flying 
Birds’ by the parallel displacements of the unit cell, which is filled by 
some design. This procedure is the only one possible where a pattern 
has neither rotational nor mirror symmetry. Otherwise, a pattern can 
be produced using other procedures; the initial design then (or main 
motif) is not the entire unit cell of a design, but just part of it. 

Turning to the Egyptian design of Figure 31 , the main motif here 


may be the design within the confines of the shaded rectangle at the 
bottom of Figure 34 a (it makes up 1/8 of the surface area of the unit 
cell). The main motif is given separately in Figure 36 a. 

Figure 36: Constructing 

To construct a pattern we will make use of axes BC and DE in Fig- the Egyptian ornamental 

ure 34 c and Figure 36, and the two-fold rotation axis passing through pattern, 

point A We will now fix A and BC and DE in the figure and obtain 
for the main motif (Figure 36 a) reflections relative to BC and DE and 
turns by 180° about A in any sequence and indefinitely. A turn about 
A gives us Figure 36 b, a subsequent reflection relative to BC gives us 
Figure 36 c. Next we effect a reflection relative to DE (Figure 36 d) and 
another turn by 180° about A (Figure 36 e), yet another reflection rel¬ 
ative to BC, and so on. As we go through the procedure, the pattern 
comes to life before our very eyes filling up the whole of the figure. 


Let us now take the Egyptian design with a simplified colouring 

Figure 37: Constructing a 
simplified Egyptian pattern. 


(Figure 34 c). For our main motif we can now have the double-shaded 
square in Figure 34 a. We will construct our pattern using the four-fold 
axis and axis BC (Figure 37). The reader may do this on his own. 

12 The ‘Lizards’ Design 

This design is quite interesting (Figure 32). It is essentially a mosaic 
composed of identical lizards which are densely packed within the 
space of the pattern (without gaps or overlaps). The mosaic features 
not only translational but also rotational symmetry. The translational 
symmetry of the pattern is determined by the hexagonal lattice, and 
the rotational one by the rotation axes at A , B , C, D, E, F, G, H, etc. 
(Figure 38 a). The order of the rotational axes depends on the pattern’s 
colouring. For a three-colour pattern (lizards of three different colours) 
all the rotation axes are two-fold (Figure 38 b). A single-colour pat¬ 
tern, apart from two-fold axes, has three- and six-fold rotation axes 
(Figure 38 c). The ‘Lizards’ have no mirror The elements of symmetry 
for the single-colour design are given in # 16 in Figure 35, and for the 
three-colour one in # 2. 

In constructing the single-colour ‘Lizards’, we can select as the main 
motif the triangle AGH in Figure 38 c. It is easily seen that the triangle 

Figure 38: Constructing 
the Lizard design by M. C. 


includes parts of one lizard and its area is equal to that occupied by one 
lizard. The pattern can be constructed using the six-fold rotation axes 
(A) and the three-fold rotation axis ( H ). We will now turn the design in 
Figure 38 c) through 60° about A, and then (after a complete turn) we 
will turn the resultant design about H through 120°. 

In the case of the three-colour version the main motif is specified 
not by AGH but by triangle ABC that includes components of all three 
multicoloured lizards (Figure 38 b)). The pattern can be obtained using 
the two-fold rotation axes that pass, for example, through points D, E, 
and F. 

4 Regular Polyhedra 

Since time immemorial, when contemplating the picture of the Uni¬ 
verse, man made active use of the concept of symmetry. Ancient 
Greeks preached a symmetrical universe simply because symmetry 
was beautiful to them. Proceeding from the concept of symmetry, they 
made some conjectures. So Pythagorus (the 5th century B.C.), who 
thought of the sphere as the most symmetrical and perfect shape, con¬ 
cluded that the Earth is spherical and moves over a sphere. He also 
assumed that the Earth revolves about some ‘central fire’, together 
with the six planets known at the time, the Moon, the Sun, and the 

The ancients liked symmetry and besides spheres they also used 
regular polyhedra. So they established the striking fact that there are 
only five regular convex polyhedra. First studied by the Pythagoreans, 
those five regular polyhedra were later described in more detail by 
Plato and so they came to be known in mathematics as Platonic solids. 

13 The Five Platonic Solids 

A regular polyhedron is a volume figure with congruent faces having 
the shape of regular polygons, and with congruent dihedral angles. 

It turns out that there can be only five such figures (although there 
exist an infinite number of regular polygons). All the types of regular 
polyhedra are provided in Figure 39: the tetrahedron (regular triangular 
pyramid), octahedron, icosahedron, hexahedron (cube), dodecahedron. 

The cube and octahedron are mutual, if in one of these polyhedra 

Inhabitants of even the most 
distant galaxy cannot play dice 
having a shape of a regular 
convex polyhedron unknown to 
us. - M. Gardner 


the centres of faces having a common edge are connected, the other 
polyhedron is obtained. Also mutual are the dodecahedron and icosa¬ 

It can readily be shown why only five regular polyhedra are possi¬ 
ble. Let us take the simplest face - the equilateral triangle. A polyhedral 
angle can be obtained by putting together three, four or five equilateral 
triangles, that is, in three ways. (If there are six triangles, the angles at 
the common vertex sum up to 360°.) With squares a polyhedral angle 
can only be formed in one way - with three squares. With pentagons 
too, a polyhedral angle can be obtained in one way - with three pen¬ 
tagons. Clearly, no regular polyhedral angles can be formed out of 
w-hedra with n > 6. Consequently, only five types of regular polyhedra 
can exist: three polyhedra with triangular faces (tetrahedron, octahe¬ 
dron, and icosahedron), one with square faces (cube), and one with 
pentagon faces (dodecahedron). 

14 The Symmetry of the Regular Polyhedra 

Figure 39: The five possible 
types of regular polyhedra. 

The elements of symmetry of the tetrahedron are illustrated in Figure 40. 
It has four three-fold rotation axes and three two-fold axes. Each 
of the three-fold axes passes through a vertex and the centre of the 
opposite face (for example, axis AE in the figure). Each two-fold axis 
passes through the middles of opposite edges (for example, axis FG). 
Through each two-fold axis pass two planes of symmetry (through 
an axis and one of the edges that intersect with a given axis); in the 
figure planes AGC and DFB pass through axis FG. The tetrahedron 
thus has six planes of symmetry. In addition, the tetrahedron has a 
mirror-rotational symmetry: each two-fold rotation axis also doubles 
as a four-fold mirror-rotation axis. 


The symmetry of the cube was discussed partially in Chapter 2. Re- Figure 40: The symmetry of 



call that the cube has 13 rotation axes of symmetry: three four-fold 
axes, four three-fold axes, and six two-fold axes. Interestingly, the 
octahedron has the same elements of symmetry as the cube. The octahe¬ 
dron has three four-fold rotation axes (they pass through opposite ver¬ 
tices, like axis AB in Figure 41), four three-fold axes (they pass through 
the centres of opposite faces, like axis CD), and six two-fold axes (they 
pass through centres of opposite, mutually parallel edges, like axis EF). 
Both the cube and the octahedron have nine planes of symmetry (find 
them on your own). Lastly, each three-fold rotation axis in the cube 
and the octahedron is at the same time a six-fold mirror-rotation axis. 

It has already been noted above that cube and octahedron are mu¬ 
tual polyhedra. That is why they have the same elements of symmetry. 
The mutuality of the dodecahedron and the icosahedron also implies 
that the two have the same symmetry. 

The symmetry of the dodecahedron is illustrated in Figure 42. Axis 
AB, which passes through the centres of the opposite faces, is one of 
the six five-fold rotation axes; axis CD, which passes through the op¬ 
posite vertices, is one of the ten three-fold axes; axis EF, which passes 
through the centres of the opposite mutually parallel edges, is one of 
the 15 two-fold rotation axes. The same rotation axes are present in 
the icosahedron-, only in it five-fold axes pass through opposite vertices, 
not through the centres of opposite faces, whereas three-fold axes pass 
through the face centres. 

15 The Uses of the Platonic Solids to Explain Some Funda¬ 
mental Problems 

The notion of symmetry has often been used as the framework of 
thought for hypotheses and theories of scholars of centuries past, who 
put much stock in the mathematical harmony of the creation of the 
world and regarded that harmony as a sign of divine will. They viewed 
the Platonic solids as a fact of fundamental importance, directly related 
to the structure of matter and the Universe. 

So the Pythagoreans, and later Plato, believed that matter consists 
of four principal elements - fire, earth, air, and water. According to 

Figure 41: The symmetry of 

Figure 42: The symmetry of 


their thinking, the atoms of principal elements must have the shape 
of various Platonic solids: fire atoms must be tetrahedra; earth atoms, 
cubes; air atoms, octahedra; and water atoms, icosahedra. 

The concept of symmetry coded in the five Platonic solids enthralled 
the famous German astronomer Johannes Kepler (1571-1630), who 
undertook to explain why there are just six planets in the Solar system 
(in Kepler’s days, too, only six planets were known), and why the 
radii of their ‘spheres’ (orbits) are in the ratio 8 : 15 : 20 : 30 : 

115 : 195 (according to Kepler’s results). Kepler inscribed a cube into 
Saturn’s sphere. Next into that cube he inscribed another sphere, that 
of Jupiter. Into Jupiter’s sphere he inscribed a tetrahedron, and into 
the tetrahedron the Martian sphere, and into the latter a dodecahedron. 
Finally, he inscribed the Earth’s sphere. Then followed in succession 
an icosahedron, inscribed into the Earth’s sphere, Venus’s sphere, an 
octahedron inscribed into the latter, and lastly Mercury’s sphere. It is 
easily seen that Kepler’s scheme employs all five Platonic solids. Part 
of the scheme is depicted in Figure 43. Kepler calculated the radii of 
the planetary spheres in accordance with his scheme to find that the 
ratio of these radii were in good agreement with the observations. This 
striking coincidence made Kepler believe in his underlying assumption. 
He thought that he had succeeded in explaining the structure of the 
entire Solar system on the basis of a geometrical system using spheres 
and the five Platonic solids, thus directly relating the existence of six 
planets to the existence of the five Platonic solids. Kepler wrote: 

The great joy I experienced from that discovery cannot be put 
into words. I did not regret any of the time spent and felt no fa¬ 
tigue. I was not afraid of cumbersome calculations while seeking 
to find whether my hypothesis was in agreement with Coper¬ 
nicus’s theory of orbits, or whether my joy was to vanish into 

Kepler’s enthusiasm turned out to be premature. The coincidence he 
hit upon was accidental and, as was shown by later observations, quite 
rough. And to top it off, there are actually nine planets, not six, in the 
Solar system. 



Figure 43: Scheme of the 
Solar System based on the 
regular polyhedra devised 
by Kepler. 

16 On the Role of Symmetry in the Cognition of Nature 

The two just-discussed examples of unsuccessful applications of the 
Platonic solids to explain fundamental issues of nature suggest that 
symmetry alone is not sufficient. Yet the uses of symmetry to ponder 
the world around us are significant. 

The Platonic solids simply furnish an example of how symmetry 
may drastically limit the variety of structures possible in nature. We 
will now elaborate this important idea. Suppose that in some distant 
galaxy, live highly developed creatures who, among other things, are 
fond of games. We may be totally ignorant of their tastes, structure of 
their bodies, and their mentality. We know, however, that their dice 
have any of the five shapes - tetrahedron, cube, octahedron, dodeca¬ 
hedron, and icosahedron. Any other shape of a dice is impossible in 
principle, since the requirement that any face must have equal proba¬ 
bility of appearance dictates the use of a regular polyhedron, and there 
are only five regular polyhedra. 


A measure of order introduced by symmetry shows up, above all, in 
limitation of the variety of possible structures and reduction of the 
number o f possible versions. 

Later in the book we will discuss some limitations imposed by 
symmetry on the diversity of structures of molecules and crystals. The 
modern American popularizer of science Martin Gardner writes: 

Perhaps some day physicists will discover some mathematical 
constraints to be met by the number of elementary particles and 
basic laws of nature. 


Symmetry In Nature 

17 From the Concept of Symmetry to the Real Picture of a 
Symmetrical World 

We have seen that the concept of symmetry has often been used by 
scientists through the ages as a guiding star in their speculations. 

Recall the Pythagoreans who concluded that the Earth is spherical 
and moves over a sphere. Pythagoras’s ideas were used by the great 
Polish astronomer Copernicus when he was working on his theory 
of the Solar system. According to Copernicus, celestial bodies are 
spherical because the sphere is ’a perfect, comprehensive shape, having 
no corners, the most capacious’. He wrote: 

We like to look at symmetrical 
things in nature, such as 
perfectly symmetrical spheres 
like planets and the Sun, or 
symmetrical crystals like 
snowflakes, or flowers which 
are nearly symmetrical. - R. 

All bodies tend to assume that form; this can be noticed in water 
droplets and other liquid bodies. 

Here Copernicus meant free falling drops, which are known to take on 
a nearly spherical shape. In actual fact, he anticipated the deep analogy 
between a water drop falling under gravity and the Earth falling (or 
rather orbiting) in the gravitational field of the Sun. 

On the other hand, scholars of earlier times were inclined to ex¬ 
aggerate a bit the role of symmetry in the picture of the world. They 
sometimes forced their admiration for symmetry on nature by artifi¬ 
cially squeezing nature into symmetrical models and schemes. Recall 
Kepler’s scheme based on the five regular polyhedra. 

The modern picture of the world, with its rigorous scientific justifi- 


cation, differs markedly from earlier models. It excludes the existence 
of some ‘centre of the world’ Gust as some magic power of the Platonic 
solids) and treats the Universe in terms of the unity of symmetry and 
asymmetry. Observing the chaotic mass of stars in the skies, we un¬ 
derstand that beyond the seeming chaos are quite symmetrical spiral 
structures of galaxies, and in them symmetrical structures of planetary 
systems. This symmetry is illustrated in Figure 44 showing the Galaxy 
and a magnified and simplified scheme of the Solar system. 

Figure 44: Symmetry of 
the Galaxy and the Solar 

The nine planets move around the Sun in their elliptical orbits 
which are nearly circular. The planes of the orbit (save for Pluto) 
within high accuracy coincide with the Earth’s orbital plane, the so- 
called ecliptic. For example, Mars’s orbit forms an angle of 2° with the 
ecliptic. Also coinciding with the ecliptic are the planes of all the 13 
satellites of the planets, including the Moon. 

Even more than in the picture of the universe, symmetry mani¬ 
fests itself in an infinite variety of structures and phenomena of the 
inorganic ’world and animate nature. 


18 Symmetry in Inanimate Nature. Crystals 

When we look at a heap of stones on a hill, the irregular line of moun¬ 
tains on the horizon, meandering lines of river banks or lakeshores, the 
shapes of clouds, we may think that symmetry in the inorganic world 
is rather rare. At the same time, it is widely believed that symmetry 
and strict order are hostile to living things. It is no wonder that the 
lifeless castle of the Snow Queen in the fairy tale by Hans Christian 
Andersen is often pictured as a highly symmetrical structure shining 
with polished mirror faces of regular shape. Who is right then? Those 
who view inanimate nature as a realm of disorder, or, on the contrary, 
those who see in it the predominance of order and symmetry? 

Strictly speaking, both schools are wrong. To be sure, such natural 
factors as wind, water, and sunlight affect the terrestrial surface in a 
high- ly random and disorderly manner. However, sand dunes, pebbles 
on the seashore, the crater of an extinct volcano are as a rule regular in 
shape. Of course, a heap of stones is rather disorderly, but each stone 
is a huge colony of crystals, which are utterly symmetrical structures 
of atoms and molecules. It is crystals that make the •world of inanimate 
nature so charmingly symmetrical. 

Figure 45: Snowflakes 
have six-fold rotational 

Inhabitants of cold climates admire snowflakes. A snowflake is a tiny 
crystal of frozen water. Snowflakes are of various shapes, but each has 


a six-fold rotational symmetry and also a mirror symmetry (Figure 45). 

All solids consist of crystals. Individual crystals are generally tiny 
(less than a grain of sand), but in some cases they grow to considerable 
sizes, and then they appear before us in all their geometrical beauty. 
Some naturally grown crystals are given in Figure 46. It can be seen 
from the figure that crystals are polyhedra of fairly regular shapes hav¬ 
ing plane faces and straight edges. The figure shows topaz (aluminium 
fluosilicate), beryl (beryllium aluminium silicate), smoky quartz (silicon 



The beryl crystal shown in the figure is heliodor, one of the crys¬ 
talline varieties of that compound. Other varieties are aquamarine 
(blue-green), emerald (green), and vorobyevite (pale red). The colour 
is conditioned by impurities. So, the yellow of heliodor is due to Fe 3 + 
impurities. Quartz comes in a variety of forms. The clearest and trans¬ 
parent variety of quartz is rock crystal , the second clearest is smoky 
quartz, shown in the figure. There are also violet amethyst, red sar¬ 
donyx, black onyx, and grey chalcedony. Quartz is also grindstone, flint, 
and common sand. 

The symmetry of crystals is clearly seen in Figure 47: (a) common 
salt, (b) quartz, and (c) aragonite. The latter is one of the naturally 
occurring varieties of calcite ( CaCO 3 ). 

Figure 48 represents three crystalline forms of diamond : (a) octahe¬ 
dron, (b) rhombic dodecahedron, and (c) hexagonal octahedron. The 
outer symmetry of a crystal comes from its inner symmetry, that is, its 
ordered arrangement of atoms (molecules) in space. In other words, 
the symmetry of a crystal is related to the existence of the space lattice 

Figure 46: Some naturally 
grown crystals, notice the 
plane faces and straight 

Figure 47: The symmetry of 
some common crystals. 


of atoms - the so-called crystalline lattice. 

19 Symmetry in the World of Plants 

In his book The Ambidextrous World, M. Gardner writes: 

On the earth life started out with spherical symmetry, then 
branched off in two major directions: the plant world with 
symmetry similar to that of a cone, and the animal world with 
bilateral symmetry. 1 

The symmetry of a cone, which is characteristic of plants, is seen 
essentially in any tree (Figure 49). The root of a tree absorbs water and 
nutrients from the soil, that is, from below, whereas all the other vital 
functions occur in the canopy, that is, above the ground. Therefore, 
the directions ‘upwards’ and ‘downwards’ for the tree are significantly 
different. At the same time the direction in a plane perpendicular to 
a vertical are essentially indistinguishable: from every quarter the 
tree receives air, light, and water in equal measure. As a result, we 
have a vertical rotation axis (the cone’s axis) and vertical planes of 
symmetry. Note that the vertical orientation of the cone’s axis, which 
characterizes the symmetry of a tree, is determined by the direction 
of gravity 2 . That is why the general orientation of the stem of a tree is 
normally independent of the slope of the ground or the Sun’s altitude 
in a given latitude. 

True, trees can be encountered now and then, such that their stems 
are not just non-vertical, but bent in a snake-like manner, and their 
canopy may be lop-sided. It would seem that any symmetry here is out 
of the question. And still the concept of a cone at all times correctly 

Figure 48: The symmetries 
of crystalline forms of 

1 The term ‘bilateral symme¬ 
try’ is widely used in biology. 

It means mirror symmetry. 

Figure 49: The conical 
symmetry of trees. 

2 Biological experiments on 
board the Soviet orbital station 
Salyut-6 have shown that under 
weightlessness, the spatial 
orientation of sprouts, leaves, 
and root of wheat and peas is 


reflects the nature of the symmetry of the tree, its essence. After all, for 
any tree we can indicate the base and the top, and at the same time for 
the tree the notions of left and right, back and front are invalid. 

Remarkable symmetry is inherent in leaves, branches, flowers, 
and fruit. Figure 50 gives examples of mirror symmetry, which is 
characteristic of leaves, although is also encountered in flowers, which 
are generally described by rotational symmetry. 

Figure 51a depicts a flower of St. John’s wort (Hypericum), which 
has a five-fold rotation axis and no mirror symmetry. In flowers, ro¬ 
tational symmetry is often accompanied by mirror symmetry (Fig¬ 
ure 51 b). An acacia leaf, shown in Figure 52 a, has both mirror and 
translational symmetries. And a hawthorn branch (Figure 52 b) can be 
seen to have a glide axis of symmetry. 

Figure 53 shows a wild flower known as silverweed (Potentilla 
anserina). The flower has a five-fold rotation axis and five planes of 
symmetry. The high degree of orderliness in the arrangement of indi¬ 
vidual leaves on stems imparts to the figure some likeness to borders 
discussed above. 

Figure 50: Symmetry 

Figure 51: Five-fold ro¬ 
tational symmetry of a 
flower of St. John’s wort 
(Hypericum), and mirror 


Figure 52: The symmetry of 

In the rich world of flowers rotation axes of symmetry of various 
orders occur. But the most widespread is the five-fold rotational sym¬ 
metry. This symmetry is to be found in many wild flowers ( bluebell, 
forget-me-not, geranium, stellaria, pink, St. John’s wort, silverweed, etc.), 
in fruit tree flowers (cherry, apple, pear, mandarin, etc.), in flowers 
of fruit and berry plants ( strawberry ; blackberry, raspberry, guelder 
rose, bird-cherry, rowan-tree, hawthorn, dog-rose, etc.), and in some, 
garden flowers ( nasturium, phlox, etc.). It is sometimes argued that 

Figure 53: The symmetry in 
the silverweed wildflower 
Potentilla anserina. 


plants’ ‘love’, which is known to be impossible in principle in periodic 
structures, can be explained as a safeguard of the plant’s individuality. 
Academician N. Belov maintains “that the five-fold axis is a sort of tool 
in the struggle for existence, an insurance against petrification, the first 
stage of which would be ‘catch’ by a lattice.” 

20 Symmetry in the World of Animals 

The five-fold rotational symmetry also occurs in the world of animals. 
Examples are the starfish and the urchin (Figure 54). Unlike the world 
of plants, however, rotational symmetry is rare in the world of animals. 
As a matter of fact, we may find it only in some denizens of the sea. 

Figure 54: The five-fold 

Insects, fishes, birds and other animals generally exhibit a difference rotational symmetry in the 

between forward and backward directions, which is incompatible with Starfish and Sea Urchin, 

rotational symmetry. The Push-Pull invented in a famous Russian 
fairy tale (Figure 55) is a striking animal in that its front and rear 
are absolutely symmetrical. The direction of motion is an essentially 
distinguishable direction, about which no animal is symmetrical. In 
that direction an animal moves for its food and escapes from danger. 

The symmetry of living creatures is also dictated by another direc¬ 
tion - the direction of gravity. Both directions are significant, since 
they define the plane of symmetry of a creature (Figure 56). Bilateral 
(mirror) symmetry is characteristic of nearly all members of the animal 


Figure 55: The Push-Pull, 
a mythical animal whose 

This symmetry is especially apparent in a butterfly (Figure 57). The front and rear are abso- 

symmetry of left and right is present here with nearly mathematical lutely symmetrical, 



Figure 56: The bilateral 
plane of symmetry for 

It can be said that any animal consists of two enantiomorphs - its left different animals 
and right halves. Also enantiomorphs are paired organs, one of which 
is in the right and the other in the left half of the body, such as ears, 
eyes, horns, and so on. 


Figure 57: The bilateral 
symmetry of a butterfly. 

21 Inhabitants of Other Worlds 

Many works of science fiction discuss the possible appearances of 
visitors from other planets. Some authors believe that extraterrestrials 
may differ markedly in their appearance from ‘earthlings’; others, on 
the contrary, believe that intelligent creatures throughout the entire 
Universe must be very much alike. The question concerns us only in 
the context of symmetry. Whatever the extraterrestrial looks like, his 
appearance must exhibit bilateral symmetry, because on any planet a 
living creature must have a distinguishable direction of motion and on 
any planet there is gravity. The extraterrestrial may be like a dragon 
from some fairy tale, but not like a Push-Pull, by no means. He cannot 
be left-eyed or right- eared. He must have an equal number of limbs on 
either side. Symmetry requirements reduce drastically the number of 
possible versions of the extraterrestrial’s appearances. And although 
we cannot say with certainty what that appearance must be, we can 
say what it cannot be. Recall the idea expressed in Chapter 4: symmetry 
limits the diversity of structures possible in nature. 

Order in the World of Atoms 


22 Molecules 

Chapter 5 was concerned with symmetry in nature as seen by the un¬ 
aided eye. Symmetry is also found at the atomic level. It manifests itself 
in microscopic, geometrically ordered atomic structures of molecules 
and crystals. 

Figure 58 presents schematically two simple molecules: (a) carbon 
dioxide (CO 2 ), and (b) steam (H 2 0). Both molecules have a plane of 
symmetry (the vertical line in the figure). The mirror symmetry here 
comes from the fact that paired identical atoms (oxygen atoms in C0 2 
or hydrogen atoms in H 2 0) are bound to the third atom in a similar 
way. Interchanging the paired atoms will change nothing - this will 
only amount to mirror reflections. 

In the methane molecule (CH 4 ), the carbon atom C is bound to the 
four identical hydrogen atoms H. The four C - H bonds being identical 
predetermines the spatial structure of the molecule in the shape of a 
tetrahedron , with hydrogen atoms being at the corners and a carbon 
atom at the centre (Figure 59). The symmetry of the molecule CH 4 is 
essentially the symmetry of the tetrahedron discussed in Chapter 4. 

Its elements are six planes of symmetry, each of which passes through 
the atom C and two atoms H (for example, planes LMQ and LKR in 
Figure 60), four three-fold rotation axes, each of which passes through 
the atom C and one of the atoms H (for example, axis KO in the figure). 
There are three two-fold rotation axes (for example, axes PQ and SR). 

The crystal is characterized by 
its internal structure, arrange¬ 
ment of atoms, and not by its 
outward appearance. These 
atoms combine to produce 
a sort of huge molecules, or 
rather an ordered space lattice. 
- H. Lindner 



Figure 58: The planes 
of symmetry of water 
(H 2 0)and carbon dioxide 
(C0 2 ) molecules. 

/ \ 

/ \ 

/ \ 

Figure 59: The plane of 
symmetry of the methane 
molecule (CH 4 ). 


Notice the difference between the spatial structure of the methane 
molecule given in Figure 59 and the structural formula of the molecule 
normally given in chemistry texts. 


H —C —H 


Now suppose that one of the hydrogen atoms in the molecule (for 
example, the atom at K in the tetrahedron shown in Figure 60) is re¬ 
placed by a radical OH. In that case we obtain the molecule of methyl 
alcohol (CH 3 OH). As compared with the methane molecule, this 
molecule exhibits lower symmetry (even supposing that the molecule 
retains its tetrahedral shape). It is easily seen that out of the six planes 
of symmetry only three remain - those passing through C, OH, H 
(planes LKR, KPN, KMT in Figure 61); out of the four three-fold rota¬ 
tion axes, only one remains (KO in the figure). There are no two-fold 
rotation axes now. 

Notice that the chemical formula of methyl alcohol is now written 
CH3OH, not CH4O. This is no mere chance. The form CH4O would 
mean that all four H atoms in the molecule are physically equivalent, 
which is not the case here: only three H atoms are equivalent, whereas 
the fourth stands alone, as it enters the OH radical. 

23 The Puzzle of the Benzene Ring 

The benzene molecule consists of six carbon and six hydrogen atoms 
(C 6 H 6 ). The carbon atoms are arranged in one plane to form a regular 
hexagon (the so-called benzene ring). It is well known that carbon is 
tetravalent, that is, a carbon atom provides four electrons which can 
realize four covalent bonds with other atoms. One of them is the bond 
between a carbon atom and a hydrogen atom, the other three binding 
a given atom to neighbouring carbon atoms in the benzene ring. The 
structural formula of the benzene molecule is sometimes presented as 
shown in Figure 62, where some pairs of carbon atoms are bound by 
single and others by double bonds. 


Figure 60: The plane of 
symmetry of the methane 
molecule (CH 4 ). 


Figure 61: The plane of 
symmetry of the methane 
molecule (CH 4 ). 


H C H 

c ^ c 

Figure 62: The structural 
symmetry of the benzene 
molecule (CgFIg). 


At first sight everything about the structural formula given in Fig¬ 
ure 62 is OK. But this is only true at first sight. The fact is that the 
presence of different bonds (single and double) must violate the regular 
shape of the benzene ring, since stronger (double) bonds correspond to 
smaller atomic spacings. At the same time, X-ray studies show that all 
the sides of the carbon hexagon in the benzene molecule are equal. The 
experimentally found symmetry of the benzene ring (the symmetry of 
the regular hexagon) suggests that all the C - C bonds in the ring are 

What is the nature of these bonds.? These cannot be single covalent 
bonds, otherwise one bond in each carbon atom would be free. But 
they cannot be double bonds either, since for this to be the case each 
carbon atom lacks one valence. 

The enigma of the benzene ring turned out to be exceedingly inter¬ 
esting. One of the valence electrons of each carbon atom participates 
in the formation of a bond of this atom with five atoms of the ring at 
once, and not with one of the neighbouring atoms. This implies that 
the electron is collectivized not by a pair of atoms (which is common 
for covalent bonds), but by the entire molecule (or rather the entire 
benzene ring). In other words, the benzene molecule has six electrons 
not pinned up by localized bonds between atoms, but capable of freely 
moving around the entire benzene ring. This is generally represented 
by the structural formula shown in Figure 63, where the solid lines 
denote, as usual, the localized bonds (each bond due to the collectiviza¬ 
tion of a pair of electrons by a pair of appropriate atoms), and dash 
lines denote non-localized bonds due to the collectivization of six elec¬ 
trons by the benzene ring. 

24 The Crystal Lattice 

It would seem that diamond and graphite have nothing in common. 
The diamond is unusually hard, transparent, and is a dielectric. Pro¬ 
cessed stones are used in jewelry. Graphite, on the other hand, is soft, 
laminar, opaque, electro-conductive. In a word, a far cry from a gem. 
At the same time, however, both diamond and graphite are carbon in 
its pure form. The different behaviour of diamond and graphite is only 



Figure 63: The schematic 
representation of the collec¬ 
tivization of six electrons by 
the benzene ring (Cgth). 


explained by their different crystalline structure, or different crystalline 
lattices. This is a graphic example of the important role played by the 
crystalline lattice in determining the properties of a solid. 

