**Title**

Latent,
Manifest, and Broken Symmetry: A Bottom-up Approach to Symmetry, with
Implications for Complex Networks

**Author**

**Book details**

Paperback: 210 pages

**Publisher:**Createspace Independent Publishing Platform

**Language:**English

ISBN: 978-1463766718

Product dimension: 15.2 x 1.2 x 22.9 cm

**About the book**

There is a subtle kind of symmetry called latent symmetry which manifests itself only when the conditions are right. It can occur in systems composed of equal or equivalent components. It lies dormant or latent, and becomes manifest when the components happen to have certain special mutual placements. Although the latent-symmetry idea has been around for more than a decade, not many natural manifestations have been observed to date. But a recognition of the possibility of latent symmetry enables us to formulate a comprehensive symmetry-composition principle enunciated in this book. The principle is applicable to any system composed of equal or equivalent sub-parts. And there are many such systems around. Crystals are an obvious example, the equal components being the unit cells. Several complex networks can also fall within the purview of this principle, if we take note of the approximate nature of their symmetry. This book presents such an all-inclusive view of symmetry in an accessible language.

We are surrounded
by symmetry and broken symmetry. From the Big Bang onwards, as our universe
cooled and expanded, a series of symmetry-breaking transitions occurred,
resulting in a gradual evolution of the complexity of life we see today. By now
it is well recognized that discovering new broken symmetries (particularly
broken gauge symmetries) is the path science must take for going deeper into
the mysteries of Nature. At a very fundamental level, laws of physics are all
about symmetry. The present edifice of science in general, and physics in
particular, would be unthinkable without symmetry. There is a lot of symmetry
even in biological systems. This book celebrates symmetry in all its forms,
including latent symmetry.

**Foreword to first edition by Prof. A. M. Glazer**

Wherever we look we see a variety of patterns and shapes that show different types of symmetry. Much of this is obvious, such as for instance when we look at the pyramids of Egypt, or crystals in a museum. However, what is not so obvious is just what exactly is symmetry and why is it so prevalent? In this unique and intriguing book, Professor Vinod Wadhawan has set about answering these sorts of questions. He takes us on a journey from very basic descriptions, such as the growth of a crystal, on to more esoteric and complex notions, demonstrating that, in fact, symmetry is even more pervasive than we thought before. Some symmetries are far from obvious, as illustrated by the idea of latent symmetry. This is said to manifest itself when one combines two or more ‘equal’ objects or systems, each with its own symmetry description, and the resulting composite system exhibits

*new*symmetry elements that were not expected from the original systems. For instance, two identical right-angled isosceles triangles can be joined together to form a square, that has an unanticipated four-fold rotational symmetry. The notion of latent symmetry is relatively new and deserves further consideration.

Not only do we have the symmetry exhibited by living organisms and physical objects, but also by ideas themselves. As such this book has a strong philosophical content that will enable the reader to gain much more insight into the phenomenon than is normally got from a typical university education. Wadhawan shows us how even the concept of randomness is intricately bound up with notions of symmetry. Even the idea of predictability is an example of symmetry in action! And then, having explained what symmetry is, emphasis is placed on what happens when symmetry is broken. In a sense, pure symmetry could even be described as rather boring, since it implies a lack of change or progress. Nonetheless, we still need to understand it. It is when symmetry is broken that fun things start to happen and new ideas, progress and phenomena are created. This book explains how this comes about and why symmetry-breaking is so important. The book is written with an eye to explaining the fundamental

concepts of symmetry, rather than go into complex mathematical proofs and lemmas, which in any case can be found elsewhere for those who like those sorts of things. This means that Wadhawan is able instead to concentrate on the philosophical importance of understanding symmetry, and how it impacts on the world that we observe. Rather like the Second Law of Thermodynamics, symmetry is seen to play a vital role in what holds the universe together. You can see then that this book covers just about everything that we know about symmetry, and possibly that which we do not!

A.M. Glazer

Professor of Physics and Emeritus Fellow of Jesus College

University of Oxford

Author of

*Space Groups for Solid State Scientists*#
**Preface to the First Edition**

#
The
symmetry of any composite system made up of equal or equivalent components
depends on at least two factors: The inherent symmetry of each component, and
the symmetry imposed on the system by the manner in which the components are
arranged with respect to one another (‘placement symmetry’). But if the
composite system is found to have a *higher*
symmetry than what can be accounted for by these two factors, then that extra,
unexpected symmetry is what I call *latent symmetry*. It is as if this additional symmetry was lying latent or
dormant in the equal or equivalent components, and became manifest only when
the components came together to form the composite system. To accommodate such
a possibility, I enunciate in this book a new *symmetry composition principle*. According to it: When the occurrence of a symmetry implies
the coexistence of two or more equal or equivalent building blocks, the overall
symmetry is either the product of the building-block-symmetry group and the
placement-symmetry group, or there is an *additional*
component which arises from the latent symmetry present in the building blocks.

The emergence of symmetry in thermodynamically open
composite systems can be traced ultimately to the second law of thermodynamics,
which is therefore the

*primary*organizing principle. How this principle operates in various diverse systems is discussed in this book. It is argued that the same explanation holds, whether it is the symmetry of a crystal, or that of a complex social network.
Symmetry of complex networks is, in fact, another
major theme of this book. That real-life networks should possess any symmetry
at all may come as a surprise. But by now we should all be reconciled to the
fact that there is something about symmetry which touches everything in our
universe. The present edifice of science in general, and physics in particular,
would be unthinkable without symmetry. There is a lot of symmetry even in
biological systems.

