Friday, 6 July 2018

Second Edition of Vinod Wadhawan’s Book on Symmetry (6 July 2018)

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

Vinod Wadhawan

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
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
July 2018


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

Wednesday, 17 January 2018


UNDERSTANDING NATURAL PHENOMENA: Self-Organization and Emergence in Complex Systems


Vinod Wadhawan

Book details

Paperback: 518 pages

Publisher: CreateSpace Independent Publishing Platform; 2nd edition (January 17, 2018)

Language: English

ISBN-10: 1548527939

ISBN-13: 978-1548527938

Product Dimensions: 6.7 x 1.3 x 9.6 inches

Shipping Weight: 2.2 pounds


 Legend for the front cover

A flower is a work of art, but there is no artist involved. The flower evolved from lesser things which, in turn, evolved from still lesser things, and so on, all the way down. For example, the symmetry of a flower is the end result of a long succession of spontaneous processes and events, as also of some simple ‘local rules’ in operation, all constrained, even aided, by the infallible second law of thermodynamics for ‘open’ systems. In fact, the second law is the mother of all organizing principles, leading to the enormous amounts of cumulative self-organization, structure, symmetry, and ‘emergence’ we see in Nature.

About the book

Science is all about trying to understand natural phenomena under the strict discipline imposed by the celebrated scientific method. Practically all the systems we encounter in Nature are dynamical systems, meaning that they evolve with time. Among them there are the ‘simple’ or ‘simplifiable’ systems, which can be handled by traditional, reductionistic science; and then there are 'complex’ systems, for which nonreductionistic approaches have to be attempted for understanding their evolution. In this book the author makes a case that a good way to understand a large number of natural phenomena, both simple and complex, is to focus on their self-organization and emergence aspects. Self-organization and emergence are rampant in Nature and, given enough time, their cumulative effects can be so mind-boggling that many people have great difficulty believing that there is no designer involved in the emergence of all the structure and order we see around us. But it is really quite simple to understand how and why we get so much ‘order for free’. It all happens because, as ordained by the infallible second law of thermodynamics, all ‘thermodynamically open’ systems in our ever-expanding and cooling (and therefore gradient-creating) universe constantly tend to move towards equilibrium and stability, often ending up in ordered configurations. In other words, order emerges because Nature tends to find efficient ways to annul gradients of all types.

This book will help you acquire a good understanding of the essential features of many natural phenomena, via the complexity-science route. It has four parts: (1) Complexity Basics; (2) Pre-Human Evolution of Complexity; (3) Humans and the Evolution of Complexity; and (4) Appendices. The author gives centrestage to the second law of thermodynamics for ‘open’ systems, which he describes as ‘the mother of all organizing principles’. He also highlights a somewhat unconventional statement of this law: ‘Nature abhors gradients’.

The book is written at two levels, one of which hardly uses any mathematical equations; the mathematical treatment of some relevant topics has been pushed to the last part of the book, in the form of ten appendices. Therefore the book should be accessible to a large readership. It is a general-science book written in a reader-friendly language, but without any dumbing down of the narrative.


In medieval times, our understanding of the world around us was primarily in the realm of religion and magic. However, it was in the 15th century that a more rational approach to the study of nature began to appear, followed in the early 18th century by the so-called period of enlightenment. Nonetheless, the role of religion continued to dominate thought right through the 19th and twentieth centuries. Yet, here we are in the 21st century, when one would have thought that rationality would be the order of the day, we are nonetheless still surrounded by irrationality, and partly religious magical thinking. You have only to type the word “crystal” into Google to see page after page on the magical healing of crystals. For we scientists, such beliefs make no sense at all and even can be seen as an attack against the scientific method itself. No doubt, Nature is observed to be complex and at times may seem to be mysterious, but that does not mean that we should give up and substitute the concept of “belief” for true scientific examination. This is why the material described in this book is so useful and important to understand today.

Vinod Wadhawan has been a crusader for rationality in thinking and public discourse for many years. Though this book has been designed as a comprehensive textbook on complexity science, it serves many other purposes as well. He explains how the known processes and understandings of complex systems can develop from often simple beginnings. While such happenings may often seem to the layman to be strange or even magical, they are generally susceptible to scientific reasoning. For example, consider the appearance in a fluid of regular hexagonal-shaped cells when the fluid is under a large temperature gradient. This beautiful phenomenon is called Bénard convection and is fully understood once one appreciates the underlying thermal convection currents in the fluid.  As Vinod quotes from others, “Nature abhors gradients”.