The crystal lattice is a natural three-dimensional pattern. As with 
plane patterns, it is dominated by some form of translational symme¬ 
try. It has been noted in Chapter 2 that there exist 14 types of spatial 
lattices that differ in their translational symmetry (14 types of Bravais 
lattices). They form seven crystallographic systems: 

§ cubic system: a = b = c, a = f) = y = 90°; 

§ tetragonal system: a = b + c, a = f = y = 90°; 

§ hexagonal system: a = b + c, a = fl = 90°, y = 120°; 

§ trigonal system: a = b = c, a = f = y + 90°; 

§ rhombic system: a + b ± c, a = f = y = 90°; 

§ monocline system: a + b + c, y + a = fl = 90°; 

§ tricline system: a + b + c, a [1 + y\ 

Here a, b , and c are lengths of the edges of a unit cell; and (a, f) and y 
are angles between the edges (see the margin figure). 

25 The Face-Centred Cubic Lattice 

Suppose that we have many balls of the same diameter. We will pack 
them densely on a plane (the brown balls in Figure 65). Over the first 
layer a second one will be placed (the red balls). It is easily seen that 
the second layer is packed as densely as the first one. Next we will lay 
a third layer. Here we have two versions: (a) the centres of the balls 
in the third layer come exactly over the centres of the balls in the first 
layer; (b) the centres of the balls of the third layer are displaced hori¬ 
zontally relative to the balls in the first layer. Let us take the second 
version (the blue balls). The resultant dense multilayer packing cor¬ 
responds to the face-centred cubic lattice (f.c.c.). In other words, the 
centres of the balls here (second version) form an f.c. c. lattice. 


Vectors defining a three- 
dimensional unit lattice. 


Figure 65: Arranging balls 
to get a dense packing. 

Figure 66 shows the cubic unit cell of the f.c.c. lattice. The sites here 
are the vertices of the cube and the centres of all its faces. We can 
readily discern in the figure the planes of the angles corresponding to 
the ball layers in Figure 65. Let plane DEF correspond to the first layer 
(the brown balls). Then plane ABC will correspond to the second layer 
(the red balls). Site K will now belong to the third layer. 

Each cell thus includes four sites (for example, sites D, P, M, and 
S in Figure 66); the remaining sites in the figure must be assigned to 
neighbouring cells, and so the cell in question is a four-site cell. 

The f.c.c. cell, just like any Bravais lattice, can also be defined 
using a one-site unit cell. Shown in Figure 67 (in red) is a one-cell 
rhombohedron-shaped cell. Crystallographers prefer using not one- 
but four-site cells, since it reflects in the most complete manner the 
elements of symmetry possessed by the f.c.c. cell. 

Figure 66: Cubic unit of the 
f.c.c. lattice. 

Figure 67: Face centred cu¬ 
bic cell showing a one-cell 
rhombohedron-shaped cell. 


Face-centred cubic lattices occur fairly often. This sort of lattice is 
to be found, for example, in aluminium, gold, copper, nickel, platinum, 
silver, and lead. The lattice of common salt (NaCl) actually consists of 
two geometrically identical interlocked f.c.c. lattices, one made up of 
Na + ions and the other of CI“ ions (Figure 68). 

26 Polymorphism 

Figure 68: Face centred 
cubic cell in common salt 

It has earlier been noted that the difference in the properties of di¬ 
amond and graphite is determined by the difference in the crystal 
lattices of these two forms of carbon. As is seen in Figure 69, the dia¬ 
mond lattice is formed by two identical interlocked f.c.c. lattices, one of 
which is displaced relative to the other by a quarter of the edge of the 
f.c.c. cell along all three coordinate axes (the white circles in the figure 
show sites of one of the f.c.c. lattices, and the filled circle shows the 
site of the other f.c.c. cell). Each carbon atom in the diamond lattice is 
the centre of a tetrahedron, whose vertices are the four nearest neigh¬ 
bours of a given atom; this is seen especially clearly in the cell shown 
in red in the figure. 

Figure 69: Structure of the 
diamond lattice. 


Note that the crystal lattices of germanium, silicon and grey tin have 
the diamond lattices. 

The graphite lattice is given in Figure 70. It is distinctly laminar in 
structure, with each layer being characterized by six-fold rotational 
symmetry. The bonds between the atoms from different layers are 
much weaker than within the same layer. 

Diamond and graphite are good examples of two different crys¬ 
talline modifications of a chemical element (or compound). This phe¬ 
nomenon is known as polymorphism. Under certain conditions a sub¬ 
stance may change from one crystal modification to the other, and the 
changes are called polymorphic transformations. If, for instance, we 
heat graphite up to 2000 K to 2500 K under a pressure of up to 10 10 Pa, 
the crystal lattice will transform with the result that graphite will turn 
into diamond. In this way artificial diamonds are produced. 

27 The Crystal Lattice and the External Appearance of a Crys 

The symmetry of the external shape of a crystal is conditioned by 
the symmetry of the crystal lattice. Ideally plane crystal faces are the 
planes that pass through the sites of the lattice. True, through lattice 
sites one can draw many different sets of parallel planes (Figure 71). 

These sets differ in their orientation in space, inter-planar spacing 
and density of packing of sites in the plane. Of especial interest are 
the most dense planes (in the figure they are shown in red). It is along 
these planes that a single crystal specimen normally fractures, and it 
is to these planes that the faces of a grown single crystal correspond. 

In general the faces of a unit cell are not parallel to those planes. One 
should not therefore expect that the shape of a single crystal specimen 
will coincide with the shape of a unit cell (compare Figure 68 and 48 
showing diamond). 

Figure 70: Structure of the 
graphite lattice. 


Figure 71: Parallel planes in 
crystal lattice. 

28 The Experimental Study of Crystal Structures 

Crystal structure cannot be seen even through the most powerful mi¬ 
croscope now available. The atomic structure of a crystal is identified 
using the diffraction of X-rays. For the latter a crystal is a diffraction 
grating produced by nature. 

Let us consider the simplest X-ray technique. A single crystal sam¬ 
ple is oriented in a particular way relative to an X-ray beam. On re¬ 
flecting from various sets of the parallel planes passing through the 
sites of the lattice, X-rays produce (on a photographic film) a picture 
characteristic for the given orientation of the single crystal, the so- 
called Laue pattern of the single crystal (from the name of the German 
physicist Laue). Each spot in the pattern corresponds to one of the 
reflected X-ray beams. Reflections will only be observed in directions 
that meet the known diffraction condition: 

2d sin 6 = n A 

where d is the separation between neighbouring parallel reflecting 
planes, 9 is the angle between the direction of the reflected X-ray 
beam and the reflecting plane (equal to a half of the angle between the 


directions of the reflected and initial beams), A is the wavelength of the 
X-rays, and n = 1, 2,... 

Figure 72 is a schematic representation of the reflection of X-rays 
from three sets of parallel planes passing through sites, supposing of 
course that the above condition is met. 

Figure 73 is an example of the Laue pattern of zinc blende (ZnS) 
single crystal for two orientations of the sample relative to the initial 
beam. One of the pictures shows a four-fold rotational symmetry and 
the other a three-fold one. The arrangement of spots on the pictures is 
a tale-tell indication of the elements of symmetry of the lattice studied. 

In addition to the diffraction of X-rays, some crystal studies are 
made using the diffraction of electrons and very slow neutrons. 

29 The Mysteries of Water 

It is common knowledge that heating reduces the density of liquids 
and subjecting them, more viscous. Water behaves differently, how¬ 
ever. Heating from 0 °C to 4 °C increases the density of water, and 
pressurizing it reduces its viscosity. 

The mystery was unravelled after the atomic structure of water 
was investigated. It turned out that water molecules interact with one 
another in a directed way (just like the carbon and hydrogen atoms 

Figure 72: Structure of the 
diamond lattice. 

Figure 73: Laue pattern 
of zinc blende (ZnS) single 
crystal for two orientations 
of the sample relative to the 
initial beam. 


in the methane molecule). Each water molecule may thus form bonds 
with four neighbouring molecules so that their centres will form a 
tetrahedron. This is shown schematically in Figure 74 where the balls 
stand for water molecules. 

This arrangement corresponds to the fairly loose, skeleton-like 
molecular structure, where each molecule has only four nearest neigh¬ 
bours. By way of comparison, for the dense packing of balls the num¬ 
ber of the nearest neighbours is twelve. 

Note that, unlike crystals, the molecular structure of water should 
be viewed as a manifestation of short-range order. Near each molecule 
the neighbouring molecules are arranged in an orderly manner, the 
order gradually diminishing with the distance from a given molecule. 

Figure 74: Water molecules 
form tetrahedral structures 
for interaction. 

The skeleton-like molecular structure of water provides a good ex¬ 
planation for its physical properties. Water increases in density when 
heated from 0 °C to 4 °C because the heating disturbs the molecular 
bonds causing the arrangement of molecules to pack more densely. Fur¬ 
ther heating reduces the density because a conflicting effect sets in: the 
thermal expansion of spacing between oxygen and hydrogen atoms in 
the water molecule. This accounts for the well-known fact that water 
has the highest density at 4 °C. 

The skeleton-like molecular structure of water (near 0 °C) also 
explains another property of water - the drop in its viscosity with 
increasing external pressure. Pressurizing, just like heating, disrupts 
molecular bonds and thus reduces viscosity. 

30 Magnetic Structures 

The orbiting of electrons in the field of the atomic nucleus may pro¬ 
duce an atomic magnetic field. Magnetic materials may have their 
atomic magnetic fields ordered. So, in a ferromagnetic magnetized to 
saturation, the magnetic fields of all atoms are oriented in one direc¬ 
tion (that of the magnetizing field), whereby the magnetic properties of 
the substance become especially apparent. 

Also of interest is the magnetic order in a special type of magnetic 


material-the so-called anti-ferromagnetics, which are currently widely 
used in logical elements and memories of modern computers. The 
direction of the atomic magnetic field in those materials alternates in 
a regular fashion from one atom to the next, with the result that apart 
from a crystal lattice a magnetic lattice is present as well. For simplicity, 
Figure 75 a shows a plane square lattice, the dash lines delineating a 
unit cell. Figure 75 b shows the same lattice for the antiferromagnetic. 
The directions of the atomic magnetic field are shown by arrows at 
sites, the dash lines delineating a magnetic unit cell. It is easily seen 
that the linear size of the unit cell is twice the size of the crystal cell. 

For a real (three-dimensional) example of an anti-ferromagnetic, let 
us take manganese oxide (MnO), its crystal lattice is given in Figure 76. 
It consists of two identical interlocked f.c.c. lattices, one of which con¬ 
tains manganese ions Mn 2+ and the other oxygen ions O 2- . Oxygen 
ions have no magnetic field. The magnetic fields of manganese ions 
in the planes are represented in the figure by the same colour (for ex¬ 
ample, red), they are oriented in the same way, whereas the magnetic 
fields of manganese ions belonging to the planes of different colour are 
oriented in opposite directions (the directions of magnetic fields are 
perpendicular to the planes chosen). 

- 1 



t H-i 

I t 11 
t 1-M 

II l I 


Figure 75: Magnetic and 
anti-ferromagnetic lattices. 

Figure 76: Structure of 
an anti-ferromagnet man¬ 
ganese oxide (MnO) crystal 


31 Order and Disorder 

In the surrounding world order and disorder are inseparable. Whatever 
the degree of order of a given atomic structure, it still has some ele¬ 
ments of disorder. Specifically, this is true of the atomic structure of 

It is to be noted first of all that atoms are by no means fixed at lat¬ 
tice sites, but are instead involved in thermal vibrations near those sites, 
the amplitude being the larger the higher the temperature. Thermal 
vibrations make individual atoms leave their sites and wander (dif¬ 
fuse) about the crystal. In the lattice some unoccupied sites (so-called 
vacancies) appear. Both vacancies and atoms at gaps between sites 
(interstices) will obviously distort the geometry of the lattice by in¬ 
fluencing the arrangement of neighbouring atoms (Figure 77). What 
is more, any real crystal may have some foreign atoms, the so-called 
impurities. These atoms may be at interstices, but may also substitute 
for lattice atoms by “driving them away” from their places. 

Substantial disorder in the lattice is caused by so-called dislocations 

Figure 77: Vacancies and 
interstices in a crystal 


- violations of the regular arrangement of atomic planes. Some idea 
of dislocations may be obtained in Figure 78, which represents the 
so-called edge dislocation. 

The presence of defects in a crystal lattice is the decisive factor 
influencing the strength and plasticity of a material. At present, special 
techniques enable filament crystals to be grown, such that their lattice 
is essentially defect-free. The strength of such crystals may be as high 
as 10 10 Pa, hundreds of times larger than for conventional crystals. 
According to modem thinking, the plasticity of a material is controlled 
by the migration of defects, above all dislocations, over the sample. 
Interestingly, if the density of the defects grows, they eventually begin 
to hinder the migration with the result that the plasticity of a material, 
drops. This is what occurs in certain kinds of processing (forging, 
annealing etc.) 

Consequently, a material can be strengthened by two opposite ways- 
either by preventing defect formation, or by hindering the migration of 
defects about the sample (that is, by increasing the density of defects). 


The first technique means growing defect-free crystals, and the second 
one a special processing of materials. 

7 Spirality In Nature 

Mirror asymmetry (also called left-right asymmetry) widely occurs in 
nature and is of principal importance for living things. A characteristic 
example of a mirror-asymmetric object is a helix or a spiral This ex¬ 
plains why left-right asymmetry is often referred to as spirality. Also 
used is the term handedness. Recall that the hand is an example of a 
mirror- asymmetric object. 

Pasteur was more right than 
many of his colleagues sus¬ 
pected when he wrote elo¬ 
quently of left-right asymmetry 
as a key to the mystery of 
life. At the heart of all living 
cells on earth are right-handed 
coils of nucleic acid. This 
asymmetric structure is surely 
the master key of life. - M. 

32 The Symmetry and Asymmetry of the Helix 

The figure in Figure 79 appears to be unmoved if turned about axis 
OO through 60° and translated along the same axis through distance 
a. The axis is a six-fold helical axis with translation period a. Helical 
symmetry is a symmetry relative to a combination of a turn and a 
translation along the rotation axis. An example of an object featuring 
helical symmetry is the spiral staircase. 

For an object having helical symmetry we can draw a helical line, or 
spiral (the dash line in Figure 80). The helical line can be constructed 
as follows. Cut a right-angled triangle out of paper (ABC in Figure 79). 
Take a circular cylinder and glue to its surface the leg BC of triangle 
ABC so that the leg coincides with the generatrix of the cylinder sur¬ 
face. Next wrap the triangle around the cylinder tightly pressing the 
paper to the cylinder surface, the hypotenuse AB will give the helical 
line. Two methods of turning the triangle around the cylinder are pos¬ 
sible, both of which are shown in Figure 79. One of them produces a 
left-handed and the other a right-handed spiral. 


The type of a helical line is identified rather simply. Let us mentally 
move along a helical line, the motion will have two components - along 
the helical axis and around the axis (the straight and circular arrows 
in Figure 79). Let us place the observer so that the structure moved 
along the helical axis away from him. If in the process the circular 
motion is clockwise, then the helical line is termed right-handed, and 
if counter-clockwise, it is called left-handed. In other words, if a point 
moving away from the observer in a helical line appears to him to 
turn clockwise, the spiral is a right-handed one, if counterclockwise, a 
left-handed one. When reflected in a mirror, a left-handed spiral will 
become a right- handed one, and vice versa. The left- and right-handed 
spirals form a pair of enantiomorphs. 

Note that speaking about the helical line or the screw, the term 
“spiral” is often used. It will be remembered that we have here a spatial 

Figure 79: Construction of a 
helical line. 

Figure 80: A helical line. 


33 Helices in Nature 

We use helices widely in technology. It is perhaps of interest that he¬ 
lices are also encountered in nature. Some examples of natural helices 
are shown in Figure 81: (a) the tusk of the narwhal, a small cetacean 
animal endemic of northern seas, is a left-handed helix, (b) the shell 
of a snail Note: Each type of shell has a certain spirality. Any “freaks”, 
•which are encountered from time to time, having the opposite spirality, 
are especially valued, (c) the umbilical cord of a new-born is a triple 
left-handed spiral formed by two veins and one artery, (d) (horns of the 
Pamir sheep are enantiomorphs (one horn is left-handed and the other 
right-handed spiral). 

Figure 81: Some natural 

In plants we may find numerous signs of helical symmetry in the ar¬ 
rangement of leaves on the stalk, branches of the stem, the structure of 
cones, some flowers, and so on. Creeping plants are remarkable helices. 
Examples are known of entangled creeping plants with different hand- 


edness, for exampl ebindweed and honeysuckle. This gives rise to bizarre 
arrangements, which repeatedly attracted poets. Crystal lattices, as a 
rule, have mirror symmetry. But there exist also mirror-asymmetric 
lattices, some of them are characterized by a helical structure. An ex¬ 
ample of a twisted crystal lattice is quartz. Its base is a tetrahedron with 
a silicon atom at the centre and oxygen atoms at its vertices. Along 
the main crystal axis the tetrahedra lie along a helical line. The quartz 
lattice may be twisted either to the left or to the right. Therefore, there 
exist two enantiomorphic forms of quartz. The left and right single 
crystals of quartz are represented in Figure 82. It is easily seen that one 
is a mirror image of the other. 

Figure 82: The handedness 
in crystals. 

Another realm of natural screws is the world of “living molecules”, 
those molecules that play an important role in biological processes. 

These molecules include, above all, protein molecules, the most 
complex and numerous of all the carbon compounds. In man there are 
up to 10 5 types of proteins. All the human organs, including bones, 
blood, muscles, sinews, and hair, contain proteins. The many ferments 
and hormones are proteins as well. The protein molecule contains 
atoms of carbon, hydrogen, oxygen, and nitrogen. Each molecule 
has an enormous number of atoms, of the order of 10 3 to 10 6 . Each 
giant molecule contains many links (amino acids) connected into a 
chain. The frame of the chains is twisted as a right- handed spiral. 

In chemistry it is called Pauling’s alpha-spiral (after the prominent 
American chemist Linus Pauling). The molecules of sinew fibres are 
triple alpha-spirals. Alpha-spirals twisted repeatedly with each other 
form molecular helices such as those found in hair fibres, in horny 
substance, and so on. 


Especially important in living nature are the molecules of deoxyri¬ 
bonucleic acid (DNA), which are bearers of hereditary information 
in living organisms. The DNA molecule has the structure of a double 
right-handed spiral which, in a sense, is the main natural helix. Let us 
take a closer look at the structure of that molecule. 

34 The DNA Molecule 

A schematic diagram of the DNA molecule is given in Figure 83. The 
molecule consists of many links called nucleotides, the links being con¬ 
nected into two chains (in the figure the nucleotides are shown by red 
rectangles). Each nucleotide contains a sugar molecule, a phosphoric 
acid molecule ( phosphate ), and a molecule of a nitrogen-containing 
compound ( nitrous compound). The nitrogenous bases of two nu¬ 
cleotide strands are linked by hydrogen bonds shown by dash lines in 
Figure 83. The structure in the figure looks like a ladder, in which the 
vertical elements are sugar-phosphate chains, and the rungs are pairs of 
nitrogenous bases. There are four nitrogenous bases: adenine (A) and 

Figure 83: A schematic dia¬ 
gram of the DNA molecule. 


H-C-H O- Jh 










Fig. 83 




Figure 84: The structural 
formulae for the ATGC 
nitrogenous bases in DNA 

guanine (G) (purines), thymine (T) and cytosine (C) (pyrimidines). Each 


rung contains either adenine and thymine (A-Tor T-A), or guanine 
and cytosine (G-C or C-G ). Figure 84 represents the structural formu¬ 
lae of the adenine-thymine pair and the guanine-cytosine pair, which 
enter the DNA molecule. There are no combinations of adenine with 
guanine or thymine with cytosine. 

Considering the above, the “ladder” schematically representing the 
structure of DNA assumes the form shown in Figure 85. The arrange¬ 
ment of the pairs AT, TA, GC, and CG along the “ladder” is the genetic 
code of a living organism. Despite the fact that there exist only four 
types of rungs, the enormous number of these rungs on the ladder 
enables DNA to include all hereditary information. 

This information is retained when cells are reproduced. The DNA 
molecule divides into two halves (along the red line in Figure 85), each 
of which is essentially a sugar-phosphate chain with nitrogenous bases 
arranged normally to the chain. Since each base may only bond to a 
specific base (A to T, T to A, G to C, C to G), then each of the halves 
will build up to form a molecule that is a complete replica of the initial 

In passing from the schematic representation of the DNA molecule 
to its real spatial structure, it is necessary to take into account the fact 
that each sugar-phosphate chain represents a right-handed spiral, so 
that on the whole the DNA molecule appears as a double right-handed 
spiral and looks like a spiral staircase, rather than a ladder (Figure 86). 

It is worth noting that all the spirals in the DNA molecules in man 
are right-handed. Among the prodigious wealth of them not a single 
left- handed one is to be found. Note that the DNA strands in one cell 
may be up to a metre in length. All in all, there are 10 11 km of DNA in 

The structure of DNA was discovered in 1953 by a team of scien¬ 
tists that included the American Watson and the Englishmen Crick 
and Wilkins. This discovery is justifiably regarded as one of the most 
remarkable contributions to biology in the 20th century. 

















































Figure 85: A schematic 
diagram of the structure of 
the DNA molecule showing 
the “ladder”. 

Figure 86: A schematic 
diagram of the structure of 
the DNA molecule showing 
the double right-handed 


35 The Rotation of the Plane of Light Polarization 

Some media possess a fascinating property: when a plane-polarized 
light beam is passed through them, the light polarization plane turns 
by a certain angle. Such media are termed optically active. The media 
may be levorotary or dextrorotary. Suppose that the beam strikes our 
eye. If the polarization planes turn clockwise, the medium is called 
dextrorotary (Figure 87 a); if counterclockwise, levorotary (Figure 87 b). 1 

^Note that the “left-handed” 
(“right-handed”) combination 
of the straight and circular 
arrows in Figure 87 differs from 
the appropriate combination in 
Figure 79. This implies that in 
a levorotary medium the polar¬ 
ization plane turns following in 
fact a right-handed helix, and in 
a dextrorotary medium it turns 
following a left-handed helix. 

Figure 87: The action of 
levorotary and dextrorotary 
media on plane-polarized 

We are not going to consider the nature of the phenomenon of po¬ 
larization plane rotation. We will only note that an optically active 
medium must possess a left-right asymmetry, which dictates the rota¬ 
tion of the polarization plane in one direction or the other. 

An example of optically active media is the quartz crystal. 

The direction of the turn of the polarization plane depends on to 
which enantiomorphic variety a given crystal belongs. Dextrorotary 
crystals are generally called right-handed, and levorotary crystals 
are called left- handed ones. The optical activity of quartz is associ¬ 
ated with the left- right asymmetry of the lattice. If a quartz crystal is 
dissolved in a liquid, no rotation of the plane of light polarization is 

It would seem that the presence of a left-right asymmetry of crystal 
structure is a necessary condition for a polarization plane to turn. You 
can imagine the surprise of the prominent 19th century physicist Jean- 


Baptiste Biot when he discovered optical activity in aqueous solutions 
of some organic compounds, for instance, sugar and tartaric acid. It 
thus followed that the left-right asymmetry could be associated not 
only with the structure of a medium as a whole but also with the 
structure of molecules in the medium. This gave rise to the terms “left” 
(levorotary) and “right” (dextrorotary) molecules. 

36 Left and Right Molecules. Stereoisomerism 

In Chapter 6 we discussed the molecules of methane and methyl alco¬ 
hol (see Figure 59). Both molecules are identical to their mirror images. 
This is only natural since they feature mirror symmetry (six planes of 
symmetry in the methane molecule and three planes of symmetry in 
the methyl alcohol molecule). 

Let us now substitute radical CH 3 for one of the three identical 
hydrogen atoms in the molecule of methyl alcohol, that is, pass from 
the structural formula 

H H 

I I 

H —C —OH CH 3 -C —OH 

I I 

H H 

This is the formula of ethyl alcohol. The tetrahedral model of this 
molecule is provided in Figure 88 a, and the model of its mirror im¬ 
age in Figure 88 b. It appears that in this case too the mirror image is 
identical to its original molecule: to obtain a mirror image we will have 
to turn the molecule of ethyl alcohol by 180° about axis AB (see the 
figure). In other words, the molecule of ethyl alcohol Gust like those 
of methane and methyl alcohol) has no enantiomorphic versions. This 
is because despite the gradual reduction of the symmetry of molecu¬ 
lar tetrahedron (in passing from methane through methyl alcohol to 
ethyl alcohol), it still remains mirror-symmetrical. It is easily seen that 
the molecule of ethyl alcohol has a plane of symmetry (plane ABD in 
Figure 88 a. 

A different situation emerges if we take, for example, the molecule 
of butyl alcohol 



Figure 88: Ethyl alcohol. 




CH 3 -C —OH 


C 2 H 5 

The spatial tetrahedral model of the molecule is presented in Fig¬ 
ure 89 a, and its mirror image in Figure 89 b. The molecule has no 
plane of symmetry, it is mirror-asymmetrical. Therefore, both it and 
its mirror image are enantiomorphs, and no turns can effect a coinci¬ 
dence of the molecules shown in Figure 89. One of the molecules can 
be termed “left”, and the other “right”. 

Thus, if the spatial structure of a molecule excludes planes of sym¬ 
metry, then it can have two forms, which are enantiomorphs. These 
forms are called stereoisomers. 

Stereoisomerism is a manifestation of left-right asymmetry in the 
world of molecules. Stereoisomers are molecules that in addition to the 
same chemical composition have the same geometrical shape, the same 
struc- tural elements, and the same inner bonds. At the same time they 
are different molecules. As different as, say, left and right shoes. The 
existence in nature of left- and right-handed molecules was suggested 
by observations of the rotation of polarization planes. 

A special case among stereoisomers are, obviously, molecules with 
spiral structures but differing in handedness. 

Figure 89: Butyl alcohol. 

37 The Left-Right Asymmetry of Molecules and Life 

Studies of the optical activity of solutions of organic compounds, ini¬ 
tiated by Biot, were carried on by the famous French scientist Louis 
Pasteur. He came to the conclusion that if in inorganic nature left- and 
right-handed molecules occur with equal frequency, in living organ¬ 
isms mirror-asymmetrical molecules occur only in one enantiomorphic 
form. Pasteur speculated that it is here that the boundary between 
organic and inorganic nature lies. Further scientific evidence supported 
Pasteur’s conjectures. 

Recall that the alpha-spiral, which determines the structure of pro¬ 
tein molecules, is invariably a right-handed spiral. The double spiral of 


the DNA molecule is also right-handed. Various mirror-asymmetrical 
molecules that enter the compositions of cells are, as a rule, repre¬ 
sented either only by left-handed or only right-handed stereoisomers. 
So, amino acid molecules in proteins are always left-handed. 

This all goes to prove that on the molecular level, a living organism 
is characterized by distinct left-right asymmetry. An organism is con¬ 
structed out of “helices”, so to speak, some being only right-handed 
and others only left-handed. 

One manifestation of that highly interesting thing is that left- and 
right-handed stereoisomers of a substance exert different actions of 
the human body. Man consumes in food those stereoisomers that corre¬ 
spond to the nature of his own asymmetry. In a number of cases modern 
chemistry is able to obtain mirror-reflected stereoisomers. And then 
we can observe unexpected reactions of the human organism to them. 
So, the “reflected” stereoisomer of vitamin C exerts essentially no ef¬ 
fect on the body. The “reflected” form of nicotine (never to be found 
in tobacco) is much less offensive. Right-handed aspartic acid is sweet, 
left-handed is tasteless. Minor additions of right-handed phenylalanine 
to food have no unpleasant implications, whereas additions of the left- 
handed form produce drastic metabolic disturbances (phenylketonuria) 
accompanied by mental disorders. 

Suppose now that after a long space journey a man stepped onto an 
unknown planet. Suppose further that the planet is very much like the 
Earth (in its atmospheric composition, climate, landscape, plants, etc.). 
And so, the astronaut holds in his hand a fragrant apple just picked 
from an extraterrestrial appletree. But should he eat the apple? The 
enantiomorphic form of organic compounds in it is unknown. It is 
quite possible that an innocent-looking apple may appear to be biolog¬ 
ically poisonous for an Earthling. In other words, mirror-asymmetrical 
molecules of a foreign plant world may turn out to be incompatible 
with the mirror-asymmetrical human organism, just as a left-threaded 
nut is incompatible with a right-threaded bolt. 

The beautiful children’s book Through the Looking Glass by Lewis 
Carroll contains a scene that today has deep scientific meaning. About 
to pass through the looking glass into the looking-glass house, Alice 
asks her kitten: 



How would you like to live in Looking-Glass House, Kitty? I wonder if they’d 
give you milk in there? Perhaps looking-glass milk isn’t good to drink? 

As a matter of fact, milk includes many mirror-asymmetrical com¬ 
pounds, such as fats, lactose (milk sugar), and proteins. On passing 
over from the conventional world to the looking- glass world, all the 
asymmetrical molecules should have turned from some stereo isomers 
to other ones. Therefore, it is highly unlikely that the looking-glass 
milk would have been wholesome for the kitten. Although, if we were 
consistently to follow the situation described by Carroll in his book, we 
are to assume that in the looking-glass world both Alice and the kitten 
themselves would turn into their respective mirror images. And in that 
case the looking-glass milk would, of course, be for them as palatable 
and useful as the conventional, “unreflected” milk had been before. 

We will conclude the chapter with the words of M. Gardner from his 
book The Ambidextrous World: 

One of the most remarkable and least mentioned characteristics of life as we 
know it is the ability of an organism to take compounds from its immediate 
environment, many of which are symmetrical in their molecular structure, and 
to manufacture asymmetrical carbon compounds that are right- or left-handed. 

Plants take symmetrical inorganic compounds such as water and carbon dioxide 
and from them manufacture asymmetrical starches and sugars. The bodies 
of all living things are with asymmetrical carbon molecules, as well as the 
asymmetrical helices of proteins and nucleic acids. 

Part II 

Symmetry Around Us 

There is a steady system in all, 

Full consonance in nature; 

It is only in our phantom freedom 
That we sense discord with her. 

]5pt] F. Tyutchev 
(Translated by Jesse Zeldin) 

To see a World in a Grain of Sand 
And a Heaven in a Wild Flower 
Hold Infinity in the palm of your hand 
And Eternity in an hour. 