We are surrounded not only by symmetry, but also
broken symmetry. In fact, we see more of broken
symmetry than intact symmetry. From the Big Bang onwards, as our universe
cooled and expanded, a series of symmetry-breaking transitions occurred,
leading eventually to the complexity of life we see today. This book is an
attempt to explain, in an accessible language, the interplay between latent,
manifest, and broken symmetry.

Vinod
Wadhawan

Bengaluru

August
2011

**Preface to the Second Edition**

The book has been revised and updated substantially. In particular, gauge symmetry, which was discussed only briefly in the first edition, has been given the prominence it deserves. A new chapter has been added to deal with it in some detail.

Another
new feature of this edition is the introduction of my notion of

*potential symmetry*. It is similar to latent symmetry, but not identical to it. Latent symmetry is a kind of potential symmetry which becomes manifest symmetry when the conditions are just right. But potential symmetry is not always latent symmetry; in fact, it is only rarely so. Introduction of the notion of potential symmetry enables us to enunciate what I call*the fundamental theorem of symmetry*. It says that any spontaneously occurring symmetry of an object or system comprising of equal or equivalent subparts is nothing but a self-organized manifestation of the potential symmetry residing in its subparts.
Vinod
Wadhawan

Bengaluru

July
2018

**Contents**

#
**Foreword xi**

#
**Preface xiii**

**1. Overview 1**

#
**2. Symmetry Fundamentals 9**

# 2.1 Definition of symmetry 9

## 2.2 Analogy and classification are symmetry 11

2.3 Reduction
is symmetry 11

2.4 Reproducibility is symmetry 13

2.5 Predictability is symmetry 14

2.6 The symmetry principle 15

2.7 Thermodynamics and the symmetry principle 16

2.8 Ugly symmetry 17

**3. Group-Theoretical Description of Symmetry 21**

3.1 Discrete groups 21

3.2 Coset decomposition of a group 23

3.3 Lagrange theorem for subgroups 25

3.4 Symmetry group of a crystal 25

3.5 Continuous groups 27

3.6 Permutation groups 27

3.7 Special unitary groups 27

3.8 Topological space, open sets 28

3.9 Morphisms, categories 29

3.10 Semigroups, groupoids 30

3.11 Lie groups 32

**4. Network Theory 39**

4.1 Mathematical networks 39

4.2 Clustering coefficient 42

4.3 Permutation symmetry in graphs and networks 43

4.4 Real-life networks 45

4.5 Scale-free networks 46

**5. Self-organization and Symmetry 47**

5.1 Growth of a crystal as an ordering
process 47

5.2 Similar linkage patterns and symmetry 49

5.3 Symmetry as a secondary organizing principle 50

5.4 Symmetry and biology 52

**6. The Different Types of Exact and Approximate Symmetry 59**

6.1 Crystallographic symmetry 59

6.2 Space symmetry and time symmetry 60

6.3 Permutational and more general symmetries of graphs 60

6.4 Approximate symmetry of graphs 61

6.5 Symmetry in real-life networks 62

6.6 Structural

*vs*. statistical equivalence and latent symmetry 69

**7. Symmetry of Composite Systems 71**

7.1 The Curie principle 71

7.2 The Curie-Shubnikov principle 73

7.3 Interplay between dissymmetrization and symmetrization 77

7.4 The Hermann theorem of crystal physics, and its applications 77

7.5 Hexply configurations for nanocomposites 79

**8. Gauge Symmetry 81**

8.1 Introduction
81

8.2 Gauge-symmetry groups 84

8.3 Noether’s theorems 86

**9. Phase Transitions and Broken Symmetry 93**

9.1 Liberal
meanings of the term ‘phase transition’ 93

9.2 Spontaneous breaking of symmetry 94

9.3 The Landau theory of phase transitions 95

9.4 Ferroic phase transitions and domain structure 97

9.5 Prototype symmetry 98

9.6 The symmetry compensation law 98

9.7 Continuous broken symmetries 99

9.8 Discrete broken symmetries 105

9.9 Broken symmetry and biology 105

9.10 The principle of local activity 108

**10. Particle Physics, Cosmology, and the Search for New Symmetries111**

10.1 The Standard Model of Particle
Physics 111

10.2 Beyond the Standard Model 121

10.3 Origin of our universe 124

**11. Latent Symmetry, Potential Symmetry, and the Symmetry Composition Principle 129**

11.1 Latent symmetry and potential
symmetry 129

11.2 The distinction between potential symmetry and latent symmetry

11.3 The fundamental theorem of symmetry 133

11.4 The symmetry composition principle 133

11.5 Placement symmetry 136

11.6 Latent symmetry and algorithmic information 137

**12. Group-Theoretical Determination of Latent Symmetry 139**

12.1 Formal definition of latent
symmetry 139

12.2 Litvin’s partition theorem for latent symmetry 140

12.3 Latent symmetry and domain-average engineered ferroic materials 144

12.4 An example of how ignorance about latent symmetry can

lead
to errors 145

12.5 The role of placement symmetry in revealing latent symmetry 148

12.6 Concluding remarks 150

**13. Symmetry of Complex Networks 151**

13.1 Latent symmetry in complex
networks 151

13.2 Measures of symmetry of networks 154

13.3 Origins of symmetry in complex networks 156

13.4 The similar-linkage-pattern model for symmetry 157

13.5 The free-energy landscape for biological networks 158

13.6 Social networks and the meaning of cohesive energy 160

**14. Afterword 163**

**Bibliography 167**

**Index 179**

**Acknowledgements 187**

**About the Author 189**