Here you will read about a mixture, or better a fusion, of philosophical and scientific ideas, in a rather accessible language. After all the field of physics was, and is to this day in Scotland, known as Natural Philosophy. This soon gets us into a discussion of determinism and whether free will exists, subjects that have before them centuries of discussion. One of the means of rationalising the ways of nature is through the now generally well-accepted ideas inherent in thermodynamics, especially the Second Law for open systems. The law itself is not provable, but as with so many examples in science, leads to conclusions that can be tested. Despite this lack of direct proof, the laws of thermodynamics have stood the test of time and we do not know of exceptions. Wadhawan makes considerable use of this Law in explaining the phenomena associated with changes from simple to complex behaviour.

An important message that suffuses the book is that most complex systems are far too complex to be understandable in terms of the usual reductionistic approach of conventional science. One just cannot set up and solve a tractable number of differential equations for catching the essence of most of the complex systems. One has to look beyond reductionism, and attempt a holistic approach. Very often, difference equations come to the rescue. The useful tip in this book seems to be: work with difference equations if you cannot work with differential equations in a meaningful way for trying to comprehend a complex system. The book gives a pride of place to the subject of cellular automata for this reason.

So, in the first part of the book the reader is treated to a whole range of topics from concepts of evolution, relativity, quantum theory through the fundamental ideas of symmetry, particle physics, chaos theory, and causes of complexity in nature. Vinod then takes us on a tour of pre-human evolution of complexity, addressing knotty questions such as the meaning of life (but not 42 as in the Hitchhiker’s Guide to the Galaxy!), and the fundamental basis of the Darwinian view of the evolution of species. Darwinian evolution is a subset of dynamical evolution. Dynamical evolution, controlled as well as aided by the second law of thermodynamics for open systems, is at the heart of what the science of complex systems is all about. This fact is brought out very clearly in the book.

In the next part of the book we meet the evolution of complexity during human existence, including the founding of various algorithms, robotics and functions of the human brain. Many of these problems today remain unresolved, of course, but such is the nature of the scientific method that constant progress is actually being made in their understanding. In this we are currently living through a remarkable period of rapid developments of ideas. Appearance of humans on the scene has led to a rapid increase in the rate of evolution of complexity. Even more significantly, our remarkable progress in the field of artificial intelligence has brought up a critical situation in which our robots are already getting better than us in more and more aspects. As pointed out by Vinod, the self-evolution of robots can occur exponentially rapidly, whereas we humans are hardly evolving on that time scale.  There remains the question as to whether the human brain itself can be reproduced artificially. We are now making considerable progress in understanding how the brain works and one can argue that surely there will come a time when science will enable a complete artificial intelligence to be built, complete with the ability to reason, to think and perhaps even develop a conscience. Precisely what this means is a hot topic of current debate. Perhaps the relatively new field of quantum computing will open this door; but, of course, prediction of the future is difficult and likely to be wrong! What is certain though is that artificial intelligence is advancing at such a remarkable rate that the old science fiction view of robots is beginning to look seriously realistic. The other day I watched a small machine running around independently mowing the lawn in a neighbour’s garden. I saw with astonishment how it carefully manoeuvred itself around objects such as a chair on the lawn. Look at mobile phones. My first computer had 8K store on a magnetic drum, but today’s mobile phones are several thousand times more powerful and are capable, for instance, of allowing photographs to be taken at phenomenal resolution. This has been a triumph of the development of many fields, including lens design, new materials and new software techniques, let alone the ability to make telephone calls. Who would have thought of such  phenomena outside of the world of scientific fiction a few years ago?

Clearly the future belongs to robots. If they turn out, for instance,  to be made mainly of inorganic materials, they will outlast all humans, and this even raises the question as to whether humans as a species will continue to exist or even if they need to exist. These are deep, possibly troubling, but certainly exciting prospects to consider, both as a matter of practicality and of ethics. While we still have some control on robots, we should apply our minds to what kind of a future we want for ourselves. And good decisions in this regard require a basic minimum understanding of the science of complex systems by a wide cross section of society. We are living through a very special time. This is where this book comes in.

It can be seen that Vinod Wadhawan has set himself a momentous and daunting task in putting together into a single book so many apparently diverse concepts and ideas that might at first seem to be so disparate as to be intractable. But in fact, we see that there are common threads, often called simply the Laws of Nature by some. These laws are rapidly becoming ever more understood and a careful reading of this book will help us with our observations of the world around us, so that though we may continue to ask “why?”, sometimes we will come up with a rational explanation.

This book is epic in the sense that it covers so much ground that one is left somewhat dizzy. And yet, it all makes sense once one realizes how it is possible for something that is complex, for example a flower, to evolve via natural processes from humble beginnings. After all, starting with single-cell creatures such as amoebae we follow a complicated but rational evolutionary path to arrive at the most complex organizations that we know of – ourselves.    So, if you follow the logic of this book, starting with the basic concepts of thermodynamics, symmetry, quantum theory and so on, you will be treated to many many thought-provoking ideas, which will likely challenge your own preconceptions and leave you thirsting for more.