- William Blake 

8 Symmetry and The Relativ¬ 
ity of Motion 

The concept of symmetry is not confined to the symmetry of objects. It 
also covers physical events and laws that govern them. The symmetry 
of physical laws resides in their unchangeability, or rather invariance, 
under one or another of transformations related, for example, to the 
conditions under which the phenomenon is observed. Scientists got in¬ 
terested in the issue of symmetry in the laws of physics in connection 
with the studies that had led to the development of special theory of 
relativity. The symmetry here is the symmetry (invariance) of the laws 
of physics under transition from one inertial reference frame to another 
inertial reference frame, or the symmetry in relation to uniform motion. 

The laws governing natural 
events are independent of the 
state of motion of the reference 
frame in which these events 
occur if the frame travels 
without acceleration. - Albert 

38 The Relativity Principle 

Suppose that a railway carriage moves smoothly and uniformly. You 
travel in the carriage and want to find out whether it is moving or it 
is at rest. Could this be done without looking out of the window? The 
reader must know the answer. In that case it is in principle impossible 
to find this out because all the physical processes occur in the same 
manner in a carriage at rest and in a carriage in uniform motion. 

This is known as the principle of relativity for inertial reference 
frames. Recall that a reference frame is said to be inertial if a body in 
it moves uniformly when not exposed to the action of external forces. 
Any two inertial reference frames are in uniform motion relative to 


each other. In the above example of the train carriage one inertial 
frame is related to bodies at rest on the ground, the other is related to 
the carriage, which moves uniformly and rectilinearly. We can here 
ignore the rotation of the Earth and its revolution about the Sun. The 
principle of relativity can be formulated as follows. Any process in 
nature occurs in the same manner in any inertial reference frame; in all 
inertial frames a law has the same form. 

As applied to mechanical phenomena the relativity principle was 
established by Galileo. The principle was generalized to apply to all 
processes in nature, including electromagnetic ones, by the foremost 
physicist of the 20th century Albert Einstein (1879-1955), the father 
of relativity theory. Drawing on the very essence of the relativity 
principle, Einstein postulated that the velocity of light in a vacuum must 
be the same in all inertial reference frames. Einstein’s discovery was a 
great break- through in science, since it subjected all the age-old views 
of space and time to an overhaul. 

39 The Relativity of Simultaneous Events 

It follows from the invariance of the speed of light with respect to 
translations from one inertial reference frame to another that two 
spatially separated events that are simultaneous in one frame may be 
non-simultaneous in the other. Let us take a simple example. 

Figure 90: Two inertial 
frames of reference with 
relative velocity v. 

Consider two inertial frames of reference xyz and x'y'z'. Let frame 
x'y'z' be travelling relative to frame xyz along x- and x'-axes with a 
speed v (Figure 90). In frame x'y'z' there are a light source A and two 



detectors, B and C, that lie at equal distances from A along the x'-axis. 
Light source A sends out two light impulses simultaneously, one in 
the direction of B, the other in the direction of C. Since AB = AC and 
both signals travel with the same speed, the observer in x'y'z' will see 
detectors B and C operate simultaneously. 

Let’s turn now to the observer in xyz. In this frame of reference the 
light signal that travels to the left will have to cover a smaller distance 
from production to registration than the signal that travels to the right. 
The speed of light in xyz and x'y'z' being the same, for the observer in 
frame xyz detector B will operate earlier than C. 

40 The Lorentz Transformations 

We will stay with our inertial frames of reference xyz and x'y'z' 
shown in (Figure 90. Let an event occur at a time t at a point (x, y, z) 
in the frame xyz. In the frame x'y'z' the same event occurs at the time 
t' at the point (x\ y', z'). The space and time coordinates of the event 
in the frames xyz and x'y'z' are related by 

r - 7 it 

where c is the speed of light in empty space. These relationships are 
called the Lorentz transformations (from the Dutch physicist Hendrik 
Lorentz who first derived them). 

The symmetry of physical laws with respect to changes from one 
inertial reference frame to another are mathematically expressed in 
that the relevant mathematical expressions must conserve their form if 
x, y, z, tin them are replaced by x', y', z', t' according to (1). Put an- 


other way, the mathematical expressions of physical laws must possess 
symmetry with respect to the Lorentz transformations. We will illus¬ 
trate this important statement with reference to the physical law that 
states the constancy of the speed of light in all the inertial coordinate 

Suppose that a light signal originates from point x = y = z = 0 
at t = 0 along the x-axis in the system xyz. At a time t it will be 
registered at the point x = ct. y = z = 0. If the speed of light is the 
same in all the inertial systems, then, substituting x/f = c into (1), we 
arrive at x't' = c. We will now see that this is so. Dividing the first of 
(1) by the last one gives 


f 7 

■ vt 


- V 






c 2 t 



Using (1), we can easily demonstrate the relative nature of the simulta¬ 
neousness of events. Let two events have in frame xyz the space-time 
coordinates xi, fi and X 2 , (in this case we can neglect the coordi¬ 
nates y and z), and in frame x' y' z' the coordinates x' v t and x' 2 , tf 
respectively. From (1), we get 

{t-2 - h) - ~ 2 (x 2 -X!) 

Suppose that the events are simultaneous in frame xyz, but at different 
spatial points. This means that t -2 = t\, %2 + X\. It is easily seen that 
in this case t' 2 + t[; in other words, in frame x'y'z' the events are not 

If the speed of the relative motion of xyz and x'y'z' (speed v) is 
much lower than the speed of light, then relationships (1) simplify to 

x' = x - vt 

y' = y 


Z = Z 

t' = t. 



These relationships are known as the Galilean transformations. 

They reflect the principle of relativity in classical mechanics due to 
Galileo. From the viewpoint of the special theory of relativity, the 
Galilean transformation equations are a special case of the Lorentz 
transformation equations, which is valid at v <K c. 

It is worth noting that, while recognizing the relative nature of 
spatial coordinates (x r = x — vt), the Galilean transformation equations 
at the same time assumed that time is absolute ( t' = t). The notion 
of absolute time is deeply rooted in the human mind. We are in the 
habit of thinking that the phrase “the event is occurring now, at this 
moment” has the same sense for all reference systems and for the 
whole of the Universe. 

Physically, the fundamental difference between the Galilean and 
Lorentz transformations lies in the fact that in the first case we ignore 
the finiteness of the propagation of light velocity signals, whereas in 
the second case we take it into account. When dealing with relatively 
slow motions the approximation is quite justified. As follows from (1), 
with finite speeds we have to forego the absolute nature of time and con¬ 
sider the spatial and temporal coordinates jointly. This is a consequence 
of the symmetry of physical laws with respect to changing from one iner¬ 
tial reference frame to another, which manifests itself when we take into 
account the finiteness of the speed of light. 

41 The Relativity of Time Periods 

Suppose two events occur at the same point in frame x'y'z' separated 
by a time r'; in symbols: x' 2 - x’ x = 0, t 2 - f = rr'. Suppose further 
that frame x'y'z' is connected, say, with a spacecraft travelling at a 
velocity v relative to the Earth, and the above-mentioned events are the 
“astronaut left his chair” and “the astronaut returned to his chair”. The 
frame x'y'z' is called the rest frame for these events, since they are, as 
it were, at rest, i.e., occur at the same point in space. The time period 
between the two events in the rest frame is called the proper time. 

Let us now turn to a frame xyz connected with the Earth. The above 
events, if considered in frame xyz, i. e. from the Earth, will occur at 


different spatial points: x 2 and X\. The events are separated by the time 
span r = f 2 — ii by the clock of the terrestrial observer. Using ((1)), we 
can easily find that 


x' + vt 

t = 

t = t 2 ~h = 

Consequently, the time between two events depends on the choice 
of the reference frame. This time is minimal in the rest frame for the 
given events (proper time). The time increases by a factor of (1 — 
u 2 /c 2 ) -1 / 2 in the frame that moves at a speed v relative to the rest 
frame. 1 

Suppose that the speed of the ship is fairly close to the speed of 
light; for example, v/c = 0.9999 (maybe such ships will be available in 
the future). In this case r ~ 70r'. The astronaut has only left his chair 
for 20 minutes, and on Earth the clock showed all of 24 hours. 

However, it can easily be surmised that this situation is reversible. 
You, on Earth, will take half an hour to read this section, and on the 
spaceship this time span will again be 24 hours. The situation will only 
be irreversible if the ship returns back to Earth in the long run. But this 
is a separate topic that lies beyond the scope of the book. 

Tn connecting the rest 
frame with a spaceship, we 
consider the motion of the 
Earth relative to the ship. It 
will be recalled that it makes no 
sense to find out which of the 
two inertial frames is actually 
moving and which is at rest, 
since there only exists relative 

42 The Speed in Various Frames 

Before leaving the subject of the special theory of relativity we will 
consider how the speed of a body changes when we go over from one 
inertial frame to another. We can easily feel the contradiction between 
the classical composition of speeds and the postulate that speed of 
light is constant in all inertial frames. The special theory of relativity 
requires that the classical rule be replaced by another, more general 



Let a body move in a frame xyz uniformly with a speed V along 
the x-axis, and let the speed of the body be v' in a frame x'y'z' that 
moves with a speed v relative to xyz (see Figure 90). Considering that 
V = x/t and V' = x'/t', from (1) we directly obtain the rule 

If v <3C c, then the denominator becomes unity, and so we arrive at 
the classical rule: 

V = V' + v 


The Symmetry of Physical 

The symmetry of the laws of physics with respect to the Lorentz trans¬ 
formations (or relative to changing from one inertial frame to another) 
is one of the most striking examples of this kind of symmetry. Also 
there are other forms of symmetry of physical laws. 

43 Symmetry Under Spatial Translations 

On a wide board we will install several physical devices: a mathemat¬ 
ical pendulum, a pair of communicating vessels, an electric circuit 
consisting of a battery, a switch, connecting wires, and three identical 
ammeters, two of which are connected in parallel. Let us now check 
that our elementary physical laboratory functions in full conformity 
with the laws of physics: the period of the pendulum’s swing is con¬ 
trolled by its length according to the formula T = 2 nsqrtl/g, the 
level of water in both vessels is the same, the reading of either of the 
parallel-connected ammeters is one half that of the third ammeter. 

Let us now transfer our laboratory to another room. Clearly, it will 
function there precisely as before. This simple example is a graphic 
illustration of the invariance of the laws of physics under spatial transla¬ 

When learning about the 
laws of physics you find that 
there is a large number of 
complicated and detailed 
laws, laws of gravitation, of 
electricity and magnetism, 
nuclear interactions, and so on. 
But across the variety of these 
detailed laws there sweep great 
general principles which all the 
laws seem to follow. Examples 
of these are the principles o f 
conservation, certain qualities 
of symmetry ... - Albert 

Just to get an idea of how important this type of symmetry is, let 
us try and imagine what would happen if physical laws were changed 
by spatial translations. You move to another flat and to your surprise 


find that your TV set, which has functioned perfectly, now will not 
work. You shift a clock on your table and it either stops or goes wrong. 
Swimmers perform differently in different swimming pools since the 
water resistance changes from place to place. Results obtained in an 
experiment carried out in Moscow University cannot be checked at 
Oxford using exactly the same equipment. And so on and so forth. It 
is easily seen that the demise of the symmetry of physical laws with 
respect to spatial translations immediately leads us to a picture of some 
absurd, unreliable world. 

To be sure, when speaking about translational symmetry we should 
be aware of the fact that relative translations of objects may affect 
the intensity and nature of their interaction. Naturally, transferring a 
pendulum to the Moon would change its period. Moving a clock may 
indeed render it inoperative if it is put on a strong magnet. Shifting a 
TV set farther away from a transmitting aerial will adversely affect its 
operation or put it out of the aerial’s range. In shifting something from 
one point in space to another, one should consider the environment, 
or rather the degree of its influence on the functioning of the device in 

Feynman writes: 

It is necessary, in defining this idea, to take into account every¬ 
thing that might affect the situation, so that when you move the 
thing you move everything. 

The invariance of physical laws under spatial translations can be 
shown referring to the example of the law of gravitation: 

F = 



R 2 

To be more specific, we will assume that mi and m 2 are the solar and 
terrestrial masses respectively. Then R is the separation between the 
centres of the celestial bodies, and F is their mutual attraction. 

The constant G is known as the gravitational constant. Suppose 
now that the positions of the Earth and Sun relative to some point 
within the Galaxy chosen as the origin of coordinates are given by 
the position vectors r\ and r 2 respectively. The attracting force F for 


Figure 91: Spatial transfor¬ 

a given time can be found from the law of gravitation, where we put 
R = \?2 — ?i | (Figure 91). After a spatial translation we will have to add 
to ^ and r 2 the translation vector r (see the figure). It is easily seen 
that this does not change R in the expression 

R=\(r 2 + r)-(r 1 +r)\ = \r 2 - n\ 

It follows that a translation does not affect the law of gravitation. The 
invariance of the laws of physics under spatial translations is normally 
described by the term the homogeneity of space. 

44 Rotational Symmetry 

The laws of nature are invariant not only under translations but also 
under rotations in space. It is immaterial to a TV set in which direction 
its screen faces (although this is of consequence for the viewers). Ex¬ 
perimental results are not influenced by the fact that the experimental 
set-up is oriented to north, not to east (save for geophysical measure¬ 
ments). Together with our Earth we are all involved in the complex 
motion around the Sun, this motion is a combination of rotations and 
translations. Should physical laws not be invariant under rotations 
and translations in space, they would change in time with a period of 
a year. In that case the finding obtained in one or another of physical 
experiments would, in addition, depend on the month in which the 
experiment would be carried out. 

The invariance of the laws of nature under rotations is normally 
described by the term the isotropicity of space. 

The notion ofisotropicity of space has not come easily to humanity. 


In ancient times it was widely believed that the Earth is flat and so the 
vertical direction is an absolute one. When the concept of the spheric¬ 
ity of the Earth took root, the vertical direction became a relative 
one (varying from point to point on the terrestrial surface). It is well 
known that for a long time the Earth was thought to be the centre of 
the Universe. Within the framework of this model of the world, not all 
of the directions in space were considered identical, but only those that 
passed through the centre of the Earth. In other words, that model only 
provided for spatial isotropicity at one point, the Earth’s centre; for 
any other point in space it was always possible to indicate physically 
different directions, for example the direction to the Earth’s centre and 
one perpendicular to it. Transferring the centre of the Universe from 
the Earth to the Sun will, clearly, leave the situation unchanged, that 
is, some directions will remain singled out. And only a negation of 
all centres of the Universe squares with the idea of the isotropicity of 

45 Symmetry in Time 

One often hears that today’s physics has nothing to do with the 
physics of earlier times. Even special terms are used to distinguish 
between them- “classical physics” and “modern physics”. This terminol¬ 
ogy reflects the developments in the science of physics, which just as 
any other science does not and cannot be at a standstill. 

The absolutely natural historical process of the development of 
physics in no way suggests that the laws of physics change over time. 
One of the most important symmetries of physical laws is their con¬ 
stancy in time, or rather their invariance under a shift in time. 

The law of gravitation put forward by Newton describes the mutual 
attraction of bodies that does not change with the passage of time; the 
attraction had existed before Newton and it will exist in the centuries 
to come. The laws governing the behaviour of ideal gas, which were 
established in the 17th and 18th centuries, are widely used in modern 
science and engineering. No wonder that today’s schoolchildren study, 
say, Archimedean principle, which was discovered in the 3rd century B. 
C. It will never occur to anybody that a TV set may go wrong because 


the physical laws governing the behaviour of the electron beam in 
electric and magnetic fields will change with time. 

If the laws of nature changed with time, then each piece of research 
in physics would only have an “up-to-the-minute” significance. Each 
research worker would have to start from scratch, and there would 
be no continuity between generations of scientists, without which 
science cannot exist and progress. In a world without symmetry in 
time, the same causes would have certain effects today and other 
effects tomorrow. 

The invariance of the laws of nature under shifts in time is normally 
called the homogeneity of time. 

46 The Symmetry Under Mirror Reflection 

Figure 89 presents the left-handed and right-handed molecules of butyl 
alcohol. One molecule is the looking-glass companion of the other; 
when reflected, each atom of the left-handed molecule coincides with 
the appropriate atom of the right-handed molecule, and vice versa. 
Suppose that the atom-by-atom mirror reflection was accomplished not 
with a single molecule but with a macroobject as a whole, for example 
with some physical device. As a result, we would obtain the looking- 
glass twin of the initial device that, up to the handedness, yields an 
exact replica not only of the appearance but also of the internal struc¬ 
ture of the device, down to the atomic structure. If the original device 
contained right- handed helices or spirals, they will turn into left- 
handed ones; and right (left)-handed molecules or groups of molecules 
will turn into left (right)- handed ones. Such a mirror image may in 
principle exist as a real object (not only as a mental image). 

Suppose then that we have the mirror image of a conventional 
clock, we will call it a “mirror” clock. It is quite obvious that such a 
clock will function in exactly the same way as the original clock. True, 
the hands of the “mirror” clock will turn in the opposite direction 
to the original clock, and so the face will look different as well. We 
might as well think of a “mirror” TV set, a “mirror” electric network, 
a “mirror” optical system, and so on. All of these must work in similar 


ways to the conventional devices, networks, systems, and so on. This 
means that we here have another symmetry of the laws of nature, the 
invariance under mirror reflection. 

The following is a graphic illustration of such a symmetry. Suppose 
you sit in a cinema theatre, the back wall of which is replaced by a 
large flat mirror. If you turn about and watch the film in the mirror, 
you will notice nothing unusual physically. The events in the mirror 
screen will happen exactly as on the usual screen. True, on the mirror 
screen you will have difficulties reading inscriptions that might appear 
on it, and furthermore, a familiar landscape or familiar asymmetrical 
objects will appear unfamiliar. But the latter effect of nonrecognition 
associated with the replacement of “right” by “left” has nothing to do 
with the laws of physics. 

Up until 1956 physicists treated mirror symmetry as being similar 
to those corresponding to homogeneity of space and time, isotropicity 
of space, invariance under the Lorentz transformations. Put another 
way, they held that mirror symmetry is inherent in all the laws of 
physics, without exception. In 1956 the American physicists Lee and 
Yang suggested that invariance under mirror reflection must not apply 
to the group of laws that describe the decay of elementary particles. 
This prediction was proved by a direct experiment in 1957. Much to 
their surprise, physicists found that mirror symmetry is inherent not in 
all phvsical laws and that in some phenomena nature exhibits left-right 
asymmetry. We will take a closer look at that most interesting issue 
in Chapter 14. We will only note here that the discovery of mirror 
asymmetry in the decay of elementary particles seems to provide a 
clue to the striking fact of the asymmetry of living molecules (to be 
discussed in some detail in Chapter 7). 

47 An Example of Asymmetry of Physical Laws 

In order not to leave you with the impression that the laws of physics 
are invariant under any transformation, we will furnish an instructive 
example of transformations under which the laws of physics are nonin¬ 
variant. Such an example is the transformations that involve changes 
in the spatial scale, or rather similarity transformations. 



All the laws are noninvariant under similarity transformations. In 
other words, the geometrical similarity principle is, strictly speaking, 
inapplicable to the laws of physics. 

True, the notion of similarity has struck deep roots in human mind. 
So it enjoys wide use in literature and the arts. Suffice it to remember 
Jonathan Swift, who sent his Gulliver first to Lilliput and then to the 
giants of Brobdingnag. The same idea lay at the foundation of con¬ 
jectures ventured at the turn of the century that the atom is a solar 
system in the microscopic world. 

It would seem that if we were to build a new set-up, each part of 
which would be several-fold larger (or smaller) than the appropriate 
part of the original set-up, the new installation would function in 
exactly the same way as the original one. It is not for nothing that 
aerodynamic and hydrodynamic structures are tentatively tested on 
scale models. 

It is well known, however, for those dealing with the tests that 
models should not be scaled down too much. It was Galileo who first 
established that the laws of nature are not symmetrical with respect to 
changes in scale. He came to that conclusion while speculating about 
the strength of bones of animals as their size is increased. Similar 
reasoning is provided in Feynman’s book The Character of Physical 
Laws. We can construct a toy gothic cathedral out of matches. Why 
then can we not construct a similar cathedral out of huge logs? The 
answer is: should we undertake the project the resultant structure 
would be so high and heavy that it would collapse. You might object 
that when comparing two things one should change everything that 
constitutes the system. Since the small cathedral is subject to the pull 
of the Earth, to be consistent we would have to subject the larger 
cathedral to the pull of the Earth increased appropriately, which would 
be even worse. 

From the point of view of modern physics, the invariance of the 
laws of nature with respect to similarity transformations has a clear 
and comprehensive explanation - atoms have a size whose order of mag¬ 
nitude is absolute, identical for the whole of the Universe. Atomic size 
is determined by the universal physical constant - Planck’s constant 
H (ft = 1.05 X 10 —34 J s); being given by the relationship ft 2 /me 2 , where 


m and e are the mass and charge of the electron, respectively (the min¬ 
imum known mass at rest and the minimum known electric charge in 

This relationship yields 10 -10 m for the linear size of the atom. It 
follows, by the way, that reducing the linear dimensions of some real 
installation having a volume of 0.1 m, say, billion-fold, we will then 
be left with only about one hundred atoms! Clearly, no workable 
apparatus can be manufactured out of such a small number of atoms. 

A striking example of the asymmetry of physical laws with respect 
to scale changes is the fundamental fact that sufficiently strong scaling 
down puts the laws of classical mechanics, specifically Newton’s laws, 
out of business. The laws of the microworld, quantum mechanics, take 

10 Conservation Laws 

48 An Unusual Adventure of Baron Miinchhausen 

You may have read the story of Miinchhausen mired in a bog with his 
horse. The galant baron relates: ’’Now the whole of my horse’s body 
has sunk into the stinking mud, now my head too began to sink into 
the bog with only the plait of my wig sticking above the water. What 
was to be done? We would have perished if it had not been for the 
prodigious strength of my hands. I am awfully strong. And so I took 
hold of my plait, jerked with all my might and easily pulled out of the 
bog both myself and my horse, whom I squeezed tightly with my legs 
like with pincers.” The reader can easily catch the esteemed baron in 
the lie. Really, according to Newton’s third law (action and reaction) 
the plait acts on the hand with a force equal in magnitude and opposite 
in direction to the force with which the baron’s hand acts on the plait. 
And since the hand and the plait are parts of the same physical system 
(the baron), the resultant force exerted by the baron on himself will 
clearly be zero. 

There exist several basic laws 
of nature that have the math¬ 
ematical form of conservation 
laws. A conservation law states 
that in a closed system some 
physical quantity, for example, 
the total momentum or energy, 
remains constant at all times. J. 

It is thus in principle impossible to lift oneself by the hair. In a more 
general context, this prohibition can be viewed as a consequence of the 
momentum conservation law. According to this law, the momentum of a 
system will not change as a result of the interaction of the components of 
the system with one another. As applied to our case, this means that the 
baron’s momentum directed downwards into the mire cannot change 
as a result of the interaction of the baron’s hand with his plait. 

If the system is not subject to external influences (it is called closed 


system), then there are no reasons for the system’s momentum to 
change. In that case, the total momentum does not change with time 
or, as it is conventionally put, it is conserved. The law of conservation 
of the momentum of a closed system is one of the three important 
conservation laws. The two other laws are the laws of conservation of 
energy and of angular momentum. 

49 The Problem of Billiard Balls 

The game of billiards provides a good opportunity to illustrate the 
action of conservation laws for momentum and energy. We will take 
a cue into our hands and try and push one of the billiard balls so that 
it rolls exactly along the line connecting the centre of this ball with 
the centre of another ball. This sort of collision is called a head-on 
collision. It is interesting that the first (striking) ball comes to rest at 
the moment of collision no matter what its initial speed. The second ball 
starts moving along the line with a speed exactly equal to that of the 
first ball before the collision (Figure 92). 

Before collision 

This result can be computed using the laws of conservation of 
energy and momentum for the balls involved in the collision. Let v be 
the speed of the striking ball, the second ball being at rest before the 
collision. Further, let Vi and be the speeds of the first and second 
balls, respectively, after the collision (at the moment we do not know 
that the first ball will come to rest after the collision). Billiard balls 
collide in an elastic manner, which means that no energy is lost in the 
collision. For an elastic collision the law of conservation of energy is 

mv 2 mv\ mv \ 

~ 2 ~ ~ ~T~ + ~ 2 ~ 

where m is the ball’s mass. It follows that in the collision the total 
kinetic energy of the balls is conserved. 

We will now turn to the law of conservation of momentum. Recall 
that the momentum of a body with mass m and speed v is mv. Mo¬ 
mentum is a vector quantity, and so we have to take into account its 
direction. In a head-on collision we have the only direction singled out 
physically (the red dash line in Figure 92). Clearly, the momenta of the 

After collision 

Figure 92: Before and after 
in a head-on collision. 



balls after the collision can only be directed along this direction. The 
one-dimensional nature of the problem enables us to consider, in the 
law of conservation of momentum, solely the numerical values of the 
momenta, that is, to write the law in its scalar form: 

mv = mv i + mv 2 

Combining both laws, we arrive at the set of equations in zq and v 2 

2 2,2 

V = V\ + 

V = Vl + v 2 

Squaring both sides of the second equation gives 
v\ + 2zqw 2 + v\ = v 2 
From the first equation we obtain 

2zq v 2 = 0 

Physically, it is clear that v 2 += 0 (the second ball cannot remain 
at rest after the impact). We therefore conclude that zq = 0 and hence 
v 2 = v. If billiard balls collide not in a central way ( off-centre collision), 
then after the impact both balls will move apart in different directions. 
It is remarkable that in all cases the balls will move apart at a right 
angle, which can easily be tested in practice. But this result can be 
predicted from the laws of conservation of energy and momentum for 
colliding balls . 

We will denote by v the speed of the striking ball, and by zq and 
v 2 the speeds of the balls moving off after the impact at an angle a. 
(Figure 93). Show that a = 90°. 

The law of conservation of energy will have the same form as for a 
central collision: 

but the law of conservation of momentum must now be written in a 
vector form: 

Before collision 

After collision 

Figure 93: Before and after 
in a side collision. 


mv = mvi + mv 2 

Figure 94 presents vector mv as a sum of vectors mvi and mv 2 . Let 
us consider the triangle ABC (|AC| = mv, \AB\ = mv i, | BC\ = mv 2 ) 
and apply the law of cosines 

\AC\ 2 = \AB\ 2 + \BC\ 2 - 2\AB\ • \BC\ cos f 


Figure 94: Resolution of 
vectors in a collision. 

where f is the angle between the sides AB and BC. This relationship 
can be written as 

v 2 = v\ + v\ + 2v\v 2 cos a 

(considering that f = 180° - a, and hence cos f = — cos a). 

This is the most convenient form of the law of conservation of mo¬ 
mentum for the two colliding balls. Since from the law of conservation 
of energy v 2 = v 2 + v | we have 2viv 2 cos a = 0. And since v\ + 0 and 
v 2 £ 0, then cos a = 0, that is a = 90°. 

Figure 95: Conservation 
of momentum on a billiard 

Shown in Figure 95 is a specific situation on a billiard table. The di¬ 
rections from ball 2 to the two pockets (see dash lines) form a straight 
angle. Therefore, it is better to drive ball 1 against ball 2, not 3, since a 
good stroke may send both ball 2 and ball 1 to the respective pockets. 


50 On the Law of Conservation of Momentum 

Let us take a closer look at the law of conservation of momentum. In 
our thought experiment we will replace the billiard table with a large 
flat surface on which n balls move about colliding randomly with one 
another. We will suppose that in the general case the masses of the 
balls are different: m 1; m 2 ,..., m„. We will denote the speeds of the 
balls at time t as vi(t), v 2 (t),..., v n {t)- We will then have the sum 

midi(t) + m 2 v 2 (t) + ... + m n v n (t), 
which can be conveniently represented as 


^ mvi{t) 


According to the law of conservation of momentum for a system of 
colliding balls, this sum must remain unchanged in time. Although 
individual summands here change in collisions, the sum as a whole 
remains a constant value (is conserved). The law of conservation of 
momentum can be written as 


^ mivft) = p 

where p is the total momentum of the balls. The vector p is constant, so 
that the collisions of balls with one another do not affect its direction 
and magnitude. The above equation implies that the total momentum 
of some system (in our case, a system of balls) does not change when 
parts of the system interact with one another. 

In classical mechanics the law of conservation of momentum can 
be derived from Newton’s third and second laws. The derivation is 
instructive, and so we will reproduce it here. Suppose that within 
a system of balls, two balls with masses raj and m 2 collide. Ball m 2 
exerts a force fi on ball mi, and ball raj exerts a force f 2 on ball m 2 . 
According to Newton’s third law, 

fi = ~h 

( 1 ) 


From Newton’s second law, we have 

fi = miai, f 2 = m 2 a 2 

where a\ and a 2 are the accelerations of m\ and m 2 , respectively. Let 
At be the duration of the collision, and Avi and Av 2 the changes of 
the speeds of the balls during the collision. For small enough At we 
can assume that 

_ Av± Av 2 



a 2 — 


As a result, equation (1) reads 




Av 2 

= - m2 a r 

We will now substitute Atq = uj — v-[ where v' x and v\ are the speeds 
of the ball mi before and after the collision, respectively. In a similar 
way we obtain Av 2 = v- v 2 . Now equation (1) reads 

mi (v[ - vi) = —m 2 (v' 2 - v 2 ) 


miv[ + m 2 v 2 = miVi + m 2 v 2 

The last relation implies that as a result of the collision of the balls 
their total momentum does not change. We thus arrive at the law of 
conservation of momentum. 

51 The Vector Product of Two Vectors 

Before we proceed to consider the law of conservation of angular 
momentum, we will take up a purely mathematical concept of the 
vector product of two vectors. Suppose that a vector a is to be multiplied 
in a vector way by a vector b, or it is required to find a vector c that 
is the vector product of the initial vectors. In symbols we have c = 

(a X b). Let us translate one of the multipliers, say b, parallel to itself 
so that both vectors (a and b) have common origin. Denote by S the 
plane passing through a and b and by cp the angle between them. The 
magnitude of c will be given by 

c = ab sin (p , 


and its direction is perpendicular to S. True, there are two directions 
perpendicular to the plane; these are mutually opposite. To have the 
required direction for c we will turn the first multiplier (a) to the 
second multiplier ( b) through the smaller angle. The direction of c will 
then obey the so-called right-handed screw rule (Figure 96 (a)). It is 
easily seen that transposing the multipliers changes the sign of their 
vector product: 

(a X b) = — {h X a) 

(compare Figure 96 (a) and (b)). 

Note that the direction of c is an arbitrary notion, since it is asso¬ 
ciated with the convention to use the right-handed screw. Nothing 
forbids us in principle from starting of with the left-handed screw, 
not the right-handed one. Vectors, whose direction is not conditioned 
physically, but is arbitrarily related to the right (left)-handed screw, 
are called axial vectors. On the contrary, conventional vectors whose 
direction is defined physically are called polar vectors. The vector 
product of two polar vectors is an axial vector. 

Notice that an axial vector is reflected in a mirror in a different way 
from a conventional (polar) vector (see Figure 97, where a is a polar 
vector, c an axial vector). The handedness of the axial vector can be 
seen in Figure 98, where the axial vector c is regarded as the vector 
product of the polar vector a and the polar vector b. We can grasp the 
handedness of c by reflecting a and b, and noticing the change in the 
direction of rotation from a to b (see circular arrow in Figure 98). 

c t = bxa 


Fieure 96: Vector addition. 

52 Kepler’s Second Law - the Law of Conservation of the An- Flgure 97: Polar and axial 

vectors on reflection. 

gular Momentum of a Planet 

The second of Kepler’s three laws describing planetary motion is 
known as the law of areas: an imaginary straight line connecting a 
planet with the Sun ’’sweeps” in the planet’s orbital plane equal areas 
in equal time periods. Figure 99 (a) shows the elliptical orbit of some 
planet, with the Sun being at focus O. The areas of the figures AOB and 
COD shaded in the figure are equal; therefore, by Kepler’s second law, 



c t 

Figure 98: Direction of re¬ 
sultant vector under mirror 

the planet covers segments AB and CD in equal times. Accordingly, the 
closer the planet approaches the Sun, the higher is its orbital velocity. 

The law of areas discovered by Kepler corresponds to the law of 
conservation of the orbital momentum of the planet. We will clarify the 
concept of angular momentum with reference to Figure 99 (b). 

Suppose at a time t a planet lies at a point A on its orbit. We will 
draw from O to A a vector r(t) - the position vector of the planet at 
time t. The planet has a velocity v{t) and momentum mv(t) (where m 
is the planet’s mass). The vectors r(t) and v(t) make an angle ip(t). The 
orbital angular momentum of the planet, M,is the vector product of the 
position vector r by the momentum vector mv: 

Figure 99: Illustration of 
Kepler’s law. 

M = (r x mv(t)) 

Its magnitude is determined by the product rmv sin <p, and its direction, 
according to the right-handed screw rule, is normal to the orbital plane 
(in Figure 99 this is the direction into the page). 

Here r and mv are conventional (polar) vectors, and M is an axial 



We will now show that Kepler’s law of areas corresponds to the law of 
conservation of the orbital angular momentum. Let A and F be two close 
points on the orbit at which the planet lies at t and t + At. respectively 
(Figure 99 (c)). The time increment At is assumed to be small enough 
- such that the arc AF could be replaced by a segment and the orbital 
speeds of the planet at A and F could be considered essentially the 
same. The area of the shaded triangle AOF in Figure 99 (c) will be AS. 
Then from vertex F we will drop the perpendicular FE to the extension 
of AO. It is easily seen that 

■\FE\ = ^\AO\-\AF\-sin <p(t) 

= v(t)At, then 
= ^r(t)v(t) sin cp(t) 

AS _ M 
At 2m 

The left-hand side of this equation includes the areas “swept out” by 
the position vector of the planet in a unit time, and on the right-hand 
side we have M, that is, the orbital angular momentum of the planet. 
According to Kepler’s law, the quantity AS/At does not vary with 
time. It follows that M, too, must be independent of time. 

True, Kepler’s second law and the law of conservation of the orbital 
angular momentum of a planet are not completely equivalent. The law 
of conservation of momentum contains more information than Ke¬ 
pler’s law of areas, since its content is the conservation of not only the 
magnitude but also of the direction of angular momentum in space. The 
conservation of the direction of angular momentum accounts for the 
fact that the orientation of the orbital plane of a planet is unchanged in 

AS= i|AO| 

Since \AO\ = r{t) and |AF| 






53 Conservation of the Intrinsic Angular Momentum of a Ro 
tating Body 

Apart from the orbital angular momentum, a planet also has an in¬ 
trinsic angular momentum. Whereas orbital angular momentum is 
associated with the motion of a planet in its orbit, the intrinsic angular 
momentum arises as it rotates about its own axis. The intrinsic angular 
momentum of a planet is also conserved. Its direction makes an angle 
with the orbital plane, which does not change in the course of time. 

The conservation of intrinsic angular momentum is responsible for the 
constant alternation of night and day; the conservation of the direc¬ 
tion of that momentum is responsible for the unchanged (for a given 
latitude) variation of the length of day during the different seasons. 

The constancy of angular velocity and direction of rotation of a 
gyroscope or common top is also related to the conservation of the 
intrinsic angular momentum of these bodies. When a figure skater 
pirouetting on ice extends his arms to come to a halt swiftly, he takes 
advantage of the law of conservation of intrinsic angular momentum. 
Extending the arms sidewards shifts some mass of the skater away 
from the axis of rotation, which by the law of conservation of angular 
momentum is compensated for by a reduction in the angular velocity 
of rotation. We will explain this using Figure 100. 

For simplicity, in Figure 100 instead of the skater we will consider 
a system of two masses mi and m 2 slid on a massless rod. The masses 
rotate on it about point O. In the initial situation both masses lie at the 
same distance r from O (see Figure 100 (a)). When the system is set in 
rotation, the angular velocity co is related to the orbital velocity v by the 

Figure 100: Conservation 
of momentum of a rotating 



relation a> = vjr. Then the magnitude of the total angular momentum 
of the system will be 

M = r (mi + rri 2 )v = (mi + m. 2 )cor 2 

We will further suppose that m2 shifts to r 1 (see Figure 100 (b)). This 
more or less corresponds to the skater extending his arms. Now the 
angular momentum will be 

Mi = a>i(mir 2 + m 2 r 2 )• 

Since ri > r and Mi = M (the angular momentum is conserved), 

coi < a). 

This law finds widespread use in engineering. So, this is the princi¬ 
ple behind the gyroscope, a sort of top whose spin axis stays constant 
in space. Gyroscopes are used in gyrocompasses, automatic guidance 
devices, large gyros are used on some ships to achieve stability against 

Symmetry and Conserva 
tion Laws 


The law of conservation of energy had been used in mechanics before 
Galileo. So in the late 15th century the great Leonardo da Vinci postu¬ 
lated the impossibility of perpetuum mobile. In his book On True and 
False Science he wrote: ”Oh, seekers of perpetual motion, how many 
empty projects you have created in those searches.” The laws of conser¬ 
vation of momentum and angular momentum were formulated later, 
in the 17-18th centuries. It was not, however, till the beginning of the 
20th century that the laws assumed prominence. The attitude to them 
changed radically only after it had been discovered that these laws 
were related to the principles of invariance. Once this relation had 
been revealed, it became clear that conservation laws are predominant 
among other laws of nature. 

Today it is difficult to find 
a paper devoted to funda¬ 
mental issues of physics that 
would not make mention of 
invariance principles and its 
author would not in his argu¬ 
ments draw on the concepts 
of the existence of relations 
between conservation laws 
and invariance principles. - E. 

54 The Relationship of Space and Time Symmetry to Con¬ 
servation Laws 

The relationship can be formulated as follows: 

1. The law of conservation of momentum is a consequence of the homo¬ 
geneity of space, or rather a consequence of the invariance of the 
laws of physics under translations in space. Momentum can thus be 
defined as a physical quantity whose conservation is a consequence 
of the above-mentioned symmetry of physical laws. 


2. The law of conservation of angular momentum is a consequence of 
the isotropy of space, or rather a consequence of the invariance of 
the laws of physics under rotations in space. Angular momentum 
is a physical quantity that is conserved as a consequence of the 
above-mentioned symmetry of physical laws. 

3. The law of conservation of energy is a consequence of the homogeneity 
of time, or rather a consequence of the invariance of the laws of 
physics under translations in time. Energy is a physical quantity 
that is conserved as a consequence of the above-mentioned symme¬ 
try of physical laws. 

The three-dimensionality of space predetermines the vector nature 
of momentum and angular momentum, and so the laws of conser¬ 
vation of momentum and angular momentum are vector laws. The 
one-dimensionality of time predetermines the scalar nature of energy 
and the corresponding conservation law. 

The relationship of conservation laws to space-time symmetry 
means that the passage of time or a translation and a rotation in space 
cannot cause a change in the physical state of the system. This will 
require that the system interact with other systems the corresponding 
conservation law. 

55 The Universal and Fundamental Nature of Conservation 

Such a way of looking at conservation laws may appear to be unusual 
to the uninitiated reader. Significantly, the conservation laws can be 
obtained without applying the laws of motion directly from symmetry 
principles. The derivation of the law of conservation of momentum 
from Newton’s second and third laws, provided in Chapter 10, should 
in this connection be regarded as a special consequence. 

It follows thus that the region of applicability of conservation laws 
is wider than that of the laws of motion. The laws of conservation of 
energy, momentum and angular momentum are used both in classical 
mechanics and in quantum mechanics, whereas Newton’s laws of 
dynamics are out of place in quantum mechanics. Wiegner wrote: 



For those who derive conservation laws from invariance prin¬ 
ciples, it is clear that the region of applicability of these laws 
lies beyond the framework of any special theories (gravitation, 
electromagnetism, etc.) that are essentially separated from one 
another in modern physics. 

Clearly, the region of applicability of conservation laws must be as 
wide as that of appropriate invariances. This enables one to think of 
the laws of conservation of energy, momentum and angular momen¬ 
tum as universal laws. 

The relationship of conservation laws to invariance principles also 
suggests that any violation of these laws, should it occur, would point 
to a violation of appropriate invariance principles. So far no experi¬ 
mental evidence is available to indicate that the laws of nature may 
turn out to be non-invariant under translation in time, and under 
translation and rotation in space. This circumstance, together with the 
aforementioned property of universality, makes the laws of conserva¬ 
tion of energy, momentum, and angular momentum really fundamental 

It follows from the fundamental nature of conservation laws that 
we should select as the conserved quantities fundamental physical 
quantities: energy, momentum, and angular momentum. Note that in 
classical mechanics these quantities appear as functions of the velocity 
and coordinates of a body. So the energy, momentum, and angular 
momentum of a billiard ball can be represented in the form 

E = 

mv, M = (r x mv) 

whence it follows in particular that 

( 1 ) 

E = 

2 m 

, M = (rxp) 

( 2 ) 

It might appear that we can conclude from expressions of type (1) 
that the role of fundamental quantities is played by velocity and coor¬ 
dinates. But if we go over from the classical mechanics of billiard balls 
to the quantum mechanics of microobjects, the very concept of the 
velocity of an object will then become unsuitable and expressions (1) 


become pointless. At the same time the conserved quantities (E, p , M) 
retain their meaning both in classical and quantum mechanics. It is 
important that in quantum mechanics they are, generally speaking, 
not expressible in terms of each other; the second of (2) does not hold 
in the microworld, since a microobject has no states in which the 
values of momentum and coordinates are specified simultaneously. 

As regards the first of (2), it is valid only for the free motion of a mi¬ 
croobject. For a bound microobject (for instance, an atomic electron) 
the energy is quantized, with the result that for each energy level we 
cannot indicate a definite value of momentum. Quantum mechanics, 
thus, actually makes it possible to bring out the fundamental nature 
of conserved quantities, their independence in full conformity with 
the fundamentality and independence of the corresponding types of 
symmetry of the laws of physics. 

The universality of conservation laws suggests that the conserved 
physical quantities are used in various branches of physics. They can be 
described by various expressions into which enter physical quantities 
characteristic of a given field. Consider momentum, for example. We 
will write out the four expressions for the momentum 

p = mv; 


The first expression describes the momentum of a body with mass m 
and speed v in classical mechanics; the second one, in the theory of 
relativity. The third expression is the momentum of a unit volume of 
electromagnetic field in terms of the vectors of the strengths of electric 
and magnetic fields. The fourth expression gives the momentum of 
a photon in terms of its cyclic frequency a> and unit vector h in the 
direction of the motion of the photon; here Ti is Planck’s constant. 

These four formulas are a good illustration of the universal nature of 
the concept of “momentum”. As for the quantities v, E, H, and <x>), they 
are all applicable in respective disciplines of physics only. 



It is worth noting here that proceeding from Newton’s laws of 
dynamics we can obtain the law of conservation of momentum solely 
for the special case where the momentum is given by the expression 
p = mv\ in other cases Newton’s laws are clearly not applicable. 

If we start off with invariance principles, we can derive the law of 
conservation of momentum regardless of the expression describing the 
momentum in one case or another. 

56 The Practical Value of Conservation Laws 

In the introductory talk about symmetry, it was noted that symmetry is 
some general entity inherent in a wide variety of objects (phenomena), 
whereas asymmetry brings out some individual characteristics of a 
specific object or phenomenon. Permeating all spheres of physics and 
all specific situations, conservation laws express those general aspects of 
all situations, which is eventually related to the appropriate principles of 
symmetry. These laws “ignore” the particularity of a given situation, 
they “ignore” the particular mechanisms of interplay, their region of 
applicability lies beyond the framework of specific theories. It is the 
general, all-embracing character of conservation laws, which do not 
require an analysis of phenomena, that is responsible for the striking 
simplicity of these laws and unconditioned reliability of results derived 
from them. 

It is worth mentioning that not infrequently the interaction mech¬ 
anisms (details of a phenomenon) are unknown or known in only a 
fairly approximate way. In many cases the inclusion of a host of de¬ 
tails in a problem overly complicates the mathematical side of the. 
problem. Against the background of these difficulties, the simple and 
elegant conservation laws appear quite attractive. When handling a 
phenomenon, a physicist, above all, applies conservation laws and only 
then proceeds to examine details. Many phenomena to date have been 
studied on the level of conservation laws only. 

So in the billiard problem (see Chapter 10) we neglected the mech¬ 
anism of collision of balls. Otherwise, we would have had to delve 
into the details of elasticity theory. In this particular case, it turned 
out to be sufficient to utilize the laws of conservation of energy and 


momentum alone. 

57 The Example of the Compton Effect 

We will illustrate the practical value of conservation laws referring 
to an effect discovered in 1923 by the British physicist A. Comp¬ 
ton (the Compton effect). The gist of the effect is as follows: when a 
monochromatic beam of X-rays is scattered at electrons that enter the 
composition of the target substance, the wavelength of the radiation 
is increased by AA, which is found using the relation (the Compton 

. , 4 ith . 2 <p 

AA = -sin 2 - 

me 2 

where <p is the scattering angle, Ti Planck’s constant, m electron rest 
mass, c the velocity of light in a vacuum. 

To derive the Compton formula we need not know the exact mech¬ 
anism of the interaction of X-ray quanta (photons) with the electrons; 
it is sufficient to apply the laws of conservation of energy and momentum 
for a collision of a photon with an electron. In a sense, we again have to 
turn to the billiard problem, although neither the electron nor the pho¬ 
ton resemble a billiard ball, of course. Before a collision, the ball of an 
electron can be treated as being at rest as compared with the incident 
ball of a photon. The energy of the photon is E, its momentum is p. 
Assuming the collision to be off-centre, after it, the photon will be scat¬ 
tered at an angle ip, its energy being E i and momentum pi; the electron 
will be scattered at an angle 8, its energy being E e \ and momentum p e 
(Figure 101). The law of conservation of energy has the form 

E = Ei + E e 

and the law of conservation of momentum (Figure 102) 

p=pi+p e 



Before collision 

After collision 

Figure 101: The Compton 


Figure 102: The Compton 

or in the components, 

p = pi cos <p + p e cos 8 
O = pi sin <p — p e sin 8 



The form of these equations is quite general, they are applicable also 
for billiard balls. To take account of the specific feature of the photon- 
electron problem it is sufficient to express the photon energy as 

E = h co 

and its momentum, using the formula discussed earlier in the book, as 

- hi y ^ 

p = —n 


As regards the electron, its energy can be given in terms of momentum 
using the conventional relation 

E, = eL 


that is equally valid both for the billiard balls and for the free electron 
(provided that its velocity is well under the velocity of light). 

It follows that the conservation laws can be written as 

tico = hco\ + 


hid Ticoi 

— = -cos <p + p e cos 8 

c c 

- tuo 1 . . _ 

(J = -sin ip + p e sm 8 


It only remains to do some algebra. Introducing the notation A co = 
co — coi (A co is generally small, that is, A co « co), we can write the 
above set of equations in the form 

hAco = 



hco — h(co— A co) cos ip = p e c cos 8 
h(co — A co) sin ip = p e c sin 8 

Squaring the second and third equations and adding them together, we 
get rid of 8\ 

hi 1 cl? + h 2 (oo - A co) 2 - 2 h 2 oo{oo - A co) cos <p = p 2 e c 2 


or, using the first equation (the law of conservation of energy), 

2mc 2 Aco 

+ (co — A<y)“ - 2co(co - Aco) cos cp = 


We will now divide both sides of this by <y 2 and open the brackets. 
Ignoring the small term (A coj co) 2 gives 

A co 

A co 


cos cp = 

2 me 2 

A co 



2 me 2 Aco 
hco co 

The wavelength X is related to the cyclic frequency co by the expression 
X = 2nc/co. It is easily seen that 

AX = Xi - X = 

2 nc 

2nc co - co\ 

- = 2 nc - 


2 nc 

Aco/co 2 
1 - Aco/co 


o cp me . 

2sin 2 - = ——AX 

2 2 kTi 

The Compton formula follows from this immediately. 

We have thus seen that by approaching the Compton effect at the 
level of conservation laws, we can derive the Compton formula, that is, 
to find the variation of the wavelength increment with the scattering 
angle for the photon. 

Before we leave this section, it should be stressed that it would 
be wrong to overestimate the role of conservation laws. To be sure, 
they are not a panacea for all aches and pains of the physicist. So to 
determine the probability for a photon to scatter at a given angle, it 
is necessary, apart from conservation laws, to take into account the 
specific mechanism of the interaction of photons and electrons. In that 
case, the parallel with billiard balls does not work any more. 

58 Conservation Laws as Prohibiting Rules 

It is well known that conservation laws are often formulated as rules 
(laws) of prohibition. So the law of conservation of energy is, in essence, 



the law that prohibits perpetual motion. In a sense the law of conserva¬ 
tion of momentum is the law that prohibits lifting oneself by the hair. 
The law of conservation of angular momentum prohibits, for example, 
a planet from leaving its orbit and changing the angle at which its spin 
axis is inclined to the orbital plane. 

There are two reasons to approach conservation laws as laws of 
prohibition. Above all, we notice that symmetry, which introduces a 
measure of orderliness, at all times tends to reduce the number o fpossible 
versions. It has already been stressed in Chapter 4 that symmetry dra¬ 
matically limits the diversity of structures that may be encountered in 
nature. We can now complement that statement by stressing that sym¬ 
metry limits the diversity of not only structures but also of the forms 
of behaviour of physical systems. Through conservation laws (or rather 
prohibition laws) symmetry covers all the conceivable forms of the 
behaviour of a system and at times it nearly uniquely predetermines a 
certain behaviour of a system. 

Remember the billiard balls. No matter how strongly you push a 
ball to a head-on collision it is bound to come to rest, since by energy 
and momentum conservation it may not move after the impact. Also of 
interest is the case of off-centre collision: the conservation laws decree 
that after the collision the balls only move off at right angle to each 

In his book The World of Elementary Particles the American physicist 
K. Ford writes: “The older view of a fundamental law of nature was 
that it must be a law of permission. It defined what can (and must) 
happen in natural phenomena. According to the new view, the more 
fundamental law is a law of prohibition. It defines what cannot happen. 
A conservation law is, in effect, a law of prohibition. It prohibits any 
phenomenon that would change the conserved quantity.” 

It would seem that prohibitory rules represent just a simplified ver¬ 
sion of guiding rules. In actual fact, this is not so. Take road signs, for 
example. Suppose the sign shown in Figure 103 (a) is put up before 
a crossing. This sign is guiding: it explicitly directs the traffic - only 
forward. If then the sign is as shown in Figure 103 (b) (prohibitory 
sign), the traffic does not have an unequivocal direction - it may either 
continue forward or turn to the left; it is only prohibited to turn to the 

Figure 103: The Compton 


right. The meaning of the above quotation from Ford is that the con¬ 
servation laws are to be likened to prohibitory laws, and not guiding 

There is one more important reason for which conservation laws are 
to be regarded as laws of prohibition. The fact is that in the world of 
elementary particles (where, additionally, there are a wide variety of 
other conservation laws), conservation laws are derived as rules that 
prohibit the phenomena that are never observed in experiment. Sup¬ 
pose experimental data definitely indicate that some transmutations 
of particles never occur, or in other words are prohibited. This circum¬ 
stance may be used as a basis for formulating some conservation law. 
But it is not always that the invariance principle underlying the law 
is revealed; in such cases a conservation law only appears as a law of 
prohibition. We will be looking at such examples later in the book. 

The World of Elementary 

Elementary particles are the frontier of modern physics which corre¬ 
sponds to the most fundamental level of probing into the physical 
picture of the world. Naturally, it is here that the most important reg¬ 
ularities show up, which, in the final analysis, control the structure 
of matter and the character of physical processes. It is of principal 
importance, therefore, to gain an insight into the aspects related to 
conservation and invariance in the world of elementary particles. But 
before we proceed to discuss these issues, we will have to take a look 
at the currently known elementary particles. 

The elementary particles are 
not just interesting scientific 
curiosities. They represent the 
deepest-lying substructure of 
matter to which man has been 
able to probe; consequently, 
they provide one of the most 
challenging problems on the 
current frontiers of science. - K. 

59 Some Features of Particles 

We will single out three quantities characterizing particles - mass, elec¬ 
tric charge, and spin. The ensemble of characteristics will be extended 
markedly later. In addition, it will include lifetime, specific charges 
(electronic, muonic, and baryonic), isospin, strangeness, and so on. 

By mass we will understand the rest mass of a particle, that is, the 
mass in the frame of reference connected with the particle itself. The 
smallest mass is possessed by the electron (m = 9.1 X 10 -28 g); there¬ 
fore, the mass of other particles is often expressed in electron masses. 
Mass is also expressed in energy units MeV (megaelectronvolts). The 
use of energy units for mass is based on the well-known relationship 
due to Einstein: E = me 2 . In terms of energy units, the electron mass is 


0.511 MeV. 

The electric charge of particles is denoted by numerals: 0, +1, -1. In 
the first case, there is no charge (the particle is neutral). In the second 
case, the charge is equal to that of an electron, but unlike the electron, 
it is positive. In the third case, the charge coincides with the electron 
charge both in magnitude and in sign. Note that the electric charge 
of charged particles is exactly equal to the electron charge, that is, 

1.6 X 10“ 19 C. 

The spin of a particle is the specific angular momentum of a parti¬ 
cle which can be called the intrinsic angular momentum since it is not 
related to motions of the particle in space; it is indestructible, its mag¬ 
nitude is independent of external conditions. This angular momentum 
can arbitrarily be associated with the rotation of the particle about its 
own axis. An analogue of spin may be the intrinsic angular momentum 
of a planet or gyroscope as discussed in Chapter 10. The squared spin 
is given by the expression n 2 s(s + 1), where h is Planck’s constant, s 
is a number characterizing this particle, which is normally referred to 
as its spin (in the latter case spin is measured in units of Ti). Like any 
angular momentum, spin is a vector quantity. This vector is quite spe¬ 
cific, however, since its projection in a given fixed direction only takes 
on discrete values (is quantized): tis, h(s — 1),..., —Tis. The total number 
of spin projections is 2s + 1. In this connection, it is said that a particle 
with spin s may be in one of 2s + 1 spin states. 

For many elementary particles, specifically for the electron, spin is 
1/2. These particles have two spin states, one for each of the opposite 
spin directions. 

Note that all the particles of this type (for example, all electrons) 
have exactly the same mass, charge, and spin. It is in principle impos¬ 
sible for the mass of one electron to differ from that of another one by, 
say, 0.001 per cent. The values of the mass, electric charge, and spin of 
the electron are, according to the evidence available, the lowest values 
of these quantities ever encountered in nature, save for the cases where 
a particle does not have a rest mass, charge, or spin. 



60 The Zoo of Elementary Particles 

The particles in the zoo are generally classed according to their mass, 
charge, and spin. They form three families: 

The first family is the smallest - it consists only of one particle. It is 
the photon, the quantum of electromagnetic radiation (symbol y). The 
rest mass and electric charge of the photon are zero, s = 1. Note that 
according to the theory of relativity, any particle with zero rest mass 
cannot have an electric charge and in any inertial frame of reference 
it travels with the same velocity - the velocity of light in empty space. 
The photon is an example of such a particle. 

The second family consists of particles called leptons. Up until 1975 
four leptons were known: electron (e ), electron neutrino (p e ), muon (p ), 
and muon neutrino (vf). We have already discussed the electron. The 
muon has a mass of 207m, its electric charge is negative, s = 1/2. Both 
neutrinos are indistinguishable in terms of the three characteristics 
used here (no wonder that for a long time it was believed that there 
only exists one type of neutrino in nature). Like the photon, both neu¬ 
trinos have neither rest mass nor electric charge. Unlike the photon, 
however, the spin of the neutrino is 1/2, like that of any lepton. 

In 1975 a fifth lepton, tauon (r ), was discovered. It appeared to be 
an ultra-heavy particle: its mass is about 3500m. The electric charge 
of the tauon is negative. Physicists have good reasons to think that a 
tauon must have a companion particle, a tauon neutrino (y T ). Counting 
the third type of neutrino, the number of leptons becomes six. 

The third family consists of particles called hadrons (from the Greek 
for “large”, “massive”). Hadrons are numerous: several hundred of 
them are known. 

The hadron family divides into two subfamilies: the mesons and 
the baryons. The mesons either have no spin or have integer spins, 
whereas the baryons have half-integral spins. Among the hadrons 
(both mesons and baryons) there are many particles that decay quickly, 
their lifetime is only 10 -22 s to 10 -23 s; these particles are called reso¬ 
nances. If we don’t include the resonances, then up until 1974 physi¬ 
cists have identified 14 hadrons, among them five mesons and nine 



The five mesons include two pions (the neutral pion 7T° and the 
positively charged pion n + ), the kaons (the positively charged kaon 
K + and the neutral kaon K°), the neutral eta meson All of these 
mesons are spinless particles (s = 0). Their masses are as follows: 

7i 0 = 264m, n + = 273m, K + = 966m, K° = 974m, rf = 1074m. 

The nine baryons include the nucleons (the proton p and the neutron 
n), the neutral lambda hyperon A 0 , the sigma hyperons (the neutral 
one E° and the charged ones E + and E ), the xi hyperons (the neutral 
one 5° and the negatively charged one S“), the negatively charged 
omega hyperon f l~. The omega hyperon has the spin 3/2; the other 
baryons have s = 1 /2. The masses of the above baryons are as follows: 
p = 1836.1m, n = 1838.6m, A 0 = 2183m, E+ = 2328m, E° = 
2334m, E“ = 2343m, S° - 2573m, E~ = 2586 m, IT = 3273m. 

Note that the particles of one group in the meson or baryon sub¬ 
family have similar masses. So with pions they only differ by 3 per 
cent, with kaons, by 0.7 per cent, with nucleons, only by 0.14 per cent. 
The members of a group mainly differ by the electric charge. In this 
connection the mesons or baryons of one group can be viewed as one 
particle characterized by several charge states. With this approach the 
baryons E + E° E“ are one particle (sigma hyperon) that may be in 
three different charge states. (The small difference in masses of the 
charge components is due to the difference in the sign of the electric 
charge.) Pion, kaon, nucleon, and xi hyperon have two charge states 
each. Figure 104 presents all the above elementary particles, with the 
exception of those with zero rest mass. The vertical axis in the figure 
shows the mass; the horizontal one, the electric charge. The yellow 
rectangles combine those mesons or baryons that may be treated as 
different charge states of a particle. 

61 Particles and Antiparticles 

The diagram of Figure 104 contains 17 particles. As a matter of fact, the 
number of particles to be considered is to be doubled. The fact is that 
each particle, with the exception of the photon, the neutral pion, and 














© « 








i t _ 










fib E* SIGMA 



4 A' 









1 n 















-- 1 - 






-/ 0 +/ 

Figure 104: Elementary 
particles showing leptons 
and hadrons. 

the eta meson, corresponds to an antiparticle. So for the electron we 
have its antiparticle, the positron (e + ); there are two antineutrinos, the 
electron one (v e ) and the muon one (v^), and so on. The photon, the 
neutral pion, and the eta meson do not have antiparticles. We can say 
that each of these particles is identical to its antiparticle. Such particles 
are referred to as truly neutral ones. 

Including the antiparticles, the number of particles at hand becomes 
37. The diagram in Figure 105 contains all of these particles, save for 
zero mass ones (photon, two neutrinos and two antineutrinos). The 
particles in the figure are represented by the blue colour, the antipar- 


Figure 105: Elementary par¬ 
ticles showing leptons and 
hadrons and antiparticles. 

tides by the red colour, and the truly neutral particles by the green 



colour. The diagram is far from comprehensive: it does not include 
the short-lived hadrons (resonances) and the hadrons discovered since 
1974 (the so-called charmed hadrons). 

Particles and their antiparticles have equal spins and masses, but the 
signs of their electric charges are opposite. So, unlike the proton, the 
antiproton is negative. But what if a particle is not charged? What is 
the difference, say, between a neutrino and antineutrino, or a neutron 
and antineutron? We will answer the question in more detail later in 
the book, for the moment we will only note that elementary particles 
are characterized not only by electric charge, but also by a number of 
other charges, which will be considered in Chapter 13. Particles and 
antiparticles have opposite charges. 

It is seen in Figure 105 that antiparticles are normally denoted 
by a bar over the symbol for a particle. So A 0 is the antiparticle for 
the lambda hyperon A 0 , or antilambda hyperon. Since the charges 
of a particle and its antiparticle have opposite signs, in a number of 
cases (electron, muon, charged mesons) no bar is used. The electron 
is denoted by e ; the antielectron (positron) by e + . So e + , p + , n~, K 
are symbols for antiparticles, whereas e~, p~, n + , K + are symbols for 
particles. But in the case of the sigma hyperon £ + , we cannot denote 
the antiparticle by £“, because there exists the sigma hyperon £“. It is 
absolutely necessary here to use for the antiparticle the symbol £ + . It 
should be remembered that the antisigma hyperon £ + has a negative 
electric charge. 

But whatever the similarity between a particle and its antiparticle, 
they differ in a fundamental way: particle and antiparticle annihilate 
in merging with the result that several mesons or photons are formed. 
So £ + and £ + are mutually destroyed, whereas the encounter of £ + 
and £“ is quite peaceful. The mutual destruction process (or in the 
language of physics, annihilation) may be an indication of a meeting of 
a neutron and antineutron, and not of two neutrons. 

Let us take some examples of events (reactions) of annihilation of 


particle and antiparticle: 

e + e + -> y + Y 
p+p^>n + +n + n° 
p + p —> n + + n + + n~ + n~ 
p + p —> n + + n + + tt~ + n~ + n° 

The first reaction is the annihilation of an electron and a positron 
yielding two photons. The other three reactions are the various cases 
of annihilation of a proton and an antiproton. 

62 Particles, Antiparticles and Symmetry 

The very fact that antiparticles exist is closely connected with sym¬ 
metry. Without antiparticles the equations of theoretical physics 
describing the various types of elementary particles would be nonin¬ 
variant under the Lorentz transformations, that is, under transfer from 
one inertial frame to another. Put another way, the existence of antipar¬ 
ticles in addition to particles is related to the invariance of the laws of 
physics under transfer from one inertial frame of reference to another. 
This, unfortunately, lies beyond the scope of this book. 

We live in a world of particles, antiparticles being fairly rare visitors. 
One can, however, imagine another world, the one built of antiparticles. 
We will call it the antiworld. Antimatter in the antiworld consists of 
antiatoms and antimolecules. So the antiatom of hydrogen has the 
nucleus of an antiproton, with a positron orbiting in its field. In the 
antiatom of helium, two positrons orbit in the nuclear field, the nucleus 
consisting of two antiprotons and two antineutrons. 

An encounter with the antiworld is a favourite theme of science 
fiction. After many years of flight at a speed near the speed of light a 
spaceship arrives at an unknown planet. The ship starts orbiting the 
planet and the astronauts start studying the planet’s surface, using a 
most advanced optical apparatus that enables them to discern things 
up to one metre across from a height of several hundred kilometres. 
They find that the planet is inhabited and, moreover, the intelligent 
creatures look very much like people. At the same time, the astronauts 



endeavour to contact the dwellers of the planet. After having spent, 
say, a fortnight in orbit, the astronauts become confident that they 
have found a similar civilization which strikingly resembles their own 
terrestrial one. Meanwhile, the space travellers and the extraterres¬ 
trials reach a measure of mutual understanding. The astronauts are 
kindly invited to land on the planet at a specified region. The terres¬ 
trial ship starts its engines and begins descending, approaching the 
lower atmosphere of the hospitable planet. And now a catastrophe 
happens- the shocked extraterrestrials see, high up in the sky where 
the spaceship was just visible, a blinding flash ... 

Today’s science fiction reader or fan has already twigged that the 
hospitable planet is a piece of antiworld. Upon entry into the antimat¬ 
ter atmosphere, the terrestrial spacecraft was immediately destroyed 
due to the annihilation of the particles of the ship with the antiparti¬ 
cles of the atmosphere. 

In this science fiction story there is one moment that is worth ex¬ 
amining. It is not surprising that the astronauts made the fatal mistake: 
the antiworld of the planet must have appeared as natural as the world 
of their Earth. The exchange of information by radio between the 
ship’s crew and the planet could also give no grounds to suspect that 
they belonged to opposite worlds. World and antiworld are absolutely 
symmetrical; the laws of nature are invariant under replacement of all 
the particles by their respective antiparticles. This invariance of the laws 
of physics is normally referred to as charge invariance (or C-invariance). 
It is not surprising then that when viewing from a distance a world 
built of antiprotons, antineutrons, positrons, etc., the crew mistook it 
for a conventional world, the one built of protons, neutrons, electrons, 
etc. By the way, they contacted that world by the agency of photons, 
that is, truly neutral particles, which are the same for both the conven¬ 
tional world and the antiworld. 

But the charge invariance of the laws of nature is not a completely 
rigorous invariance: there are processes in which it does not hold. And 
so, in principle, the space travellers could have made some experiments 
on board their ship to clarify the nature of the world before them to 
determine whether it was a conventional world or an antiworld. We 
are going to take a closer look at this later in the book. 


The symmetry of the physical properties of the 'world and the anti- 
world is combined with the distinct asymmetry of the spatial distribution 
of matter and antimatter. It has been estimated that in our Galaxy, for 
each antiparticle there are more than 10 17 particles. It follows, in par¬ 
ticular, that the probability for a spacecraft to hit upon a star system 
made from antimatter is infinitesimal. 

This asymmetry has not yet been explained. It is to be assumed 
that either the Universe as a whole is charge asymmetric or matter 
and antimatter have spatially separated into isolated regions, which 
interact with one another exceedingly weakly. Both assumptions pose 
a variety of fundamental questions which, so far, scientists have not 
been able to answer. 

63 Neutrino and Antineutrino (Left and Right Helices in the 
World of Elementary Particles) 

The spin of a neutrino is always aligned against (antiparallel to) the 
neutrino’s momentum. This means that if we visualize the spin 
through the rotation about the neutrino’s axis, the latter is always 
aligned with the direction of motion of the particle. If we follow a 
neutrino flying off, it will spin counterclockwise (Figure 106 (a)). The 
antineutrino’s rotation axis is also parallel to the direction of motion. 
Unlike neutrino, however, the receding antineutrino rotates clock¬ 
wise (Figure 106 (b)). In other words, the neutrino can be compared 
to a left-handed helix and the antineutrino to a right-handed helix. 

This holds good both for the electronic and for the muonic neutrino 

It might appear that this model of neutrino (antineutrino) contra¬ 
dicts the principle of invariance with respect to transfers from one 
inertial frame of reference to another one. Suppose that a left-handed 
helix is flying past us. In other words, it is a left-handed helix in the 
laboratory frame of reference. Suppose further that we try to catch up 
with the neutrino in a spacecraft travelling at a velocity higher than 
the neutrino’s velocity. Then in the frame associated with the ship the 
neutrino will now move not away from us but toward us, the coun¬ 
terclockwise rotation of the neutrino will not change in the process. 






Figure 106: The symmetry 
of dodecahedron. 



This means that in the ship’s frame of reference the neutrino will be a 
right-handed helix, not a left- handed one. 

This reasoning is invalid, however. The fact is that in any inertial 
frame, a neutrino travels at the velocity of light (recall that the neu¬ 
trino’s rest mass is zero); therefore the ship cannot have a velocity 
higher than that of the neutrino. And so the neutrino remains a left- 
handed helix in any frame of reference. 

Note. The attentive reader may have noticed that such a model of 
neutrino (antineutrino) does not agree with the invariance of the laws 
of physics under mirror reflection. A mirror reflection turns the left- 
handed helix into a right-handed one, and so a neutrino must turn into 
an antineutrino. The reader is absolutely right. We will be looking at 
this issue in more detail in Chapter 14. 

64 The Instability of Particles 

One of the most important features of elementary particles is that most 
of them are unstable. This means that particles decay spontaneously, 
without any induction, yielding other particles. Exceptions are the 
photon, neutrino, electron, and proton (with respective antiparticles); 
these subatomic particles do not decay, they are stable. 

A free neutron, for instance, decays spontaneously to produce 
three stable particles, namely, a proton, an electron and an electron 
antineutrino 1 : 

Yl —► p ~\~ 6 H - V e 

A positively charged pion decays to yield an antimuon and a muor 

7I + —> )J + + Vjj 

the resultant antimuon decays, in turn, to give a positron, a muon 
antineutrino, and an electron neutrino 

^Free neutrons are not 
stable, but atomic neutrons 
are stable. In unstable nu¬ 
clei, neutron may decay, the 
phenomenon being called 
the p -radioactivity of atomic 

n + —> e + + + v e 


A neutral pion disintegrates into two photons 

^ -»■ r + y 

Figure 108 at the end of the chapter tabulates the principal decay 
schemes for unstable particles. 

The decay of an elementary particle is a phenomenon that is in need 
of some explanation. To begin with, we will consider the lifetime of 
a particle before decay. For definiteness, we will discuss the neutron. 
Suppose that at t = 0 we have «o free neutrons (assuming no » 1). In 
the course of time, the neutrons will become ever fewer due to decay, 
that is, n{t) will be a decreasing exponential function 

n{t) = n o ex P (!) 

(here e = 2.718 ... is the base of the natural logarithm). The curve 
of this function is shown in Figure 1. The constant ?, which has the 
dimensions of time, is called the lifetime of the neutron. This is the 
time during which the number of neutrons will decrease e-fold. For 
neutrons t = 10 3 s. 

It is remarkable that the lifetime t does not characterize the time of 
existence of an individual neutron. An individual neutron may live just 
a minute or a day. It is in principle impossible to foresee the moment 
at which a concrete neutron will disintegrate. It only makes sense 
to speak about the probability of decay. The factor e~^ T on the right- 
hand side of (1) is the factor describing the probability for individual 
neutrons to decay during time t. It is immaterial in that case how long 
the neutron has lived by t = 0; for all the neutrons, the probability 
that they may live for time t is absolutely the same. It can be said that 
neutrons do not age. 

Despite the fact that an individual neutron may in principle live 
indefinitely, the number of neutrons in a large ensemble falls off with 
time following a definite law. Speaking about the lifetime, we do not 
mean the lifetime of an individual neutron, but the time during which 
the total number of neutrons decreases markedly (to be more exact, 



How then do we treat the very act of decay? One should not believe 
that if a neutron decays into a proton, an electron, and an antineutrino, 
it means that before the decay the neutron constituted some combina¬ 
tion of the above particles. The decay of a subatomic particle is by no 
means a decay in the literal sense of the word. This is an act of transfor¬ 
mation of the initial particle into some set of new particles in which the 
initial particle is annihilated and new particles are produced. 

It is worth noting that having discovered the ^-radioactivity of 
atomic nuclei (the emission of electrons by nuclei), scientists decided 
at first that electrons enter the composition of nuclei. It was not until 
some time later that they understood that the electrons of /J-radiation 
are born at the time of decay of neutrons of ^-radioactive nuclei. 

Another argument against the literal interpretation of the term “de¬ 
cay of a particle” is the fact that many particles may decay in different 
ways. So in the overwhelming majority of cases (more than 99 per 
cent), a positive pion decays following the above-mentioned scheme: 

7Z + -A fj + + Vjj 

In other cases, however, it decays differently: 

n + -a e + + v e 

About a half of sigma hyperons E + disintegrate by the scheme 

E+ -*■ p + 71°, 

whereas the other half by 

E + —> n + n + , 

For a specific E + -hyperon, it is impossible to predict how it will disin¬ 
tegrate, let alone the time of the disintegration. 

Let us return to the lifetime of particles. As stated above, for the 
neutron this time is 10 3 s. For comparison we will quote the lifetimes: 
for muons, about 10 -6 s; charged mesons, about 10 -8 s; hyperons, about 
10 -10 s. As compared with neutrons, all these particles are like butter¬ 
flies that live for only one day. And still, in terms of the microworld, 
they must be brought under the heading of long-lived particles. There 


are particles that live far shorter - 10 16 s (the neutral pion) and even 
10 -23 s (the so-called resonances). 

Worthy of special attention are the five subatomic particles that 
are stable and do not decay at all. Three of them (the photon and two 
neutrinos) move with the velocity of light in any inertial frame of 
reference. The infinite lifetime of these particles is, in essence, a conse¬ 
quence of the theory of relativity (see Chapter 8). More surprisingly, 
two particles with nonzero rest masses-electron and proton-also have 
infinite lifetimes. Their stability is based on symmetry, which is here 
expressed by specific conservation laws (see Chapter 13). It is to be 
stressed that the stability of the electron and the proton is crucial for 
the existence of stable atoms, and hence for our entire world. 

65 Inter-conversions of Particles 

In the world of elementary particles a wide variety of interconversions 
occur, as a result of which some particles are destroyed and others 
are born. Decays of unstable particles and annihilations in collisions of 
particles with antiparticles are examples of such interconversions. 

Interconversions may also occur when particles collide with particles. 
The following are some examples of processes occurring when two 
protons collide with each other: 

p+p —> p + n + tt + 

p+p —> p + A°+? + 
p+p->p + E++?° 
p + p —* n + A°+?+ + t r+ 

p +p -* p + S°+?°+?+ 

?+? —>?+?+? + ? 

It can easily be found that the total rest mass of the particles born in 
these processes is larger than the double rest mass of a proton by a 
factor of 1.07, 1.36, 1.4, 1.43, 1.73, and 2, respectively. This suggests 
that in the above processes the kinetic energy of the protons involved 
in the collision must be sufficiently high. This energy goes into the 


production of the difference of the total intrinsic energies between the 
born and annihilated particles 2 . This difference is A me 2 , where Am is 
the difference of the total rest masses of the particles after and before 
the process. By increasing the kinetic energy of the protons, it is pos¬ 
sible to observe the processes in which the number of particles born 
increases. In principle you can conjure up a really fantastic picture: 
two protons collide with enormous energy to produce a galaxy! 

Suppose we wish to “split” protons by bombarding them with 
photons gradually increasing the energy of the latter. Instead of the 
splitting of protons we would observe various interconversions, for 
example the following: 


Y +p —* n + n + 

Y + p p + JT + + 7I~ 

Y +p -> p+p+p 

2 The intrinsic energy of a 
particle is the energy associated 
with its rest mass, that is, the 
energy me 2 . For more details 
see Chapter 13. 

This example demonstrates that interconversions make futile any 
attempts to split some particles by bombarding them with others. In 
actual fact, we observe not the splitting of the particles bombarded but 
the birth of new particles. New particles are born at the expense of the 
energy of the colliding particles. 

Interactions of particles are studied in a chamber in which charged 
particles leave a distinct track. Widely used are chambers filled with 
liquid hydrogen in superheated state. A charged particle passing 
through the chamber causes the hydrogen to boil leaving a clearly 
visible track of small bubbles. Such chambers came to be known as 
bubble chambers. If the chamber is placed in a fairly strong magnetic 
field, the tracks of particles will be curved, and particles with opposite 
signs will curve into opposite directions. 

Shown in Figure 107 (a) is a fascinating photograph of tracks taken 
in 1959 in the new liquid-hydrogen bubble chamber (72 inches). The 
chamber was bombarded by an antiproton beam. The picture shows a 
rare event-an antiproton on colliding with a proton produces a lambda 


Figure 107: (a) A photo¬ 
graph of a bubble chamber 
showing tracks of elemen¬ 
tary particles, (b) Schematic 
diagram showing the 
tracks and identification of 

hyperon and an antilambda hyperon 

p + p -> A 0 + A 0 

The deciphering of this photograph is given in Figure 107 (b). The 
antiproton collided with the proton at point A Two electrically neutral 
(therefore invisible in the picture) particles-A 0 and A 0 were born. At 
point ? the antilambda hyperon disintegrated to yield an antiproton 
and a positively charged pion: 

A 0 —> p + n + 

Notice that the tracks of these particles (p and n + ) diverge, which is 
due to the fact that their charges have opposite signs (the chamber 
had been placed into a magnetic field perpendicular to the plane of the 
photograph). At point C the antiproton collided with the proton, the 



annihilation following the conventional scheme: 

p + p —* n + + n + + n~ + n~ 

The tracks of the pions and the antipions diverge in the photograph. At 
point D the lambda hyperon decayed: 

A 0 —> p + n~ 

Note that the events recorded in the photograph include four gener¬ 
ations of particles. The first generation is represented by the initial 
antiproton; the second one, by the hyperon A 0 and the antihyperon 
A 0 . The third generation particles are the products of disintegration 
of second-generation particles. Lastly, the fourth generation includes 
the particles born as a result of the annihilation of the proton and the 
secondary antiproton. 

Interconversions of elementary particles enable us to gain a better un¬ 
derstanding of the properties of the particles themselves. It is specifically 
these studies that made it possible to establish the existence of two 
types of neutrino (electron and muon neutrinos). Also, these studies 
made significant contributions to our understanding of conservation 
laws and invariance principles. All of these issues will be considered in 
later sections. 

In conclusion, we will provide a table of elementary particles (Fig¬ 
ure 108). Note that the table contains two types of kaons K°, which 
have significantly different lifetimes: short-lived kaons K s and long- 
lived kaons K±. 


Table 1 Elementary Particles 






Particle name 








Decay products 



















































[ft »,r,> 


T * 




£10- 11 








(7 y) 

5T * 

5T _ 



+ / 

2.6 10 * 

A + 




+ / 

1.2 10 ' 

1 ft * >■, t/i * 



K,: 0.9- 
10 10 







fl* I Xx°T®> 





Eta meson 




~10 " 

fyyl (x°x°x°) 









+ / 












Lambda hyperon 






2.5 10 10 

<px) (nx°) 

£ + 




+ / 

0.8 10" 

(px°) (nr*) 










io - H 





v - 




1.5-10 10 



Xi hyperons 






3 10 10 












1.3 10" 

a‘x > a x e i 


(\‘K ) 

Figure 108: Principal de¬ 
cay schemes for unstable 
elementary particles. 

13 Conservation Laws and Par¬ 


66 Conservation of Energy and Momentum in Particle Re¬ 

In relativity theory the energy £ of a particle having a rest mass m and 
momentum p is given by 

E = (1) 

The dependence E(p) is plotted in Figure 108. At zero momentum 
(p = 0) the expression (1) yields the well-known Einstein’s formula 

t = me 

The energy me 2 is the intrinsic energy of a particle (the energy pos¬ 
sessed by the particle in the frame associated with the particle). Sup¬ 
pose that the particle’s momentum is not zero but small as compared 
with me. In that case, the right-hand side of (1) becomes 

The strong hint emerging from 
recent studies of elementary 
particles is that the only 
inhibition imposed upon the 
chaotic flux of events in the 
world of the very small is that 
imposed by the conservation 
laws. Everything that can 
happen without violating a 
conservation law does happen. 
- K. Ford 

Figure 109: Energy mo¬ 
mentum relationship for 
elementary particles. 

me 2 


1 + 

1 ( pc T 

2 V me 2 / 



me 2 

The first term describes the intrinsic energy of the particle, the second 


one is the well-known expression for kinetic energy. At sufficiently 
large momenta (p » me), the expression ( 1 ) becomes 

E ^ pc 

For particles with zero rest mass the relationship E = pc clearly holds 
at any momentum (see the dash line in Figure 109). 

Suppose that a particle with rest mass m\ decays into a particle 
with mass m 2 and a particle with mass m 3 , the energy and momentum 
of these latter being £ 2 , p 2 and £ 3 , p 3 , respectively. In the frame of 
reference associated with the initial particle the laws of conservation of 
energy and momentum have the forms: for energy 

mic 2 = yj(m 2 c 2 ) 2 + (p 2 c) 2 + yj(m 3 c 2 ) 2 + (p 3 c ) 2 
for momentum 

0 = p2 + p3 

It follows that for the decay to take place the following inequality must 

mi > (m 2 + m 3 ) 

Or, in other words, in a decay the total mass of the products must be 
smaller than the rest mass of the initial particle. 

Significantly, in a decay the magnitude of the momentum of a decay¬ 
ing particle is immaterial. No matter how we accelerate, say, a charged 
pion, all the same it can only decay into particles such that their total 
rest mass is smaller than the rest mass of the pion. With collisions the 
situation is different. By increasing the momentum of the particles 
involved in a collision, processes can be realized in which particles 
will be born with the total rest mass larger than that of the colliding 

Consider the process production of an electron-positron pair by 
collision of two photons 

r + y -> e + e+ 

The initial particles here have no rest mass at all, nevertheless their 



collision produces two particles with rest mass. 

Here is an example of radiation-to-matter conversion. 

Let pi and p 2 be the momenta of the photons, and p 3 and p 4 the 
momenta of the electron and the positron. The laws of conservation 
will then read: 

Pic + P 2 C = -\J {me 2 ) 2 + (p 3 c) 2 + yj(mc 2 ) 2 + (p 4 c) 2 
for energy, and 

pl +p2 = p3 + p4 

for momentum. 

It is easily seen that the total energy of the photons must be larger 
than 2me 2 . If the photons with equal momenta collide head on, then 
p 3 + p 4 = 0. This means that an electron-positron pair may have 
essentially zero momenta, and so the required total energy of the 
photons appears to be minimal and equal to 2 me 2 . 

67 The Conservation of Electric Charge and Stability of the 

In all the processes occurring in the world of elementary particles, the 
law of conservation of electric charge holds: the total electric charge of 
the primary particles is exactly equal to the total electric charge of the 
secondary particles. 

The form of symmetry underlying this conservation law is more 
subtle than those concerned with spatial and temporal translations, 
that is, the laws of conservation of energy, momentum, and angular 
momentum But to formulate this principle we would have to resort 
to quantum mechanics. We will, therefore, confine ourselves to the 
remark that underlying the law of conservation of electric charge is the 
symmetry of physical laws with respect to changes in the magnitude of 
the intensity of electric field. The familiar statement that the magnitude 
of a potential has no physical meaning (and it is only the potential 
difference that matters), thus amounts to the statement that electric 
charge is conserved. 


The stability of the electron is one of the most important conse¬ 
quences of the law of conservation of electric charge. Since the elec¬ 
tron is a particle with smallest nonzero rest mass, its decay can only 
give rise to zero rest mass particles (recall that this conclusion follows 
from the laws of conservation of energy and momentum). But all the 
zero-mass particles are electrically neutral. Accordingly, the decay of 
an electron is forbidden by the law of conservation of electric charge. 

68 The Three Conservation Laws and Neutrino 

The existence of the neutrino had been predicted long before it was 
found experimentally, and the prediction was based on energy conserva¬ 

It was found that the energy of an electron produced in the beta 
decay of a nucleus turns out to be different in different decay events 
and at all times it is smaller than half the total energy released in the 
process. Experimenters placed a beta-active sample within a heat- 
insulating lead chamber whose walls did not let in a single electron. 
Accurate measurements have shown that the chamber heats up to a 
lesser degree than might be expected from the heat budget. It was 
suggested that in the beta fissure of nuclei the conservation of energy 
is invalid. The prominent Swiss physicist Wolfgang Pauli (1900-1958) 
came up with another explanation of the enigma of beta decay. Assum¬ 
ing that the law of conservation of energy is also valid in the microworld, 
in 1930 he came to the conclusion that in beta decay in addition to 
the electron some neutral particle is produced, which could not be 
recorded by the experimental set-up. It was this particle that carries 
away the energy that is obtained if from the energy liberated in the 
process we subtract the energy carried away by the electron. The fa¬ 
mous Italian physicist Enriko Fermi (1901-1954), the author of the 
theory of beta decay, christened this neutral particle “neutrino”, which 
is the Italian for “small neutron”. So in the list of subatomic particles, 
appeared another entry, the neutrino. For a long time this particle had 
actually been a phantom, a particle only deduced from symmetry. 

Neutrino was found experimentally in 1956. Frankly, by that time 
nobody questioned the very fact of the existence of the “elusive” neu- 


trino. This was because the neutrino hypothesis was based not only on 
the law of conservation of energy but also on the laws of conservation 
of momentum and angular momentum. Let us take a simple example, 
the beta decay of a neutron; the neutron decays following the equation 
mentioned above 1 

n —» p + e~ + v e 

We will consider this decay in the frame of reference associated with 
the neutron. For sufficiently slow neutrons, this frame is essentially the 
lab frame. If this decay did not produce an antineutrino, then from mo¬ 
mentum conservation, the proton and the electron would have to move 
off in opposite directions, as is shown in Figure 110 (a). It was found, 
however, that the proton and the electron scatter in a different manner 
( Figure 110 (b)). This suggests that another particle is produced here 
whose momentum determined the observed picture (see Figure 110 (c)). 
The law of conservation of momentum dictates that the vector sum of 
the. three momenta (the proton, electron, and third particle, that is, the 
antineutrino) is zero in the frame of reference associated with the neu¬ 
tron. The angular momentum must also be conserved here. Before the 

' Note that the decay 
scheme includes not a neutrino 
but an antineutrino (electron). 
This was revealed later. 

Figure 110: Neutrino cre¬ 
ation during neutron decay. 

decay the angular momentum was determined by the neutron’s spin 
s = 1/2. If the decay products only consisted of a proton (s = 1/2) and 
an electron (s = 1/2), the law of conservation of angular momentum 
would not hold in this case. In fact, the proton and the electron may 


have either parallel or antiparallel spins, and so the total spin may be 
either 1 or 0, and by no means 1 /2. For the angular momentum to be 
conserved, another particle is necessary, such that its spin is 1/2. This 
explains Figure 111, where the momentum vectors are shown in blue 
and the angular momentum vectors are shown in red (notice that the 
antineutrino is a right-handed helix). 

Figure 111: Angular mo¬ 
mentum conservation 
during neutrino creation. 

69 Experimental Determination of Electron Antineutrino 

In 1956 the American physicists Cohen and Reines obtained direct 
experimental proof of the existence of the antineutrino (and hence 
neutrino). They used the process 

v e + p —» n + e + 

This sort of a process is highly unlikely. It is only known that the neu¬ 
trino and the antineutrino interact with matter extremely weakly-they, 
essentially unhindered, pierce all barriers, the entire globe and even the 
Sun. To record such a particle it is necessary to use a sufficiently dense 
beam and advanced experimental techniques. 

Cohen and Reines used an antineutrino beam from a high-capacity 
nuclear reactor. Into this beam they placed a special-purpose detector 
consisting of several layers of water separated by a scintillator capa¬ 
ble of detecting individual photons (Figure 112 (a)). There is a tiny 
probability for an antineutrino to interact with a proton in the water 

Amineuirino flux 

Figure 112: Discovering the 
neutrinos experimentally. 



to cause the above process that yields a neutron and a positron. It 
should be stressed that these events are, if any, exceedingly rare. It is 
for this reason, that the scintillator was utilized. A produced positron 
is stopped and undergoes an annihilation with an atomic electron 

e + + e~ —> y + Y 

As a result, two photons are created that fly off in opposite directions 
so that they can be recorded simultaneously in two adjacent scintillator 
layers (Figure 112 (b)). As for the neutron, it diffuses in a layer of water 
for a relatively long time (about 10 -6 s), until it is caught by a cadmium 
nucleus (some cadmium is added to the water). After having absorbed 
the neutron, the cadmium nucleus emits a photon or photons, which 
are caught by a scintillator layer (Figure 112 (b)). Consequently, the 
scintillator must respond to an antineutrino-proton collision with three 
pulses: first a pair of simultaneous pulses are recorded using adjacent 
scintillators, and then, in about 10 -6 s, another pulse. Both of the first 
two pulses correspond to a 0.5 MeV photon, and the third pulse to a 
10 MeV photon. In Cohen and Reines’s experiment this specific picture 
of pulses was actually observed (approximately three times per hour). 
Thereby the existence of the antineutrino was proved. 

At the time of the experiment just described, nobody suspected 
that there are two forms of neutrino (antineutrino), the electron and 
the muon neutrino. It is only natural that nobody knew that it was 
the electron neutrino that was found. The second type of the neu¬ 
trino (muon) was discovered in 1962 by a group of experimentalists at 
Columbia University, USA. After it had been established that there ex¬ 
ist two types of neutrino, two independent specific conservation laws 
were formulated - the conservation of electron and muon numbers. 

70 Electron and Muon Numbers. Electron and Muon Neu¬ 

The electron and muon numbers are specific charges of elementary par¬ 
ticles. They have nothing to do with the electric charge. The electron 
number of the electron and the electron neutrino is assumed to be 1, 
and that of their antiparticles, -1. With all the other particles and an- 


tiparticles the electron number is zero. The muon number of the muon 
and the muon neutrino is 1, that of their antiparticles, —1. With all 
other particles (antiparticles) the muon number is zero. 

In each process the total electron number of the reactants must be 
conserved. It follows, for example, that the creation of an electron must 
be accompanied with the creation of either an electron antineutrino or 
a positron, or with the annihilation of an electron neutrino: 

h —> p + e~ + v e 

Y + Y e ~ + e+ 

n v e —^ p H - 6 

Apart from the electron number, in each process the total muon num¬ 
ber must also be conserved. So the creation of a muon must be accom¬ 
panied by the creation of a muon antineutrino or the annihilation of a 
muon neutrino: 

n -> p +v^ 
n + v^ -r p + p~ 

The conservation of electron and muon numbers in the decay of a 
muon dictates that in addition to an electron, an electron antineutrino 
and a muon neutrino must be produced: 

p~ e~ + v e + Vp 

Specifically, the discovery of conservation of electron and muon num¬ 
bers is associated with the solution of one problem that for a long time 
was unanswered-the so-called p - e — y-problem. It was noted long ago 
that nobody ever observed the muon reaction p~ —> e~ + y. The answer 
came from the laws of conservation of electron and muon numbers: 
in this scheme neither the electron nor the muon number is conserved. 
This reaction thus appears to be forbidden twice (by two conservation 

The existence of two independent conservation laws (for electron and 
muon numbers) is closely related to the existence of two different neu¬ 
trinos (antineutrinos). The electron neutrino (antineutrino) takes part 
in processes where an electron or positron is created or annihilated, 


whereas a muon neutrino (antineutrino) takes part in other processes, 
ones in which a muon or an antimuon is created or annihilated. 2 

That the electron and the muon neutrinos are two absolutely differ¬ 
ent particles was established in 1962 in the aforementioned experiment 
at Columbia University. A powerful beam of high-energy protons was 
directed from an accelerator at a target to produce a beam of pions n + 
and antipions n ~. Pions and antipions are known to decay in 99 per 
cent of all cases into antimuons and muons \T , and hence muon 
neutrinos and antineutrinos: 

2 If the existence of the third 
type of neutrino (tauon) will be 
established experimentally, we 
will have to introduce another 
conserved number, the tauon 

7Z + —> (J + + Vjj 

n + —> + v„ 

With the help of a 10-m thick iron wall, all the particles were trapped 
except for the above-mentioned muon neutrinos and antineutrinos. 
Into a flux of these neutrinos and antineutrinos a hydrogen-containing 
target was placed to study the products of processes occurring in rare 
collisions of a muon antineutrino with a proton. If the muon antineu¬ 
trino were identical with the electron one, the following processes 
might be observed with equal probability: 

Vp + P -»■ n + p + 
v e T p —> n + e + 

This was not the case, however. During 300 hours the experimenters 
recorded 30 tracks of antimuons fi + and found no sign of positrons. 
Thereby the existence of two different types of neutrino and antineu¬ 
trino was clearly established. 

71 The Baryon Number and Stability of the Proton 

We will now turn to the various processes involving baryons but 
not antibaryons. It was found that in these processes the number of 
baryons is always unchanged. For example, in the process 

n —> p + e~ + v e 


the baryon n decays just to give birth to another baryon p. In the 

p+p-r n + A° +K + tt + 

two baryons p annihilate and two baryons ( n and A 0 ) are produced. 
Consequently, the annihilation of some baryons is compensated for by 
the production of others, the total number of baryons remaining the 

Let us assign a specific number to each baryon, which is assumed to 
be unity. We will call it the baryon number. Photons, both neutrinos, 
an electron, muon and mesons have no such number (or it is zero). 

The fact that the number of baryons in various processes remains 
unchanged can be viewed clearly as the law of conservation of baryon 

Further, we will remember that there are antibaryons. If the baryon 
number of the baryons is unity, then for antibaryons it must be set 
at ?1 (recall that the antiparticles have the opposite signs of all the 
numbers). From baryon number conservation, processes must occur 
with pair creation or annihilation of an antibaryon and baryon. Such 
processes are actually observed. For example, 

p + p —> 7T + + n~ + 71° 

p + p —> A 0 + Lambda. 0 
p+p -rp+p+p+p 

The law of conservation of baryon number is at present considered to 
be well established. According to that law, in any process the differ¬ 
ence between the number of baryons and the number of antibaryons 
remains unchanged. Accordingly, for the entire Universe too, the dif¬ 
ference between the total number of baryons and the total number 
of antibaryons is unchanged It is worth noting here that by the law 
of conservation of electron number the difference between the number 
of particles of an electron family and the number of their antiparticles 
remains unchanged. In consequence, the law of conservation of muon 
number leaves unchanged the difference between the number of particles 
of the muon family and the number of their antiparticles. Curiously, 
there is no similar law for photons or mesons. 


If the stability of the electron stems from the law of conservation 
of electric charge, then the stability of the proton follows from the law 
of conservation ofbaryon number. Among the baryons, the proton has 
the smallest mass, therefore there cannot be any baryons among its 
decay product If this were not the case, the decay of a proton would 
lead to an uncompensated annihilation of a baryon. But such a process 
is prohibited by the law of conservation of baryon number. 

In the rather large family of baryons only the proton is a stable par¬ 
ticle. All the remaining baryons (neutron and hyperons) are unstable; 
each of them disintegrates to produce a lighter baryon. As regards 
the proton, according to the law of conservation of baryon number, it 
simply cannot decay into anything. The mesons, muon and electron, 
which lie along the mass scale, have a zero baryon number. 

The world around us (and hence we as well) could not exist if pro¬ 
tons and electrons were not stable. This goes to prove the exclusive 
role of conservation laws, specifically the law of conservation of 
baryon number and electric charge. 

72 Discrete Symmetries. CPT -Invariance 

We have considered seven conservation laws: for energy, momen¬ 
tum, angular momentum, electric charge and three numbers (electron, 
muon, and baryon). The first four are related to the well-known prop¬ 
erties of the symmetry of physical laws. One can expect that the re¬ 
maining conservation laws (for electron, muon, and baryon numbers) 
express some symmetries, but we do not yet know which symmetries. 

The above conservation laws are said to be absolute laws: they hold 
true at all times, in all the transformations of elementary particles. 3 To 
the seven absolute conservation laws we must add the eighth one- the 
law of conservation of CPT-symmetry. 

CPT-symmetry is a conservation law for the combination of three 
sufficiently clear symmetries: the symmetry with respect to the re¬ 
placement of all particles by their respective antiparticles (the so-called 
charge-conjugation symmetry or C-invariance), mirror symmetry (gen¬ 
erally called P-invariance), symmetry in time, that is, symmetry with 


3 Later in the book we will 
get acquainted with the laws 
that are not absolute. They hold 
in some transformations and do 
not in others. 


respect to time reversal (the so-called T-invariance). 

CPT-symmetry means that if all particles were simultaneously 
replaced by appropriate antiparticles and mirror-reversed, and then 
the sense of time were reversed, the laws of physics would remain 
unchanged and all the physical processes would proceed as before. 

This statement is usually referred to as the CPT-theorem. The CPT- 
theorem is so firmly entrenched in the foundations of physics that, if it 
were to turn out not to be true, physical theory would be in shambles. 
“All hell will break loose”, was how Abraham Pais once expressed it. 

Turning to the symmetries underlying this conservation law, we 
note first of all that the symmetries we dealt with earlier in the book 
are associated with the conservation of energy, momentum, angular 
momentum, and electric charge, and they are continuous symmetries. 

In all of them a change that leaves the laws of physics unchanged can 
be made arbitrarily small; this change can be brought about smoothly, 
gradually, in other words, continuously. Apart from continuous sym¬ 
metries there exist other symmetries that are considered in relation 
to changes that inherently cannot be continuous. They are associated 
with jumps, or are said to be discrete. All three symmetries involved 
in the CPT-theorem are discrete. It is clear that an object cannot be 
partially reflected in a mirror; the reflection is either possible or not. 
Likewise, one cannot replace a proton by an antiproton partially, the 
replacement either occurs or does not. The same is true of the reversal 
of the sense of time. 

C-symmetry was covered in Chapter 12, P-symmetry in Chapter 9. 
We only have to discuss T-symmetry. 

Everyday experiment indicates that time flows in one direction 
only. It might appear that in the world around us there is no invariance 
under time reversal. If we run a motion-picture film backward, events 
on it will look grotesque: people will walk backwards, the fragments of 
a broken vase will come together to form a whole vase, a swimmer will 
not dive into the water but, on the contrary, he will be expelled from 
the water feet first, and so forth. 

An examination of transformations of subatomic particles shows, 
however, that both senses of time are physically equivalent. So, besides 



the process e~ + e + —s > y + y the reversed process is possible: y + Y > 
e + + e~. And apart from the process p + p —> p + n + n + the reversed 
process p + n + n + -a p + p is possible. True, “possible” by no means 
implies “equiprobable”. In the last example, the reversed process is less 
likely than the direct one. This stems from the small probability of the 
meeting of three particles at once. 

It is the low probability of reversed processes that accounts for the 
apparent noninvariance of physical laws under mental time reversal. 
The laws of physics as such are symmetrical with respect to the future 
and the past. But for any specific chain of events, a certain sequence, 
as a rule, turns out to be more likely than the opposite order. 

Returning to CPT-symmetry, we will note that this means CPT- 
invariance, that is, invariance under three simultaneous operations: 
replacement of particles by antiparticles, mirror reflection and time 
reversal. In this connection, recall that energy can be defined as a 
quantity whose conservation is a consequence of invariance under a 
shift in time, momentum is a quantity conserved as a consequence of 
invariance under spatial translations, angular momentum is a quantity 
conserved as a consequence of invariance under spatial rotation (see 
Chapter 11). CPT-symmetry is the product of three qtumtities: charge- 
conjugation parity (C-parity), space parity (P-parity)f and time parity 
(T-parity). Each of these is a conserved quantity that corresponds to 
some discrete symmetry, namely charge conjugation, mirror reflection, 
and time reversal. 

It is only natural that the issue of conservation laws for charge- 
conjugation parity, space parity, and time parity presents itself. It has 
been found recently that, unlike CPT-symmetry, these conservation 
laws are not absolute. This interesting issue will be the subject of the 
following chapter. 

^ Space parity is often 
referred to as just parity. 

14 The Ozma Problem 

73 What Is the Ozma Problem? 

In 1900 the American writer of children’s books, Lyman Baum, wrote 
his famous book Wonderful Wizard of Oz. The land of Oz was ruled by 
prince Ozma. Another of Baum’s characters was a servant called the 
Long Eared Hearer who could hear sounds thousands of miles away. In 
1960 the American astronomer Frank Drake started a project of using a 
powerful radio telescope to search for radio messages from the Galaxy 
in hope of picking up signals from intelligent inhabitants of distant 
planets. A long-time admirer of Baum and his Oz books, he called his 
project Ozma and his radio telescope the Long Eared Hearer. This story 
is behind the term the ‘Ozma problem’ suggested by Gardner. 

Suppose that we exchange radio messages with inhabitants of some 
distant planet. Our signals are certain coded pulses, that is, sequences 
of pulses of various intensities. Using the universal laws of mathe¬ 
matical logic as well as the laws of physics which apply to the entire 
Universe, we can arm ourselves with patience and achieve a measure 
of understanding with extraterrestrials. If, for instance, we were to 
transmit a sequence of numbers representing the masses of nuclei of 
helium, lithium, beryllium, boron, carbon, etc., divided by the proton 
mass, we could expect that the extraterrestrials would guess that this 
sequence describes the periodic system of elements. After all, the ra¬ 
tios of nuclear masses to proton mass are the same throughout the 

Assume we have already estab¬ 
lished fluent communication 
with Planet X. How can we 
communicate to Planet X our 
meaning of left and right? 
Although an old problem, it 
has not yet been given a name. 
I propose to call it the Ozma 
problem. - M. Gardner 

It is rather tempting to convey to the inhabitants of other planets 


visual images in the form of plane (two-dimensional) figures. Suppose 
we send out a sequence of pulses that is a coded description of the 
simple figure shown in Figure 113 (a) - a rectangular figure open in its 
right side. To begin with, we ask our extraterrestrial correspondent 
to prepare a rectangle divided into twenty square units-five lines with 
four units per line. By scanning the figure from top to bottom, left to 
right (in accordance with the numbering of units in Figure 113 (a)), 
we send out the sequence of pulses shown in Figure 113 (b): a more 
intensive pulse corresponds to a darker unit. We ask the distant cor¬ 
respondent to copy our operations on his rectangle: to scan the units 
line-by-line left-to- right and colour them according to the sequence of 
pulses transmitted. 

And here emerges a fundamental problem: our correspondent has 
no idea as to what we understand by left- and right-handedness, and 
so he does not understand what is meant by “to scan a line from left 
to right”. If he hits upon the right direction of scanning, he will clearly 
end up with a contour with a gap in the right side. If he scans the lines 
in the opposite direction, he will end up with a contour with a gap 
in the left side, not in the right one (Figure 113 (c)). It is unknown in 
what direction the extraterrestrial will actually scan his rectangle, and 
so it is unknown which (left or right) figure he obtained. To be sure, 





























12 11 











1 2 3 4 5 6 7 8 9 10 11 12 13 14 IS 16 17 18 19 20 

Figure 113: Asking extra¬ 
terrestrials for their sense of 
left or right. 

to explain our concept of handedness it would be more convenient to 
transmit some kind of object that possesses reflection asymmetry, for 



instance, a right screw. This is, however, absolutely impossible, since 
we can only make use of radio communication. He might be asked to 
look at some asymmetric constellation in the skies. But constellations 
do not look the same when observed from Earth and when observed 
from some distant planet. 

There is thus no asymmetric object, no asymmetric structure which 
we could examine together with our correspondent from space. And 
so the question presents itself: under these conditions, is there any way 
of getting across to the planet our concept of left and right? It is this that 
is known as the Ozma problem. It is one of the most challenging and 
exciting problems of communication theory. This was conceived long 
before humanity began speculating about contacts with extraterrestrial 

74 The Ozma Problem Before 1956 

If we had to explain the meaning of left and right to an Earth dweller, 
not an extraterrestrial, we would only have to say that rotation from 
left to right corresponds to the motion of the hands of a clock. But for 
an extraterrestrial this explanation will not do. There is no knowing 
the direction in which the hands (if any) of an extraterrestrial clock 

Many natural compounds are known that turn the polarization 
plane of a light beam passed through them always in the same direc¬ 
tion: to the right or to the left. This is because some compounds occur 
naturally on Earth only in the form of certain (left or right) stereoiso¬ 
mers. One can expect, however, that under extraterrestrial conditions 
these compounds occur in the form of other stereoisomers than those 
of Earth. 

The animate world abounds in helices of either handedness (see 
Chapter 7). A wide variety of biological spirals are, however, of no 
help here. After all, the fact that all living things on Earth have their 
DNA molecules twisted only to the right, by no means suggests that in 
extraterrestrial beings they form right helices also. 

The many manifestations of mirror asymmetry are not sufficient to 


solve the Ozma problem. It is necessary for the vertical asymmetry to 
manifest itself in the very laws of nature. Physicists have long thought 
that all the natural laws, without exception, are invariant under mir¬ 
ror reflection. This amounts to admitting that the Ozma problem is 
insolvable in principle. 

Handling one problem in particle physics, the American physicists 
Lee and Yang in 1956 put forward the hypothesis that space parity fails 
in processes of particle decay. Chien-Shiung Wu staged an experiment 
to test the Lee-Yang hypothesis. The upheaval came on 15 January 
1957 when it was reported that in particle decays the laws of nature are 
not invariant under reflection. 

75 The Mirror Asymmetry of Beta-Decay Processes 

The Wu experiment studied the beta-decay of radioactive 

60 /~\ 60 \t' 60 | — | — 

Co : Co > Ni : +e + v e 

Let ? be the intrinsic (spin) angular momentum of the cobalt nucleus, 
p e is the momentum of the electron born in the decay, 8 is the angle 
between ? andp e (Figure 114 (a)). Figure 114 (b) depicts the reflection 
of the process of Figure 114 (a) in mirror S. Recall that the angular mo¬ 
mentum ? is an axial vector, and the momentum p e is a polar vector; 
therefore, ? and pe are reflected in a different way (see Chapter 10 and 
Figure 97). As a result, the angle between the vectors changes. If be¬ 
fore the reflection it was 8, then after the reflection it will be 180° - 8 
(Figure 114 (b)). 

The idea of the Wu experiment is fairly simple. In the decay of a 
Co 60 nucleus, an electron may fly off at any angle 8 to the direction of 
the nuclear angular momentum. If the beta-decay process is invariant 
under reflection, then the probability for the electron to be shot out at 
angle 8 must equal that for 180° — 8, since the processes in which an 
electron is emitted at 8 and at 8 are mirror reflections of each other. If 
these probabilities turn out to be different, then we have a violation of 
the invariance of the process under mirror reflection. Accordingly, it is 
necessary to measure and compare the above-mentioned probabilities. 


Figure 114: The mirror 
asymmetry in /1-decay of 
Cobalt (Co 60 ). 



To measure the probability for the electron to be shot out at one 
angle or another, it is necessary to consider the decay of a sufficiently 
large number of Co 60 nuclei. Under normal conditions spin momenta 
of nuclei are oriented chaotically; in this case it was only required 
that for the majority of nuclei in a sample these momenta be oriented 
in a certain direction, ft was this that constituted the main practical 
difficulty. To this end, Madam Wu cooled the cobalt sample nearly to 
the absolute zero of temperature (T < 0.03 K) to reduce all the joggling 
of its molecules caused by heat and placed it in a strong magnetic field. 
As a result, most nuclei in the sample aligned with the applied field, ft 
then remained only to measure and compare the numbers of electrons 
emitted along and opposite the field (Ni and IV 2 , respectively). If there 
was mirror symmetry, these numbers had to be equal. However, the 
experiment indicated convincingly that N 2 > IVi. It turned out that the 
probability of an electron to be shot out against the nuclear spin is larger 
than along this direction. Thereby it was proved that the beta-decay of 
Co 60 nuclei has no mirror asymmetry. Later, other experiments were 
carried out to study other beta-decay processes. In all of them the 
invariance of natural laws under mirror reflection (or P-invariance) 
was found to be violated. 

76 The Mirror Asymmetry in Decay Processes and the Ozma 

Now, it seemed the Ozma problem was solved. So, to explain to an 
extraterrestrial the meaning of left and right, we can do the following. 

We may ask him to manufacture a solenoid, place into it a cooled 
sample of radioactive cobalt and to count the electrons emerging from 
either solenoid end. We will then ask our space correspondent to align 
the solenoid so that his eye looks along its axis in the direction of the 
maximal emission of electrons (Figure 115). In this case, the direction 
of the motion of electrons along the coils will be from left to right (as 
the hands of a terrestrial clock). 

Note that here the direction of motion of conditional positive 
charges in the conductor will correspond to the direction from right 
to left, and 


according to the right-handed screw, the ordering magnetic field 
(and hence the vector of the spin moment of cobalt nuclei) will be 
directed to the observer. 

77 The Fall of Charge-Conjugation Symmetry 

CPT invariance (see Chapter 13) suggests that if any of the three invari¬ 
ances (here P-invariance) is violated, then at least one more invariance 
must be violated as well. It turned out that apart from mirror symme¬ 
try, charge-conjugation symmetry (C-invariance ) must also be violated. 

In other words, nature’s laws show noninvariance not only under re¬ 
placement of left to right, but also under replacement of particles by 

We will illustrate this by turning to the case of muon neutrino 
and muon antineutrino. As noted (Chapter 12), the neutrino is like a 
left- handed helix, and the antineutrino, like a right-handed one. It is 
easily seen that this model of neutrino presupposes a violation both of 
P-invariance and C-invariance. 

We will write the decay process for a pion n + in the form that takes 
into account the left-handedness of neutrino: 

n + -» + VfjL ( 1 ) 

Since in the decay of a pion the antimuon and the neutrino are shot out 
in opposite directions (in the rest frame for the pion), then because of 
angular momentum conservation, both particles must appear as helices 
of the same handedness-in this case as left-handed helices (Figure 116). 
If the process possessed a mirror symmetry, then apart from the decay 
according to (1), since in mirror reflection left-handed helices turn into 

Figure 115: Solving the 
left-right problem by look¬ 
ing at maximal emission of 


Figure 116: The decay of a 
pion and its results. 



right-handed ones, we would also have the decay process 

n~ -> //+ + V^R 

But such a decay is impossible, since the neutrino may only be a left- 
handed helix. If the charge-conjugation symmetry were not violated, 
then in addition to (1) we would have 

hi + VrR 

which is also impossible, since the antineutrino is a right-handed helix. 
Curiously enough, if we carry out a mirror reflection and a particle- 
to-antiparticle replacement simultaneously, then instead of (1) we will 
have the decays 

-» Pr + VrR 

which are actually observed. This example may be used as an illustra¬ 
tion of the interesting idea put forward in 1957 by the Soviet physicist 
L. Landau (1908-1968) and independently by Lee and Yang. They sug¬ 
gested that the so-called combined parity (CP-parity or a product of C- 
and P-parities) is conserved. 

78 Combined Parity 

Let us return to the beta-decay of Cobalt-60 nuclei in an external mag¬ 
netic field. It was shown in Figure 114 how the process is affected by 
mirror reflection. In addition to mirror reflection, Figure 117 also takes 
into account the replacement of the particles by their antiparticles. 

This figure includes four positions. A is the initial position in which the 
spin moment of the cobalt nucleus ( vecM ) is aligned with the external 
magnetic field ( H is the magnetic field intensity). B is the reflection of 
A in plane S; this leaves the direction of vecM, just like that of vecH 
(both are axial vectors), unchanged, whereas the direction of p e ( po¬ 
lar vector) changes. As a result the angle 0 between vecM and p e in 
the initial position becomes 18O“?0 in B. We have already discussed 
this. Now suppose that we replace the particles by their respective 
antiparticles: the cobalt nuclei are replaced by cobalt antinuclei, the 
electrons are replaced by positrons. Unlike a nucleus, an antinucleus 
has a negative electric charge, and so an external magnetic field will 


Figure 117: Looking at 
/5-decay with antiparticles 
and mirror symmetry. 

align the spin moments vecM in the opposite direction to vecH (see ? 
in Figure 117) Reflecting in S gives D. 

It is easily seen that if we carry out simultaneously the two trans¬ 
formations - mirror reflection and particle-to-antiparticle replacement - 
we obtain either the transition A = D or B = C. This leaves the angle 
between vecM and p e unchanged, or leaves invariant the decay process 
in question. This invariance is known as combined or CP-invariance. 
Accordingly, they attest to the conservation of combined parity (CP- 

Conservation of combined parity implies that the laws of nature 
remain unchanged not when we venture into the looking-glass land, 
and not when we venture into the antiworld, but when we cross over 
into the looking-glass antiworld. Put another way, the laws of physics 
turn out to be symmetrical under reflection in that imagined mirror, 
which at the same time effects a replacement of particles by their 
respective antiparticles (and antiparticles by particles) In the words of 
the Soviet physicist Ya. Smorodinsky, 

some test will be passed with the same result both by a left- 
handed screw made from matter and a right-handed helix made 



from antimatter. 

From the conservation of combined parity, in decay experiments the 
mirror asymmetry comes from the fact that in our comer of the Universe 
particles outnumber antiparticles markedly. Our world can be said to 
be asymmetrical with respect to mirror reflection only because it is 
asymmetrical in terms of the density of particles and antiparticles. 
Imagine that in the world the density of particles and antiparticles 
were the same (one has to suppose at the same time that the processes 
of annihilation of particles and antiparticles are forbidden in some 
sort of way). If that had been the case, Wu would have obtained a 
symmetrical result. In fact, in an external magnetic field half the nuclei 
(neutrons and protons) in the cooled sample of cobalt (anticobalt) 
would tend to align with the field, and half the nuclei (antineutrons 
and antiprotons) against the field. In this case, we would have an equal 
mixture of positions A and C in Figure 117. Clearly, reflecting yields 
the same mixture. 

79 Combined Parity and the Ozma Problem 

Since the laws of nature do not enable us to distinguish a left-handed 
helix of matter and a right-handed helix of antimatter, this suggests 
that the violation of spatial parity in decays of particles by no means 
solves the Ozma problem. It is of no help to attempt to use the Wu 
experiment to explain to an extraterrestrial our understanding of 
left and right, if we do not know a priori which he is made of, matter 
or antimatter. If he lives in an antiworld, then by repeating the Wu 
experiment and using our explanations, he would consider right what 
we consider left. The fact is that in an antiworld the solenoid coils 
carry positrons, not electrons, the electric charge of nuclei is negative, 
not positive. 

We can imagine the following fantastic situation. We have agreed 
with our extraterrestrial to meet in space and set out to our rendezvous. 
And so, having put on our space-suits, we walk out of our ships and 
move toward each other. You extend your right hand and suddenly you 
see that he, aware of the terrestrial custom of handshaking, extends his 
left hand, not his right one. Do not touch it because you are facing a 


dweller of an antiworld. 

To sum up, the conservation of combined parity gives the Ozma 
problem a new perspective. To solve this problem we will have to find 
out beforehand whether our companion is made of matter or antimatter. 

80 The Solution to the Ozma Problem 

To understand with which world (a conventional or antiworld) we are 
in contact, apart from radio signals we can make use of the neutrino 
communication channel. Our Sun is a source of neutrino; that is. of 
left- handed helices. It is, therefore, in principle sufficient to send to 
our distant correspondent a solar neutrino and ask him to compare its 
handedness with that of the neutrino sent out by the luminary in the 
world of the extraterrestrial. Unfortunately, we at present cannot even 
think of a way of sending our neutrino to a distant planet. 

And still there is a solution to the Ozma problem. Leaving out 
details, which can be far too complicated for this book, we note that 
physicists have revealed that in one decay CP-parity is not conserved. 
Considering CPT-invariance, this means that in this process T-parity 
must also not be conserved (that is, the symmetry with respect to time 
reversal must be violated). As it has been noted above, there are two 
types of neutral kaons - long-lived kaons ( Ki ) and short-lived kaons 
( K s ). The former live about 10 -8 s, the latter, 10 -10 s. In accordance 
with the conservation of combined parity, kaons Xi decay into three 
pions, and kaons Ks into two pions (the decay schemes are given in the 
table “Elementary particles”). In 1964 it was found that every now and 
then (in about one case out of 1000) a kaon K; decays into two pions, 
not three. This suggested that for kaons the law of conservation of CP- 
parity is not completely exact. It is this additional violation of symmetry 
in the laws of physics that enables, in principle, the Ozma problem 
to be solved, or rather to find of which material the extraterrestrial 
correspondent consists, matter or antimatter. 

We will have to ask him to observe the process of decay of neutral 
kaons, for example, to measure the density of the kaons K in the beam 
at various distances from the place of origin of the kaons and to report 


the results of his measurements. If the report comes from antiworld, 
then it will differ from the results of our measurements. 

15 Fermions and Bosons 

All the particles in nature are 
either fermions or bosons. 

81 The Periodic Table and the Pauli Principle Thus, there occur only anti¬ 

symmetric or only symmetric 
states of the same particles. 

The advances of atomic physics made it possible to provide a sub¬ 
stantiation for the Periodic System. According to modern thinking, 
as the number of chemical elements increases, the electron shells 
are gradually filled up. The first to be filled up are shells that have 
the strongest binds with the atomic nucleus, that is, the closest to it. 

The first (the closest to the nucleus) shell takes only two electrons to 
be filled, whereas the second and third take eight electrons, and the 
fourth and fifth, eighteen, and so on. We thus obtain the sequence: 

2, 8, 8, 18, 18,... The number of the chemical elements in the first five 
periods of the Periodic Table are exactly like these. 

The first shell may contain only two electrons, the second not more 
than eight Why? The answer to this question is two-fold. First, to 
each shell there corresponds a definite number of possible states of the 
electron-two for the first shell, eight for the second shell, and so on. 

Second, in each state there may be only one electron. This means that in 
the atom you cannot find two electrons with the same characteristics 
that define its state, such as energy, orbital angular momentum, its 
projection, spin projection. And so at least one of these characteristics 
must be different 

This rule that forbids two or more electrons from occupying the 
same state is known as the Pauli exclusion principle. In the simplest 
terms, the principle can be understood as a rule that more than two 
electrons cannot reside in the place, and these electrons must have op- 


posite spins (or must be in different spin states). Like any prohibition 
principle, the Pauli exclusion principle expresses a certain symmetry of 
natural laws. This is the so-called commutative symmetry. 

82 Commutative Symmetry. Fermions and Bosons 

This is the symmetry with respect to commutation of any two particles of 
the same type, specifically as applied to electrons. Physically, nothing 
changes if an electron in state 1 is placed in state 2, and the electron 
in 2 is placed in 1. This symmetry implies that all the electrons in 
the Universe are identical. Also identical are all the protons, all the 
neutrons, all the hydrogen atoms, all the oxygen atoms, and so on. 

Our planet consists of about 10 50 atoms. And this prodigious num¬ 
ber of atoms consists of only several dozen varieties. Furthermore, we 
are confident that the entire Galaxy, the entire Metagalaxy, and the 
entire Universe are constructed of several hundred various building 
blocks. All the chemical elements of the Universe can be arranged as a 
table that has about a hundred cells. 

This, however, does not exhaust the profound meaning of commu¬ 
tative symmetry. This symmetry dictates that all the particles in nature 
are split into two categories that behave differently in an ensemble. One 
category obeys the rule that particles of the same type, for example 
electrons, must avoid one another. According to these rules, identical 
particles may only be alone in a state. All the particles of this category 
are known collectively as fermions (from the name of the Italian physi¬ 
cist Fermi). The other category is governed by precisely opposite rules, 
which not only allow but even dictate that identical particles concen¬ 
trate densely in states. These particles are known as bosons (from the 
name of the Indian physicist Bose). 

There is a connection between the spin s of a particle and its be¬ 
haviour in an ensemble. All the particles with half-integral spin 
(s = 1/2, 3/2,...) are fermions, whereas all the particles without a 
spin or integer spin are bosons. Apart from electrons, fermions include 
other leptons, and barions. All of them obey the exclusion principle for¬ 
mulated above for electrons: if a state is occupied by a fermion, then no 



other fermion of this type may be in this state. Other bosons are photons 
and mesons. In any given state there may be any number of the same 
bosons. Moreover, the more densely is a given state populated the higher 
the probability that other bosons of this type will come to it. 

We have thus, on the one hand, clearly expressed individualism 
(leptons and barions), and, on the other hand, as clearly expressed 
collectivism (photons and mesons). In this connection it is worth 
noting that there is a marked difference between leptons and baryons, 
for one thing, and photons and mesons for the other. This is primarily 
explained by the fact that for the former there exist conservation laws 
according to which the difference between particles and antiparticles 
remains unchanged (conservation laws for electron, muon, and baryon 
numbers), whereas for the latter there are no such laws. 

Taking the Periodic Table as an example, we can clearly see just 
how fundamental is the fact that electrons are fermions. It is exactly 
the fermion nature of electrons that accounts for the peculiarities 
of the population of the atomic levels by electrons. Should the Pauli 
exclusion principle suddenly fail to apply to electrons, then in that 
case and also in all atoms all the electrons would occupy the level with 
the lowest energy. And this would destroy the diversity of elements. 
Note also that it is the fermion nature of electrons that does not allow 
atomic nuclei in a solid to approach one another too closely. At close 
distances the electron shells would overlap, in other words, many 
electrons would be in one place. But this is forbidden by the Pauli 
exclusion principle. As a result, the atoms remain separated by fairly 
decent distances from one another (about 10 -10 m or more), which is 
no less than 10 3 times larger than the size of atomic nuclei. 

83 Symmetrical and Antisymmetrical Wave Functions 

It is to be stressed that a fermion does not admit other sister fermions 
to the state it occupies, just like a boson attracts other bosons. The 
fact is in no way connected with any special forces (repulsion or attrac¬ 
tion) which act between the particles. The fermion or boson nature 
of particles is their fundamental property associated not with force 
interactions but with the symmetry under commutations of particles. 


Therefore, it would be instructive to explain even briefly how com¬ 
mutative symmetry leads to the presence of fermions and bosons in 

Note that in quantum mechanics the state of a microobject is de¬ 
scribed using some function called the wave function. It is significant 
that a physical meaning is attached not to the wave function but to 
its squared modulus, which describes the probability of finding the 
microobject in a given state. 

Let fii (1) be the wave function of particle I in state 1, and fill I) is 
the wave function of particle II in state 2. Consider the microobject 
as a system of particles I and II. The wave function of the microobject 
*3/(7, II) can be expressed as the product of the wave functions of the 
constituent particles. Since the particles are assumed to be identical, 
it is then unknown which of them is really in state 1 and which in 
state 2. We will then have to take into account both f\(I) fail!) and 
fill) M H ) ( as if the particles in the microobject were continually 
exchanging their places). Commutative symmetry requires that the 
wave function 'k(7, II) of the object meet the condition 

|^(7, II)\ 2 = |Ik (77, 7) | 2 

Out of the combinations of the above products of single-particle wave 
functions we can construct two functions that meet this condition 

T> s (7, 77) = Ml) MU ) + MI) MU) 


7,77) = Ml) MU) - MI) MU) 

The first of these is symmetrical, it does not change its sign under com¬ 
mutation: T's(7, 77) = v ks(77, 7) This function describes a system of 
bosons. The second function is antisymmetric, it changes its sign under 
commutation of particles: 'I 1 a (L II) = 7). This function de¬ 

scribes a system of fermions. This can be readily verified. If we assume 
that both particles are in the same state, for example state 1, then it 
follows from the expression for tk^ that this function vanishes. Hence 
this situation is impossible. 



84 The Superfluidity of Liquid Helium. Superconductivity 

At extremely low temperatures (under 2.19 K), He 4 forms a liquid that 
has a highly interesting property: its motion along a narrow capillary 
is characterized by the total absence of viscosity. 

Liquid helium flows undergoing no resistance from the walls. This 
phenomenon is called superfluidity. This remarkable phenomenon 
comes from He 4 atoms being bosons. Note that a system consisting 
of an even number of fermions behaves as a boson. A commutation 
of two such systems amounts to a commutation of an even number of 
fermion pairs. A commutation of each pair of fermions changes the 
sign of the total wave function. If the sign changes an even number of 
times this means that it remains the same. 

At very low temperatures, when the effect of the thermal motion of 
atoms, which scatters them over different states, becomes negligible, 
the rule manifests itself in full measure, which states that bosons must 
concentrate in one state. As a result, all the He 4 atoms concentrate 
in a state characterized by a definite momentum, and so they move 
along the capillary as a single whole. In that case, the liquid displays 
no viscosity: the presence of viscosity requires that different regions of 
the liquid travel with different velocities. 

Unlike He 4 , the atoms of He 3 are fermions. No wonder then, that 
when cooled down to 2 K, helium containing the isotope He 3 does not 
become superfluid. But the superfluid He 3 still exists. It was obtained 
in 1974 at stronger cooling-down to 0.0027 K. At such extremely low 
temperatures, a highly curious effect occurs in helium: He 3 atoms are 
paired. Each pair is clearly a boson. As a result we observe superfluid¬ 

Note one more curious phenomenon - the superconductivity of met¬ 
als It is well-known that at temperatures near absolute zero many 
metals begin to conduct electric current essentially without resistance. 
So, lead goes over to a superconductive state at 7.26 K, tin at 3.69 K, 
aluminium at 1.14 K, zinc at 0.79 K. The phenomenon of superconduc¬ 
tivity can be viewed as the phenomenon of the superfluidity of the 
electron fluid formed in a metal by conduction electrons. The fact is 


that at low temperatures electrons combine to form pairs that behave 
like bosons. This is due to the interaction of electrons with crystal ions. 
It is easily seen that superconductivity and superfluidity are essentially 
the same in nature. They are conditioned by the fact that He 4 atoms, 
pairs of He 3 atoms, and electron pairs are bosons. 

85 Induced Light Generation and Lasers 

In recent years great strides have been made in quantum electronics, a 
new area that came into life when an amazing light generator, the laser, 
was invented in 1960. The principle of the laser is the induced emission 
of light by matter. 

The gist of this phenomenon is as follows. Suppose that in a sub¬ 
stance we have excited atoms, each of which, when jumping back to 
the initial (unexcited) state can emit a photon with a definite energy. 
Under normal conditions atoms accomplish such transitions in an un¬ 
correlated manner, at different times, the photons emitted being sent 
out in different directions. This is called the spontaneous emission of 
light. One can, however, control the process to make the excited atoms 
return to the initial state at the same time, having emitted photons in a 
definite direction. This is the induced emission of light. 

The phenomenon of induced emission is directly related to the 
boson nature of photons. A photon that flies past the excited atoms, 
when its energy is equal to that of the transition in the atoms, will 
initiate massive production of new photons in the same state in which 
it is itself. This phenomenon is used in the laser. 

The Symmetry of Various 

86 The Principal Types of Interactions 

According to modern views, there are four main types of forces in 
nature, or rather four types of interactions - strong (nuclear), electro¬ 
magnetic, weak, and gravitational. 

Nuclear forces strongly bind the neutrons and the protons in atomic 
nuclei. They are responsible for a wide variety of nuclear reactions, 
specifically for those reactions that release energy in the core of a 
nuclear reactor at an atomic power station. Hadrons are responsible 
for strong interactions (baryons and mesons), while leptons do not 
participate in them. 

Electromagnetic interactions, it seems, are now the ones which occur 
most often: we encounter them when studying electric and magnetic 
phenomena and properties of matter and electromagnetic (in particular 
optical) radiation. These interactions determine the structure and 
properties of atoms and molecules. They encompass Coulomb forces, 
the forces acting on a current-bearing conductor, the forces of friction, 
resistance, elasticity, chemical forces, and what not. All the elementary 
particles, except for both neutrino and antineutrino participate in 
electromagnetic interactions. 

The state of the art in par¬ 
ticle physics does not differ 
markedly from what you 
observe when you sit in a 
concert hall just before the 
performance begins. Many 
(but not all) musicians have 
appeared on the scene. They 
are tuning their instruments. 

At times you can hear some in¬ 
teresting musical passages: the 
improvisations are to be heard 
from all directions, sometimes 
wrong notes are also heard. 

The scene is pregnant with the 
expectation of the moment 
when the first sounds of the 
symphony will be heard. 

- A. Pais. 

Weak interactions are predominant in the realm of subatomic par¬ 
ticles. They are responsible for the interactions of particles involving 


neutrino and antineutrino (specifically beta-decay processes). Further¬ 
more, they are involved in neutrinoless decays that are characterized 
by a relatively long lifetime of the decaying particles - about 10 -10 s or 
more. 1 These processes include the decays of kaons and hyperons. 

Gravitational interactions are inherent in all particles, without excep¬ 
tion, but they are of no significance for elementary particles. These in¬ 
teractions only manifest themselves on a sufficiently large scale when 
the masses involved are rather large. They are responsible, say, for the 
attraction of the planets to the Sun or for the falling of an object onto 
the ground. In what follows we will ignore gravitational interactions 

Interactions differ markedly in terms of the forces or energies in¬ 
volved. The strong interaction is about 100 times higher than the 
electromagnetic one and 10 14 higher than the weak one. The stronger 
the interaction, the faster it carries out its task. So the particles called 
resonances, whose decay occurs through nuclear interactions, have a 
lifetime of about 10 -23 s; the neutral pions, which decay through an 
electromagnetic interaction (n° —s > y + y), have a lifetime of 10~ 16 s; de¬ 
cays through a weak interaction have a lifetime of 10 -8 s to 10 -10 s. The 
strong interaction produces fast processes, the weak interaction, slow 
processes. The duration of a process is defined as a quantity that is the 
reciprocal of the probability of the process per unit time. The smaller 
the probability, the slower the process. It is to be recalled in this con¬ 
nection that the neutrino and antineutrino processes characteristic of 
the weak interaction are highly unlikely. 

Unlike the electromagnetic interaction, strong and weak interac¬ 
tions manifest themselves over extremely short distances, or rather, 
have a small range. The strong interaction between two baryons or 
mesons only shows up when the particles approach each other and 
come within a distance of only 10 -15 m. The range of the weak interac¬ 
tion is yet shorter, it is known to be within 10 -19 m. 

The most interesting difference between the types of interactions 
is associated with symmetry. All the interactions of particles are con¬ 
trolled by the absolute conservation laws discussed in Chapter 13. But 
there exist conservation laws (and pertinent symmetry principles), 
which are valid for some interactions and not for other interactions. So 
the laws of conservation of spatial and charge parity (P-invariance and 

1 Speaking about the long 
lifetime of a particle, we com¬ 
pare it with the time during 
which light covers the distance 
of the order of the atomic nu¬ 
cleus itself, that is, / k 10 1 5 m. 
The reference time taken to be 
a “unit time” in the world of 
elementary particles is about 
l/c 1(T 23 s. 



C-in variance) hold for both the electromagnetic and strong interac¬ 
tions, but they do not hold for the weak interaction. There is the rule: 
the stronger an interaction the more symmetrical it is. Put another way 
the weaker an interaction the less it is controlled by conservation laws. 
In the words of Ford, “weaker interactions turn into infringers of the law. 
and the weaker an interaction the more lawlessness”. 

87 Isotopic Invariance of Strong Interactions. The Isotopic 
Spin (Isospin) 

Suppose that all the protons in the atomic nucleus are replaced by 
neutrons, and all the neutrons by protons. The resultant nucleus is 
called the mirror nucleus of the initial nucleus. Mirror nuclei are, for 
example, the pairs: Be 7 and Li 7 , ceB 9 and Be 9 , C 14 and O 14 , and so 
on (Figure 118). It was noticed long ago that pairs of mirror nuclei 
have similar properties: essentially the same nuclear binding energy, 
the similar structure of energy level, the same spin. The similarity of 
mirror nuclei reflects a measure of symmetry of nuclear forces, namely 
the fact that the nuclear forces between two protons are the same as 
the forces between two neutrons. 

Figure 118: Some examples 
of isotpoic doublets. 

This symmetry is a special case of the so-called isotopic invariance. 


The latter means that from the point of view of the strong interaction, 
the system p — p (proton-proton) is identical with not only the system 
n — n (neutron-neutron) but also with the system p—n (proton-neutron). 
Stated another way, the nuclear forces are independent of the electric 
charge of particles. 

Associated with the isotopic invariance of the strong interaction 
is the concept of isotopic spin (isospin). It is worth recalling in this 
connection that the proton and the neutron can be viewed as two 
charge conditions of one particle the nucleon (see Chapter 12). It is 
said that the proton and the neutron form an isotopic doublet. Isotopic 
doublets are also formed by two xi-hyperons (S - , 5°) and two kaons 
IK 0 , K + ). It appeared that pions are better combined into a triplet, 
by adding to n + and tt° an antipion n~. An isotopic triplet is also 
formed by three sigma- hyperons (S , £°, E + ). As to the lambda- 
hyperon A 0 , omega-hyperon O - and eta-meson rf, to each of them 
an isotopic singlet is put in correspondence. Isotopic multiplets of 
known elementary particles come in three types-triplets (three charge 
states), doublets (two charge states), and singlets (one charge state). 
True, there exists a multiplet with four charge states. This multiplet 
is formed by A-particles which belong to short-lived baryons, called 
resonances (A - , A 0 , A + , A ++ ). The particle A ++ has a positive 
electric charge, whose magnitude equals the doubled electron charge. 

Each isotopic multiplet is characterized by a quantity called the 
isotopic spin (isospin). The magnitude of isospin 7 of a particle is related 
to the number of charge states n in the multiplet by the relationship 
n = 21 + 1. Recall that the spin s of a particle is related to the number 
of spin states of the particle in exactly the same way. The analogy 
between the isospin and the spin, although formal (spin and isospin 
are physically absolutely different), is rather profound One must recall 
that if the spin vector is in the conventional space, the isospin vector 
is considered in some fictitious space (called isotopic space or isospin 

The electron’s spin is s = 1/2, its projection in a given direction 
in conventional space takes on the values s z = +1/2 and s z = —1/2. 
The isospin of the nucleon (nucleon doublet) I = 1/2, its projection in 
some “direction” in the isospin space assumes the values 7^ = +1/2 


(for the proton) and Ig = -1/2 (for the neutron). Figure 119 contains 
the various isotopic multiplets, as well as the values of the isospin 
I and the projections of the isospin I'g. Note that for an antiparticle 
the projection of the isospin has the sign opposite to that of the cor¬ 
responding particle. When dealing with the strong interaction of 
particles, the isospin vectors of particles must be combined by the same 
rules as the spin vectors. Specifically, the isospin projection of several 
particles is the algebraic sum of the isospin projections for individual 
particles. The isotopic invariance of the strong interaction lies at the 

Table 2 Isotopic Multiplets 




n = 1 


n — 2 


n = 3 


n = 4 

V 9 A® 


K * Ai° p n H° S' 

7T + TT® tT E« E- 

A + + A*A® A“ 



0 0 


+ 10+10 0-1 

+ / 0-1+10 -1 

+ 2 +1 0-1 



0 0 


+1/2 + 1/2 + 1/2 

-1/2 -1/2 - 1/2 

+ 10-1 +10-1 

+ 3/2 +1/2 

-1/2 -3/2 


0 0 


1/2 1/2 1/2 

1 1 


foundation of the physics of isospin formalism. This invariance means 
that the laws of nature are invariant under rotations in isospin space. 
This finds its expression in the law of conservation of isospin (just as 
the invariance of the laws of nature under rotations in conventional 
space is expressed in the law of conservation of angular momentum). 
In all the strong interactions of subatomic particles, the total isospin of 
a system of particles is conserved. Note that also conserved is the total 
projection of the isospin, which in fact implies that the total electric 
charge of the particles is conserved as well. 

We will now illustrate the conservation of isospin using two pro- 

Figure 119: Table showing 
isotopic multiplets and 
values of isospin. 



p +p —* n + + D 
n+ p —> n° + D 

where D is the deuteron (the nucleus of heavy hydrogen, which con¬ 
sists of a neutron and a proton). The deuteron’s isospin is zero; there¬ 
fore, the products of reactions in the general case have the total isospin 
that is equal to the isospin of pions. that is, unity. In the first reaction 
the sum of the isospin projections will be 1/2 + 1/2 = 1, hence the 
isospin itself is 1. In the second reaction the total projection of the 
isospin is zero (—1/2 + 1/2 = 0); in this case the total isospin may 
be either unity or zero. Both values are equiprobable, therefore only 
in half of the cases a collision of a neutron with a proton can result 
in a reaction producing a deuteron. This suggests that the reaction 
n + p —> 7T° + D must be half as likely as the reaction p + p —> n + + D. 
Experiment supports this prediction made on the basis of isospin con¬ 

88 Strangeness Conservation in Strong and Electromagnetic 

In the years 1947-1955 kaons and a number of hyperons (in collisions 
of pions with nucleons) were discovered. The particles discovered 
turned out to be fairly strange. First, they came in pairs - a kaon paired 
with a hyperon. For example, 

TT~ + p —> K° + A 0 
n~ + p —> K + + £“ 
n + + p -> K + + £+ 

Second, the lifetime of new particles produced without leptons 

K + —> k + + n~ A 0 p + 7T~ 

A 0 —> n + 7t° £ + —> p + 7i° 

£ + —> n + n + £ _ —» n + n~ 


turned out to be startlingly long : 10 -s s for kaons and 10 -10 s for hy- 
perons. The fact that the decay schemes included no leptons suggested 
that these decays are associated with the strong interaction, in which 
case the lifetime of the particles must be about 10 -22 s to 10 -23 s. 

An elegant solution to both problems was found by the American 
physicist M. Gell-Mann and the Japanese physicist K. Nishijima. They 
assumed that the long lifetime of kaons and hyperons is associated 
with the conservation of some hitherto-unknown physical quantity (just 
like the stability of the proton is associated with the conservation 
of baryon number and the stability of the electron, with the electric 
charge). So, another characteristic of the elementary particles ap¬ 
peared, and not without humour it was called the strangeness. A new 
conservation law was established that is valid for strong and the elec¬ 
tromagnetic interactions: the total strangeness of the mesons and the 
baryons involved in the process is conserved. 

Figure 120 tabulates the values of strangeness S for various mesons 
(antimesons) and baryons (antibaryons). The strangeness of an antipar¬ 
ticle equals the strangeness of a respective particle with the opposite 
sign. It follows from strangeness conservation law that in collision of 

Table 3 Strangeness 



S= -2 

S= -1 





5= +2 

S = + 3 



~0 - - 


£ + £•£- 

t r* p n 

7r% 0 

K* K° 


A* K~ 

p n 



Figure 120: Table showing 
strangeness for mesons and 


a particle with zero strangeness, a lambda (or sigma) hyperon may 
only be produced together with a kaon (the total strangeness of the 
kaon and the hyperon is zero). But the production of a xi hyperon 
must be accompanied by the production of two kaons (for example, 
p+p —* p + S° +K° +K + ). Omega hyperons have been observed to be 
produced in a beam of negatively charged kaons: K~+p —> O +K°+K + 

The long lifetime of kaons is accounted for by the fact that the kaon 
is the lightest particle with a nonzero strangeness. It cannot decay 
either due to the strong interaction, or due to the electromagnetic inter¬ 
action since there is no particle to which it could transfer its strangeness. 
The kaon has one possibility: to decay by weak interaction, since in 
such interactions strangeness is not conserved. So, the decays of the 
type K° —> 7T° + 7t° + 7i° or K + —> tt + + tt° are controlled, despite 
the absence of leptons, exactly by the weak interaction, which in turn 
predetermines the long lifetime of kaons. 

The long lifetime of the lambda hyperon stems from the fact that 
this hyperon is the lightest baryon with nonzero strangeness. The 
decay of the lambda hyperons into kaons (or rather antikaons) is ab¬ 
solutely prohibited by the law of conservation of baryon number, and 
the decay into nucleons is prohibited by strangeness conservation. The 
observed decays Lambda 0 —> p + tt~ and A 0 —> n + i r° occur owing to 
the weak interaction, which does not conserve strangeness. 

The charged sigma hyperons E“ and E + , too, can only decay 
through the weak interaction. The sigma hyperon cannot decay into a 
lambda hyperon and a pion, since the mass difference of a sigma and 
a lambda hyperon is smaller than the pion mass. In the case of the 
neutral sigma hyperon, a decay is possible that conserves strangeness 
(through the electromagnetic interaction): E° —> A 0 + y. Therefore, the 
lifetime of a E°-hyperon is shorter than 10 -8 s. 

An omega hyperon and xi hyperons decay into hyperons with a 



smaller mass: 

S° -> A ° + 7T° 

£“ —> A 0 + n~ 
fT -» S° + n~ 

sr -> a - + 

-» A° + FT 

Since for the omega hyperon S = —3, and for xi hyperons S = —2, 
then in these processes, too, strangeness is not conserved, which prede¬ 
termines their slow (weak) nature. 

So far we do not know which principles of symmetry underlie the 
law of conservation of strangeness. There is no doubt, however, that 
strangeness conservation is one of the most important properties of 
the strong and the electromagnetic interactions, which accounts for the 
observed processes of interactions in the world of mesons and baryons. 
Specifically, it is of fundamental importance that strangeness does 
not conserve in weak interactions. If it were conserved not only in 
strong and electromagnetic interactions, but also in weak interactions 
(as, for instance, the electric charge, the electron, muon, and baryon 
numbers), then in addition to the electron and the proton there would 
exist eight more (!) stable subatomic particles with a nonzero rest mass: 
FC + , K°, A 0 , E + , E“, H°, S“, fl“. What structure would the atom 
have then would be anyone’s guess. 

89 Interactions and Conservations 

As it has already been noted, the highest symmetry is inherent in pro¬ 
cesses occurring due to the strong interaction. For them we have ten 
conservation laws (Figure 121): energy, momentum, angular momen¬ 
tum, electric charge, baryon number, space, charge, and time parity, 
strangeness, isospin. In principle, we can add two more conservation 
laws for the electron and muon numbers. True, in strong interactions 
these laws are fulfilled simply because there are no leptons, since the 
electron and muon numbers of all the components are zero. 

Turning to electromagnetic interactions, symmetry becomes lower 


Table 4 Interactions and Conservation Laws 


Conserved quantity 

Energy. Momentum. 
Angular momentum. 
Electric charge 

Baryon, muon, 

electron numbers 


Space parity. 

Charge parity 






4 - 







4 - 

4 - 

4 - 





4 - 

4 - 


4 - 

- isospin conservation is no longer valid. Yet more marked reduction 
is observed when we go over to the weak interaction. In the world of 
weak interactions we will have to forsake four conservations at once: 
space and charge parity, strangeness, and isospin. In some cases time 
parity is violated as well. This is somewhat compensated for by the 
conservation of CPT-parity, and in the majority of cases combined 
parity as well. 

Figure 121: Table showing 
interactions and quantities 
that are conserved in them. 

90 A Curious Formula 

Gell-Mann and Nishijima turned their attention to a rather curious fact. 
It turns out that the electric charge Q of a particle (in terms of the ratio 
of the particle’s charge to the electron’s charge), the isospin projection 
Ig, the baryon number B and the strangeness S are related by the 


following simple relationship (the Gell-Mann-Nishijima formula): 

Q = k + 

B + S 

The reader can easily prove this relationship for any meson or baryon. 
For example, for a ET-hyperon: 

Q = -1, = -1/2, B = 1, S = -2. 

, , , 1 1-2 

In this case, we have -1 =-1- 

2 2 

The Gell-Mann-Nishijima formula related the four (seemingly dif¬ 
ferent) physical characteristics for any meson or any baryon. The 
existence of such a relation suggests that there is a definite internal 
completeness of the established description of the properties of strongly 
interacting particles. 

91 The Unitary Symmetry of Strong Interactions 

Consider a system of coordinates in which the abscissa axis is the 
projection of isospin 1^, and the ordinate system is Y = B + S, a 
quantity called the hypercharge. On this plane we will position all the 
baryons with s = 1/2 : p. n. A 0 , E°, £ + , H~, S° 

The eight baryons with spin 1/2 form a hexagon in the plane Ig, Y. 
At each vertex of the hexagon there lies one baryon, in the centre two 
baryons (Figure 122). The arrangement of baryons in the plane allows 
the Q axis to be introduced. Looking at Figure 122, where all the eight 
baryons with spin 1/2 appear to be combined within a geometrically 
symmetric closed figure, one cannot ignore the fact that this is an 
example of some concealed symmetry in nature. This assumption is sub¬ 
stantiated if we place other strongly interacting particles on the plane 
Ig, Y by combining them into groups with the same spin s. It appears 
that the eight particles with s = 0, which include all the mesons and 
antimesons (K°, K + , K°, K~, jt + , k~, r]°), form (on the plane Ig, Y) 
exactly the same hexagon as the eight baryons (Figure 123). A result 
of no less interest is obtained for short-lived particles called resonances. 
Among these particles, which refer to baryons, we know nine parti¬ 
cles with s = 3/2 : A”, A 0 , A+, A++, Y*~, Y/°, Y* + , S*°, S*“ In 


Figure 122: Arrangement of 
baryons to reveal concealed 

Figure 123: Arrangement of 
mesons to reveal concealed 

the plane Ir, Y they form a rectangle shown in Figure 124, in which, 
however, one place is vacant - the vertex A. It is clearly seen in the 
figure that the missing particle (the missing baryon with spin 3/2) 
must be included in the isotopic singlet and have a negative charge 
and strangeness S = —3. You can imagine the satisfaction physicists 
derived when in 1964 the missing particle was actually found. So the 
hyperon f 1~ was added to the list of elementary particles. The eight 
baryons, the eight mesons, the ten baryons shown in Figures 122-124 are 
called supermultiplets. Each supermultiplet contains several isotopic 


Figure 124: Arrangement 
of resonances to reveal 
concealed symmetry. 

multiplets with different values of strangeness. 

The symmetry that manifests itself through a union of mesons and 
baryons into several supermultiplets is the so-called unitary symmetry. 
The explanation of the mathematical nature of unitary symmetry lies 
beyond the scope of this book, and we only note that this symmetry 
establishes the internal relationship between the particles belonging 
to various isotopic multiplets and having different strangeness. The 
fact that a fairly numerous set of mesons and baryons (including res¬ 
onances) can be compressed into a small number of eight-fold and 
ten-fold supermultiplets suggests that in the world of strongly interact¬ 
ing particles there exists a general order. 

17 Quark-Lepton Symmetry 

Up until recently physicists were bewildered by the disagreement be¬ 
tween the abundance of hadrons and a modest number of lepton types. 
Perhaps that is why a hypothesis put forward in 1964 appeared to be so 
attractive. According to this hypothesis, all hadrons consist of several 
elementary ’’building blocks” called quarks. With the passage of years 
the quark hypothesis gradually gained ground. It is formulated as the 
rule: the number of quark types must be equal to the number of lepton 
types. This rule reflects the quark-lepton symmetry, which is as yet 
quite enigmatic. 

The symmetry between 
quarks and leptons today 
appears quite significant. It 
suggests that with all their 
striking dissimilarity, there 
is something common in the 
nature of these particles. It 
is the creation of the unified 
theory of quarks and leptons 
that, it seems, is going to be 
the main effort of physicists in 

- L. Okun 

92 Quarks 

Unitary symmetry allows the existence of supermultiplets not only of 
eight or ten particles, but also other ones; specifically, supermultiplets 
are possible that contain only three particles. In the plane 7^ y these 
“particles” form a triangle shown in Figure 125 (a). The appropriate 
“antiparticles” form a triangle in Figure 125 (b). In the figure, u, d, s 
stand for the “particles” and u, d, s for the “antiparticles”. Among the 
known elementary particles (antiparticles) there are no “particles” 
that are included in the triplets shown in Figure 125. And yet these 
“particles”, called quarks, for more than 15 years now have attracted 
the attention of physicists. In 1964 Gell-Mann and Zweig pointed 
out that three quarks in combination with three antiquarks can, in 
principle, be those “building blocks” of which all the known hadrons 
(mesons and baryons) and their antiparticles are constructed. 


Figure 125: Supermultiplet 
with three particles. 

The characteristics of quarks u, d, s and their respective antiquarks 
are summarized in Figure 126. Quarks do not have an integral, but 
fractional electric charge (+2/3 or -1/3). They are fermions (spin 
1 /2); this is only natural because only out of fermions can we construct 
both fermions and bosons (an odd number of fermions gives a fermion, 
an even number of fermions gives a boson). Quarks u and d have no 
strangeness, quark s has the strangeness S = — 1 (the s-quark is, as it 
were, a carrier of strangeness). 

Hadrons are constructed out of quarks according to the following 
simple rule: the baryon consists of three quarks (antibaryon out of 
three antiquarks), and the meson, out of a quark and an antiquark. So. 
for example, pion n + has the quark structure ud, and its antiparticle 
(pion n~), the structure ud. The structure of kaons has the strange 
antiquark s(K + = us, K° = ds). The quark structure of long-lived 
baryons is represented in Figure 127. It is seen that the structure of 
most baryons includes pairs of identical quarks, and in the hyperon 
f 1~ all the three quarks are identical. Furthermore, different baryons 
may have the same quark structure (hyperons A 0 and S°). This means 
that a quark may be in different states. So we should take into account 
the possibility of two spin states of a quark. This does not need to be 
considered in the case of the hyperon f l~. The spin of this hyperon 
being 3/2, all three s-quarks are in the same spin state. Quarks are 
fermions, therefore, according to Pauli’s exclusion principle, the three 
above s-quarks must differ in some additional parameter. In quark 
theory this parameter is called “colour”. 


Table 5 Quarks and Antiquarks 

Table 6 Quark Structure of Baryons 




£ + 














According to modern views, each quark (antiquark) comes in three 
varieties, conditionally called colours. So, for example, there is a red 
s-quark, yellow s-quark, and blue s-quark. To be sure, the notion of 
quark colour should not be taken literally. 

Significantly, every baryon includes quarks of different colours (Fig¬ 
ure 128). Using the colour terminology, we can say that in each baryon 
three main colours are blended and so baryons can be viewed as colour¬ 
less (white) objects. Mesons are colourless as well, since the colour of 
an antiquark is always complementary in relation to the quark colour 
in this meson. The theory of coloured quarks (quantum chromodynam- 

Figure 126: Table showing 
varieties of quarks and their 

Figure 127: Quark composi¬ 
tion of baryons. 




s s 

Omega- hvperon 

Figure 128: The colours of 


ics) explains why we do not encounter in nature particles consisting, 
say, of two or four quarks and, specifically, individual (free) quarks. 
This is related to the fact that the hadrons (antihadrons) observed in 
nature must be colourless by all means. Clearly, we cannot produce a 
colourless combination from one, two, or four quarks. 

The theory shows quite convincingly why the hadrons occurring 
in nature must be colourless. But how stringent is the requirement of 
colourlessness? The final answer, clearly, comes from experiment The 
experimental search for free quarks has been under way for 15 years 
already, but to no avail. 

Having read that no free quarks have as yet been found, the reader 
may doubt the physical reality of the quark hypothesis and may regard 
it as just an elegant mathematical trick. In 1965 Ya. Zeldovich wrote: 

The dilemma facing physics now can be formulated as follows: 
either only the classification and symmetry properties of known 
particles are clarified, or this symmetry is a consequence of the 
existence of quarks, that is, absolutely new fundamental type of 
matter, atomism of new type. 

Another decade passed and physicists saw that the quark hypothe¬ 
sis was associated with the existence of a new type of atomism. In 
other words, by the end of the 1970s physicists no longer doubted that 
quarks in hadrons really exist. 

What made them so sure? In the first place, three quarks (plus three 
antiquarks) made it possible to build all the hadrons (antihadrons) dis¬ 
covered up until 1974. It is remarkable that this construction produced 
no extraneous objects-all the particles constructed from quarks (anti¬ 
quarks) according to the rules mentioned earlier have eventually been 
found experimentally. The quark model enabled the physicists to work 
out correctly the various characteristics of hadrons, the probabilities of 
transformations, and so on. Experiments on the scattering high-energy 
electrons at nucleons allowed quarks to be literally groped for within 
nucleons. Conclusive evidence for the validity of the quark hypothesis 
came from the discovery of new types of particles that were christened 
charmed particles. 



93 The Charmed World 

In November 1974 at the accelerator of Stanford, USA, a new particle 
of mass about 6000m and lifetime about 10 -20 s was discovered. This 
particle is now known as the //T-meson. The spin of the J /\F is unity. 
Like mesons n Q and rj°, J/'B is truly neutral. 

The new meson does not conform to any of the earlier theoretical 
schemes, it would have a lifetime approximately 1000 times shorter. 

To describe the quark structure of J/'L-meson a new quark had to be 
introduced-the so-called c-quark and a new conservable quantity called 
“charm”. Just like strangeness and parity, charm is conserved in strong 
and electromagnetic interactions, but not in weak ones. It is charm 
conservation that is responsible for the relatively long lifetime of the 

When the c-quark was introduced, the number of quark types 
became equal to four. Note that the c-quark is the carrier of charm, just 
like the s-quark is the carrier of strangeness. The electric charge of the 
c-quark is +2/3. 

The quark structure of the J/'B -meson is cc (this structure explains 
specifically the true neutrality of the meson). The structure cc is called 
the charmonium and it is treated as an atom-like system resembling 
the long-known positronium (Figure 129). Recall that the positronium 
represents an “atom” consisting of an electron and a positron orbiting 
around a common centre of mass. (The existence of the bound system 
including an electron and its antiparticle was established experimen¬ 
tally in 1951; the lifetime of the positronium is as long as 10 -7 s.) Like 
any atom, the charmonium is characterized by a system of energy lev¬ 
els. The //'F-meson corresponds to one of the levels of charmonium. 
Soon after the //\F-meson had been discovered, a number of mesons 
were found as well (T 1 , /o, Xi> + 2 , and so on), which could be cor¬ 
related with various levels of charmonium. Studies of charmonium 
properties is of great interest - they provide information about the 
interaction of quarks. 




Figure 129: The positronium 
and charmonium. 

The charms of the c-quark and c-antiquark have opposite signs. 
Therefore, the resultant (total) charm of the cc-structure is zero. It is 


said that the cc-structure has a hidden charm. Mesons with open charm 
were discovered in the summer of 1976: the D°-meson (cii-stmcture) 
and the D + -meson (cd-structure). Their behaviour appeared to be in 
complete agreement with the hypothesis of the charmed c-quark. In 
1977 the F + -meson (cs-structure) was discovered, which apart from 
charm also has strangeness. 

The discovery of charmed particles proved experimentally the 
existence of the c-quark. And since the very c-quark and its properties 
are closely linked to those of u, d, s quarks, then the quark model as a 
■whole was proved experimentally. 

Returning to the issue of unitary symmetry of the strong interac¬ 
tion, note that allowing for charm the supermultiplets of hadrons take 
the form of volume bodies (polyhedrons) in the space in which the axes 
are Ig, Y, C (recall that here C is charm, Ig is the isospin projection, Y is 
the sum of the baryon number and strangeness called the hypercharge). 
The supermultiplets presented in Figures 122-124 (see Chapter 16) are 
the sections of such polyhedrons by the plane C = 0. Given in Fig¬ 
ure 124 is an example of the polyhedron corresponding to the meson 
supermultiplet consisting of fifteen mesons. 

94 Quark-Lepton Symmetry 

It is to be stressed that the representatives of hadrons in weak inter¬ 
actions are quarks. Consider two examples: the decay of the neu- 
tron( n —r +e~ + v e ) and the collision of a neutrino with a neutron 
{v e + n —> p + e~). The decay of the neutron comes down to the decay 
of one of the d-quarks, which is a constituent part of the neutron, into 
a u-quark and leptons: 

d —» u + e + v e 

The collision of a neutrino with a neutron resulting in the production 
of a proton and an electron boils down to the collision of a neutrino 
and a d-quark that enters the composition of the neutron, with the 
result that the d-quark turns into a M-quark and in the process an 


Figure 130: A polyhedron 
corresponding to the meson 
supermultiplet consisting of 
fifteen mesons. 

electron is produced: 

v e + d —> u + e~ 

The above processes involve a pair of ev e (or ev e ) leptons and a pair 
of ud-quarks. Other forms of weak processes are also possible. So, for 
example, the lepton pair fjv may interact with the quark pair us. Any 


weak process is an interaction of a lepton pair with a quark pair. 

This removes the discrepancy between the small number of lepton 
types and the vast number of hadrons. The number of leptons must 
be compared not with the number of hadrons but with the number 
of quarks. It appears that between leptons and quarks some sort of 
symmetry exists: the number of lepton types must be exactly equal to the 
number of quark types. This conclusion follows from the theory based 
on a vast body of experimental evidence and, in particular, on the data 
on the decay of strange particles and the nonconservation of space 
parity in weak interactions. 

Before the charmed particles had been discovered, there was no 
symmetry between leptons and quarks: only three quarks (u, d, s) 
corresponded to the four leptons e~, v e , p~, vp. Therefore, the c-quark 
was a welcome arrival. 

The scheme of four leptons and four quarks suffered, however, 
from one drawback. The number of leptons (quarks) involved was 
insufficient to account for the nonconservation of combined parity in 
decays of neutral kaons (the nonconservation of combined parity was 
noted specifically in Chapter 14). At least six leptons (and as many 
quarks) were required 

95 A New Discovery 

The first evidence for the existence of a fifth lepton came in 1975. In 
1977 the hypothesis became certainty. The fifth lepton was called 
the tauon ( t~ ). Its mass was found to be 3500m. Apart from the new 
lepton another neutrino-tauon neutrino (v r ) must exist. 

In the summer of 1977 at the Fermi National Accelerator Labora¬ 
tory, USA, superheavy mesons with a mass of about 20000m (epsilon- 
messon 6) were discovered. It was found that these mesons repre¬ 
sented a structure of a quark and an antiquark of a new type. This 
quark (£>-quark) is the carrier of the quantity that is conserved in 
strong interactions. This quantity was dubbed beauty- this explains 
the abbreviation for the fifth quark. The electric charge of the b-quark 
is -1/3, its mass is about 10000m. 


At present, a search is under way for beautiful hadrons, and also for 
a sixth quark. If the third quark (s-quark) is called strange, the fourth 
quark (c-quark) is charmed, the fifth quark (b-quark) is beautiful, but it 
has been decided to call the sixth quark true quark, or f-quark. 

The scheme of the six leptons and six quarks seems to be quite 
attractive to physicists today. The future will show whether or not this 
quark-lepton scheme is a final one or the number of leptons (quarks) 
will continue to grow. 

A Conversation Between the Au¬ 
thor and the Reader About the 
Role of Symmetry 

The Ubiquitous Symmetry 

: In 1927 the prominent Soviet scientist V. Vernadsky 
wrote: “A new element in science is not the revelation of the 
principle of symmetry, but the revelation of its universal nature.” 
I think that this book provides plenty of evidence to support the 
notion of the universality of symmetry. 

Reader : The universality of symmetry is startling. Symmetry 
establishes the internal relations between objects or phenomena 
which are not outwordly related in any way. The game of bil¬ 
liards and the stability of the electron, the decay of the neutron 
and the reflection in a mirror, a pattern and the structure of dia¬ 
mond, a snowflake and a flower, a helix and the DNA molecule, 
the superconductor and the laser ... 

: The ubiquitous nature of symmetry is not only in the 
fact that it can be found in a wide variety of objects and phe¬ 
nomena. The very principle of symmetry is also very general, 
without which in effect we cannot handle a single fundamental 
problem, be it the problem of life or the problem of contacts with 
extraterrestrial civilizations. Symmetry underlies the theory of 

In the 20th century the princi¬ 
ple of symmetry encompasses 
an ever-increasing number 
of domains. From crystallog¬ 
raphy, solid-state physics, it 
expanded into the science of 
chemistry, molecular processes 
and atomic physics. It is be¬ 
yond doubt that we will find 
its manifestations in the world 
of the electron, and quantum 
phenomena will obey it as well. 

- V. Vernadsky 

.. .The thing that makes the 
world go round 

- A. Pushkin 


relativity, quantum mechanics, solid-state physics, atomic and nu¬ 
clear physics, particle physics. These principles have their most 
remarkable manifestations in the properties of the invariance of 
the laws of nature. 

Reader : It is quite obvious that here are involved not only the 
laws of physics, but also others, for example, the laws of biology. 

: Exactly. An example of a biological conservation law 
is the law of heredity. It relies on the invariance of the biological 
properties when passing from one generation to the next. It is 
quite likely that without conservation laws (physical, biological 
and others) our world simply could not exist. 

Reader : Without energy conservation the world would be just a 
holocaust of random explosions associated with random appear¬ 
ances of energy from nothing. 

: Imagine that on a nice day the laws of conservation 
of the electric charge and baryon number just cease to function. 
What would result then? 

Reader : Electrons and protons would then become unstable 
particles. And there would not be a single stable particle with a 
nonzero rest mass. Author You can easily imagine what sort of 
world we would live in. It would be a giant cluster of photons 
and neutrinos. Here and there some ephemeric formations would 
emerge only to decay quickly (in 10 -10 s to 10 -8 s) into some kind 
of photon-neutrino chaos And now imagine that suddenly the 
character of the symmetry of the wave electron function changes 
so that electrons become bosons. 

Reader : Maybe our world would then become a world of super¬ 
conductivity? Electric current would pass along wires in this 
kind of world without resistance. 

: Serious doubts emerge here in relation to wires them¬ 
selves. Having stopped to obey Pauli’s exclusion principle, elec¬ 
trons in all atoms would have to undergo a transition to the 
electron shell that is closest to the nucleus. The entire Periodic 
System would be thrown into disarray. 



Reader : Yes, indeed, symmetry permeates our world much 
deeper than it is apparent to our eyes. 

: It took several centuries to conceive this, and the 20th 
century has been especially remarkable in this respect. As a 
result, the very concept of symmetry has undergone a substantial 

The Development of the Concept of Symmetry 

: I hope that you understand how strongly today’s pic¬ 
ture of the physically symmetrical world differs from the geomet¬ 
rically symmetrical cosmos of the ancients. From antiquity to 
the present, the notion of symmetry has undergone a lengthy 
development. From a purely geometrical concept it turned into a 
fundamental notion lying at the foundation of nature’s laws. We 
now know that symmetry is not only that which is visible to our 
eyes. Symmetry is not just around us but, moreover, it is at the 
root of everything. 

Reader : So it is not without good reason that the book is divided 
into two parts: “Symmetry Around Us” and “Symmetry at the 
Heart of Everything”, isn’t it? 

: Yes, it is with good reason. The first part was devoted 
to geometrical symmetry. The second part was meant to show 
that the notion of symmetry is much deeper and that to grasp 
it one requires not so much a visual conception as thinking. As 
we pass on from the first to the second part we cover the path from 
the symmetry of geometrical arrangements to the symmetry of 
physical phenomena. 

Reader : As far as I understand, from the most general point of 
view the notion of symmetry is related to the invariance under 
some transformations. Invariance may be purely geometrical (the 
conservation of geometrical shape), but may have nothing to do 
with geometry, for example, the conservation of energy or bio¬ 
logical properties. In exactly the same way transformations may 
be geometrical in nature (rotations, translations, commutations), 


and may be of another nature (the replacement of particles by 
antiparticles, the transition from one generation to the next). 

: According to modern views, the concept of symmetry 
is characterized by a certain structure in which three factors 
are combined: (1) an object (phenomenon) whose symmetry is 
considered, (2) transformations under which the symmetry is 
considered, (3) invariance (unchangeability, conservation) of 
some properties of an object that expresses the symmetry under 
consideration. Invariance exists not in general but only in as far 
as something is conserved. 

Reader : I heard that there exists a specially worked out theory of 
symmetry with its own mathematical tools. 

: Such a theory does exist. It is called the theory of groups 
of transformations, or for short, just groups theory. The term 
“groups” was introduced by the father of this theory, the out¬ 
standing French mathematician Evariste Galois (1811-1832). 

These days any serious studies in the field of quantum physics, 
solid-state physics, and particle physics use the procedures of 
groups theory. 

But back to the notion of symmetry. We will take a closer look at 
the fundamental nature of symmetry. 

Reader : This fundamentally is well grasped if one remembers 
that symmetry limits the number of possible forms of natural 
structures, and also the number of possible forms of behaviour of 
various systems. This has been repeatedly stressed in the book 

: It can be said that there are three stages in our cog¬ 
nition of the world. At the lowest stage are phenomena-, at the 
second, the laws of nature, and lastly at the third stage, symmetry 
principles. The laws of nature govern phenomena, and the princi¬ 
ples of symmetry govern the laws of nature. If the natural laws 
enable phenomena to be predicted, then the symmetry principles 
enable the laws of nature to be predicted. In the final analysis the 
predominance of symmetry principles can be judged from the 
actual presence of symmetry in everything that surrounds us. 



Reader : But perhaps there is some contradiction here with the 
fact that the most interesting discoveries in particle physics are 
related to violations of conservation laws, that is violations of 

: The violation of P-invariance in weak interactions is 
compensated for by the conservation of CP- parity. We can speak 
about the existence of a sort of law of compensation for symmetry. 
once symmetry is lowered at one level it is conserved at another, 
larger level. But this issue is worth considering in more detail. It 
is directly related to the problem of symmetry-asymmetry. 


: Faith in the primordial symmetry (harmony) of nature 
has always been an inspiration for scholars. And today it inspires 
them from time to time to undertake a search for a unified theory 
and universal equations. 

Reader : This search is not altogether without success. Suffice 
it to mention Einstein’s theory of relativity or the discovery by 
Gell-Mann of unitary symmetry in strong interactions. 

: That is all very well. But it should be noted that discov¬ 
eries of new symmetries in the surrounding world by no means 
brings us any nearer to the cherished unified theory. The picture of 
the world, as we progress along the path of cognizing it, becomes 
ever more complicated, and the very possibility of the existence 
of such universal equations becomes ever more doubtful. In the 
book In Search of Harmony by O. Moroz there is a rather graphic 
comment: “Physicists are chasing symmetry just as wanderers 
in the desert chase the elusive mirage. Now on the horizon a 
beautiful vision appears, but when you try and approach it, it 
disappears, leaving a feeling of frustration ...” What is the matter 

Reader : Perhaps, it is that symmetry must be treated as no more 
than ideal norm, from which there are always deviations in real¬ 


: Right. But the problem of symmetry-asymmetry must 
be understood more deeply. Symmetry and asymmetry are so 
closely interlinked that they must be viewed as two aspects of the 
same concept. Our world is not just a symmetrical world. It is a 
symmetrical-asymmetrical world. The French poet Paul Valery 
(1871-1945) was right in saying: “The world has orderly shapes 
strewn over it.” He went on to observe that “events that are most 
astounding and most asymmetrical in the short run acquire a 
measure of regularity in the longer run.” 

Reader : In the preliminary remarks it was stressed that the world 
exists owing to the unity of symmetry and asymmetry. 

: The gist of the matter consists in that the unity of 
symmetry and asymmetry is the unity of dialectically opposite 
categories. It is similar, for example, to the unity of essence and 
phenomenon, necessity and chance, the possible and actual. The 
Soviet philosopher V. Gott in his book Symmetry and Asymmetry 
notes that 

symmetry discloses its content and meaning through asym¬ 
metry, which in itself is a result of changes, or violations, 
of symmetry. Symmetry and asymmetry is one of the man¬ 
ifestations of the general law of dialectics-the law of unity 
and conflict of opposites. 

Like two dialectically opposite categories, symmetry and asym¬ 
metry cannot exist independently. We have already said that in 
an absolutely symmetrical world you would observe nothing no 
objects, no phenomena. In exactly the same way, an absolutely 
asymmetrical world is impossible too. 

Reader : It seems that the more we grasp the symmetry of nature, 
the more asymmetry comes out. 

: Exactly. Therefore, any search for a unified theory or 
universal equations is bound to fail, as is any attempt to consider 
symmetry separately from asymmetry. 



On the Role of Symmetry in the Scientific Quest for Knowl¬ 

: Symmetry principles are extremely important in the 
great mystery called the scientific quest for knowledge. Any sci¬ 
entific classification is based on revealing the symmetry of the 
objects being classified. Objects or phenomena are grouped to¬ 
gether by their common features that are conserved under some 

A good example is the Periodic System of elements suggested by 
the great Russian chemist D. Mendeleev (1834-1907). From period 
to period is preserved the community of properties of elements, 
that enter a column of the table, for example lithium, sodium, 
potassium, rubidium, caesium. The behaviour of elements varies 
in the same way within a period for various periods. 

Reader : It seems to me that any classification is based not only 
on symmetry, but also on the asymmetry of properties. 

: Right. It would make no sense to note the common 
properties of lithium, sodium, potassium, if these properties were 
also possessed by other elements within a period. The symme¬ 
try of properties of appropriate elements from various periods 
is only of significance in combination with the asymmetry of 
properties of elements within the same period. Classification 
also presupposes both the conservation (community) and change 
(differences) of properties of the objects being chssified. 

Reader : Now I am beginning to understand the thesis about the 
dialectic unity of symmetry and asymmetry. This is the unity 
of conservation and change, and the unity of community and 

: Speaking about symmetry principles, we should always 
keep this unity in mind. 

Symmetry thus lies at the root of all classifications. Crystals, for 
example, are classified by the type of symmetry of the crystalline 
lattice, by the properties of atomic binding forces, by electric 


and other properties. The classification of atoms is based on the 
community and differences in the structure of their emission 

When dealing with an unknown object or phenomenon, one 
should above all identify the factors that are conserved under 
given transformations. Hermann Weyl noted that when one has 
to have something to do with some object having a structure, 
one should try and determine the transformations that leave the 
structural relations unchanged. You may hope that following 
this path you will be able to get a deep insight into the inner 
structure of the object. 

By applying symmetry to the development of scientific classifi¬ 
cations in structural studies, one can in the final analysis make 
scientific predictions. I think that some examples of such predic¬ 
tions are already known to you. 

Reader : For example, Mendeleev predicted a number of then- 
unknown chemical elements and gave a correct description of 
their properties. It is also worth noting Gell-Mann’s prediction of 
the existence of omega hyperon. 

: No less instructive is the example of the prediction 
of the displacement current. The outstanding British physicist 
James Clerk Maxwell (1831-1879) was keen enough to uncover 
in the phenomenon of electromagnetic induction discovered by 
Faraday the production of the alternating electric current by the 
alternating magnetic field. Having assumed that there exists 
also a similar inverse effect (the alternating magnetic field is 
produced by the alternating electric field), Maxwell put forward 
the famous hypothesis of the displacement current, which then 
enabled him to formulate the laws of electromagnetism. Moroz 
writes in his book In Search of Harmony. 

When we try to solve the enigma of what prompted 
Maxwell to that decisive step, what led him to the idea 
of the displacement current, the circumstances lead us to 
quite a probable answer-symmetry. The symmetry between 
electricity and magnetism. The fact that Maxwell noticed it 



could be that insight without which no great discovery can 
be made. 

Reader : Perhaps in the example of Maxwell one should speak not 
so much about symmetry as about analogy ? 

: The method of analogy is based on the principle of sym¬ 
metry It presupposes looking for common properties in different 
objects (phenomena) and the extension of this community to 
include other properties. Speaking about the role of symmetry 
in the process of scientific quest, we should pay special attention 
to the application of the method of analogy. In the words of the 
French mathematician D. Poia 

there do not, perhaps, exist discoveries either in elementary 
or higher mathematics, or even any other field, which 
could be made without analogies. 

Reader : It seems to me that Poia has somewhat exaggerated the 
role of analogies. 

: No more than the role of symmetry. The universality 
of the method of analogy, which has really been widely used 
in all the scientific disciplines without exception, is, in essence, 
the universality of the principles of symmetry. Physical models 
of objects (phenomena) are constructed exactly by the use of 
analogies. The DNA molecule is modeled as a screw. The spin of 
a particle is modeled as the angular momentum of a body that 
spins like a top about its axis. The collision of a photon with 
an electron in the Compton effect is modeled as the collision of 
billiard balls. 

Analogies between different processes allow us to describe them 
using general equations. A simple example: the swinging of the 
common pendulum, oscillations of atoms in a molecule, oscilla¬ 
tions of the electromagnetic field in an oscillatory circuit with a 
capacitance and an inductance are symmetrical (analogous) in 
the sense that all these processes are described by the same math¬ 
ematical equation (the differential equation of simple harmonic 
motions). One equation is suitable for describing the process of 


radioactive decay, the process of discharging a charged capacitor, 
the variation of air density with altitude when there is no wind, 
the decrease in the intensity of a light beam propagating through 
a medium (the differential equation of exponential decay). The 
unity of the mathematical nature of the processes at hand, which 
makes it possible to regard them as analogous, points the pres¬ 
ence of deep symmetry. 

Reader : It is just amazing how wide in its scope our talk about 
the role and place of symmetry appeared. 

: It would have been much wider if we had included the 
questions connected not only with the scientific activities of man 
but also with other aspects of his life, for example, engineering, 
architecture, arts. 

Reader : Symmetry is apparent in engineering and architecture. 
As regards painting, music, poetry, here, it seems, the dominance 
of symmetry is doubtful. 

Symmetry in Creative Arts 

: Note, above all, that the creative endeavour of man in 
all its manifestations gravitate toward symmetry. In his book The 
Architecture of the 20th Century the French architect Le Corbusier 

Man needs order, without it all his actions lose their con¬ 
cordance, logical interplay. The more perfect is the order, 
the more comfortable and confident is man. He makes 
mental constructs on the basis of the order that is dictated 
to him by the needs of his psychology-this is the creative 
process. Creation is an act of ordering. 

Reader : These are the words of an architect. It is well known that 
the principles of symmetry are the governing principles for any 
architect. In some cases the architect can do with the primitive 
symmetry of the rectangular parallelepiped, in other cases he 
uses a more refined symmetry, as, for example, in the building 


of the Council of Mutual Economic Cooperation in Moscow 
(Figure 131). 

Figure 131: Symmetry in 
the building of the Coun¬ 
cil of Mutual Economic 
Cooperation in Moscow. 

: It would be better to speak not about “primitive” and 
“refined” symmetry but about how an architect solves the ques¬ 
tion of the correlation between symmetry and asymmetry. A struc¬ 
ture that is asymmetrical on the whole may represent a harmonic 
composition out of symmetrical elements. An example is the 
Cathedral of St. Basil the Blessed on Red Square in Moscow (Fig¬ 
ure 132). One cannot help admiring this bizarre composition of 
ten different parts, however the cathedral as a whole features nei¬ 
ther mirror nor rotational symmetry. The architectural volumes 
of the cathedral, as it were, are superimposed on one another, 
intersecting, rising, competing with one another and culminating 
in a central cone. Everything is so full of harmony that it evokes 
the feeling of festivity. In his book Moscow M. Ilyin writes: 

At first glance at the cathedral one can imagine that the 
number of architectural forms used in it is extremely large. 

It soon becomes clear, however, that the builders made use 
of only two architectural motifs the eight-sided cylinder 


Figure 132: Cathedral of 
St. Basil the Blessed on Red 
Square in Moscow. 

and the semicircle. The former defines the faceted forms 
of major volumes, whereas the second is represented by a 
wide variety of forms, from wide and graceful arches of the 
ground-floor to the peaked ogee gables. 

Reader : It turns out that the symmetry of the cathedral manifests 
itself in the repetition (conservation) of the two main motifs 
throughout the structure. 

: Not just conservation, but also variations, or rather de¬ 
velopment, of them. The two main motifs do not simply recur 
in the various parts of the cathedral, but they seem to be devel¬ 
oping as the glance of the observer traces the entire structure. 
This is a highly successful solution of the problem of symmetry- 
asymmetry. It is clear that without this startling asymmetry the 
cathedral would have lost much of its individuality. 

Reader : Obviously, it is impossible to calculate preliminarily so 
successful a solution of the problem symmetry-asymmetry. This 
is a work of genuine art. It comes from the genius of the builder, 
his artistic taste, his understanding of the beautiful. 



: Quite so. It can be said that the art of architecture starts 
exactly when the architect hits upon the elegant, harmonic and 
original relationship between symmetry and asymmetry. 

Reader : True, in the modern massive construction of standard 
dwellings there is no question of the correlation between symme¬ 
try and asymmetry. 

: These days this problem takes on a new perspective. It 
is now solved not on the level of a separate house, but on the 
level of a neighbourhood or even a town. In earlier times an 
architectural ensemble distinguished by its individuality was a 
house (temple, palace, etc.). But now more frequently this role is 
played by a group of houses, for example, a district. And it is on 
this level that present-day builders are called upon to solve the 
problem of symmetry-asymmetry. 

Reader : Architecture provides plenty of examples of the dialecti¬ 
cal unity of symmetry and asymmetry. 

: In music and poetry we have the same thing. In 1908 a 
Russian physicist, G. Vulf, wrote: 

The heart of music - rhythm - consists in regular periodic 
repetition of parts of a musical piece. But regular repeti¬ 
tion of similar parts to make up the whole constitutes the 
essence of symmetry. We can apply the concept of sym¬ 
metry to musical piece all the more so since this piece is 
put down using notes, i.e., it acquires a spatial geometrical 
image, whose parts we can overview. 

He wrote elsewhere: 

Just like musical pieces, symmetry may be inherent in 
literary works, especially poems. 

Reader : As for poems, you surely mean the symmetry of rhymes, 
stressed syllables, that is, in the final analysis rhythm again. 

: Right indeed. But both in music and in poetry symme¬ 
try cannot be reduced just to rhythm and cadence. Any good 


work (musical or literary) includes some invariants of meaning 
which permeate, varying their form, throughout the work. In his 
symphony a composer generally returns to the main theme many 
times, each time varying it. 

Reader : We have already seen something of the sort in the exam¬ 
ple of the Cathedral of St. Basil the Blessed. 

: The preservation of a theme and its varying (developing) 
is here the unity of symmetry and asymmetry. And the more 
successful is an architect, a composer, or a poet in solving the 
problem of the correlation between symmetry and asymmetry, 
the higher is the artistic value of the produced piece. 

Of direct concern to symmetry is composition. The great German 
poet Johann Wolfgang Goethe argued that “any composition is 
based on latent symmetry.” To master the laws of composition 
is to master the laws of symmetry. The three principal laws 
of composition dictate the translation-identical repetition of 
elements of structure, contrasted repetition, varied repetition. 

Reader : This appears as an ornament in time. 

: Really, temporal pattern of sorts. We will always admire 
the “patterns” produced by the great Russian poet A. Pushkin. 
Here is a relatively simple, elegant Pushkinian pattern: That year 
was extraordinary, The autumn seemed so loth to go; Upon the third of 
January, At last, by night, arrived the snow. Tatyana, still an early riser, 
Found a white picture to surprise her: The courtyard white, a white 
parterre, The roofs, the fence, all moulded fair: The frost-work o’er the 
panes was twining; The trees in wintry silver gleamed; And in the court 
gay magpies screamed; While winter’s carpet, softly shining, Upon the 
distant hills lay light, And all she looked on glistened white. (Tranlsated 
by Babette Deutsch) We will not here examine the intonation 
in this excerpt. Let us reread it again and again to absorb the 
splendour of these poetical ornaments. 

Reader : Let us now turn to painting. Where does symmetry 
come in here? 

: A picture is not a colour photograph, not by any means. 



The arrangement of figures, the combinations of stances and ges¬ 
tures, the countenances, the palette, the combination of shades, 
these all are carefully thought out in advance by the painter, 
whose main concern is to affect the beholder in the way he de¬ 
sires. By using asymmetrical elements, the painter strives to 
create something that on the whole features latent symmetry. 

The Russian painter V. Surikov wrote: 

And how much time it takes for the picture to settle down 
so that it would be impossible to change anything. The 
real sizes of each object are to be found. It is necessary 
to find the lock to piece together the elements. This is 

Reader : To analyze the symmetry of an image it would be better 
perhaps to take a picture with a simple composition. 

: We can take the picture Madonna Litta (Figure 133) 
by the Italian genius Leonardo da Vinci. Notice that the figures 
of the madonna and the child are inscribed into an isosceles 
triangle, which is especially clearly perceived by the beholder’s 
eye. This immediately brings the mother and the child to the 
forefront. The madonna’s head is placed absolutely exactly, but at 
the same time naturally, between the two symmetrical windows 
in the background. Through the window we can see the gentle 
horizontal lines of low hills and clouds. All of this adds to the 
air of the peace, which is intensified by the harmony of the blue 
colour with the yellowish and reddish tones. 

Reader : The inner symmetry of the picture is apparent. But what 
can be said about asymmetry? 

: Asymmetry manifests itself, for example, in the body 
of the child, which cuts through the triangle. Furthermore, there 
is one highly expressive detail. Owing to the closeness, com¬ 
pleteness of the lines of the madonna’s figure an impression 
is produced that the madonna is totally indifferent to the sur¬ 
rounding world, and to the beholder too. The madonna is all 
concentrated on the child, she holds him gently and looks at him 
tenderly. She is thinking only about him. And suddenly all the 


Figure 133: Madonna Litta 
by Leonardo da Vinci. 

closeness of the picture is gone, once we meet the stare of the 
child. And it is here that the inner poise of the composition is 
disturbed: the serene and attentive stare of the child is directed 
at the beholder, and so the picture achieves its contact with the 
world. Just try and remove that marvellous asymmetry, turn the 
face of the child to the mother, connect their glances. This will 
kill the expressiveness of the picture. 

Reader : It so happens that each time when we, marvelling at 
a piece of art, speak about harmony, beauty, emotional inten¬ 
sity, we thereby touch on the same inexhaustible problem - the 
problem of the correlation of symmetry and asymmetry. 

: As a rule, in a museum or a concert hall we are not 
concerned with that problem. After all it is impossible at the same 
time to perceive and analyze the perception. 


Reader : The example of Leonardo’s picture has convinced me 
that the analysis of symmetry-asymmetry is still useful: the 
picture begins to be perceived more acutely. 

: We see thus that symmetry is dominant not only in the 
process of scientific quest but also in the process of its sensual, emo¬ 
tional perception of the world. Nature-science-art In all of these 
we find the age-old competition of symmetry and asymmetry. 


H. Weyl, Symmetry (Princeton, Princeton University Press, 1952) 

M. Gardner, The Ambidextrous Universe (New York, Charles Scrib¬ 
ner’s Sons, 1979) 

H. Lindner, Das Bild der modernen Physik (Leipzig, Urania-Veriag, 

R. Feynman, The Character of Physical Law (London, Pergamon 
Press, 1965) 

K. Ford, The World of Elementary Particles (New York, Blaisdell 
Publishing Company, 1963) 

A. Kitaigorodsky, Order and Disorder in the World of Atoms (Moscow, 
Mir Publishers, 1980) 

P. Davies, The Forces of Nature (Cambridge, Cambridge University 
Press, 1979) 

E. Harrison, Cosmology. The Science of the Universe (Cambridge, 
Cambridge University Press, 1981) 

About the Book 

Here is a fascinating and nontechnical 
introduction to the ubiquitous effects of 
symmetry. Starting with geometrical 
elements of symmetry, the author reveals 
the beauty of symmetry in the laws of 
physics, biology, astronomy, and 
chemistry. He shows that symmetry and 
asymmetry form the foundation of 
relativity, quantum mechanics, solid-state 
physics, and atomic, sub-atomic, and 
elementary particle physics, and so on. 

The author would like to attract the reader's 
attention to the very idea of symmetry and 
help him to discern a wide variety of the 
manifestations of symmetry in the 
surrounding world, and, above all, to 
demonstrate the most important role 
played by symmetry in the scientific 
comprehension of the world and in human 
creative effort. 

Mir Publishers