Now a few words about the author. I have personally known Vinod for a long time, ever since he came to work for a while in my laboratory. At the time he was working on a phenomenon known as ferrogyrotropy, wherein certain crystals that show chiral (“handedness” if you prefer) properties, the chiral properties can be switched by application of an external stimulus. I think he was the only person in the world then studying this phenomenon. After he left Oxford in 1980 we kept in constant contact, with Vinod playing an important role as one of my regional editors with the international journal, Phase Transitions, for which I was the general editor. I noted that every paper sent to the journal from Indian authors had been closely edited by Vinod beforehand, and so I knew that I could rely entirely automatically on his personal skill and judgement. Vinod’s ability at writing in English is commendable: he obviously has had the benefit of a classical education. Since returning to India, he has produced several books, starting with topics related to ferroic materials and smart structures, and eventually moving on to the more philosophical concepts that have to do with the science of complexity. So, we come to this his latest book, where Vinod has supplied us with many nice examples of complexity and how it arises, and as a result the reader will finish the book much more informed than at the beginning. That, after all, is the purpose of a book like this.

A. M. Glazer

Emeritus Professor of Physics and Emeritus Fellow of Jesus College, Oxford

Former Vice President, International Union of Crystallography

January 2018


I am a scientist and I take pride in the fact that we humans have invented and perfected the all-important scientific method for investigating natural phenomena. Wanting to understand natural phenomena is an instinctive urge in all of us. In this book I make a case that taking the complexity-science route for satisfying this urge can be a richly rewarding experience. Complexity science enables us (fully or partially) to find answers to even the most fundamental questions we may ask about ourselves and about our universe. We call them the Big Questions: How did our universe emerge out of ‘nothing’ at a certain point in time; or is it that it has been there always? Why and how has structure arisen in our universe: galaxies, stars, planets, life forms? How did life emerge out of nonlife? How does intelligence emerge out of nonintelligence? These are difficult questions. But, as Mark Twain is said to have said, ‘there is something fascinating about science. One gets such wholesale of conjecture out of such a trifling investment of fact’. As you will see in this book, the Big Questions, as also many others, can be answered with a good amount of credibility by using just the following ‘trifling investment of facts’:

1. Gradients tend to be obliterated spontaneously. Concentration gradients, temperature gradients, pressure gradients, etc. all tend to decrease spontaneously, till a state of equilibrium is reached, after which the gradients cannot fall any further. This is actually nothing but a nonstatistical-mechanical version of the second law of thermodynamics. [Why do gradients arise at all, at a cosmic level? The original cause of all gradients in the cosmos is the continual expansion and cooling of our universe. At the local (terrestrial) level, the energy impinging on our ecosphere from the Sun is the main factor creating gradients.]

2. It requires energy to prevent a gradient from annulling itself, or to create a new gradient. A refrigerator works on this principle, as also so many other devices.

3. Left to themselves, things go from a state of less disorder to a state of more disorder, spontaneously. This is the more familiar version of the second law of thermodynamics. Examples abound. Molecules in a gas occupy a larger volume spontaneously if the larger volume is made available to them; but there is practically no way they would occupy the smaller volume again, on their own.

4. If a system is not left to itself, i.e., if it is not an isolated system and can therefore exchange energy and/or matter with its surroundings, then a state of lower disorder can sometimes arise locally. [This is in keeping with the second law of thermodynamics, as generalized to cover ‘thermodynamically open’ systems also.] Growth of a crystal from a fluid is an example. A crystal has a remarkably high degree of order and design, even though there is no designer involved. To borrow a phrase from Stuart Kauffman, this is ‘order for free’.

5. If a sustained input of energy drives a system far away from equilibrium, the system may develop a structure or tendencies which enable it to dissipate energy more and more efficiently. This is called dissipation-driven adaptive organization. England (2013) has shown that all dynamical evolution is more likely to lead to structures and systems which get better and better at absorbing and dissipating energy from the environment.

6. The total energy of the universe is conserved. This is known as the energy-conservation principle. Since energy and mass are interconvertible, the term ‘energy’ used here really means ‘mass plus energy’.

7. Natural phenomena are governed by the laws of quantum mechanics. Classical mechanics, though adequate for understanding many day-to-day or ‘macroscopic’ phenomena, is only a special, limiting, case of quantum mechanics.

8. There is an uncertainty principle in quantum mechanics, one version of which says that the energy-conservation principle can be violated, though only for a very small, well-specified duration. The larger the violation of energy conservation, the smaller this duration is.

9. It can be understood fully in terms of the second law of thermodynamics that in a system of interacting entities, entirely new (unexpected) behaviour or properties can arise if the interactions are appropriate and strong enough. ‘More is different’ (Anderson 1972). The technical term for this occurrence is emergence. Complexity science is mostly about self-organization and emergence, and we shall encounter many examples of them in this book. To mention a couple of them here: the emergence of life out of nonlife; and the emergence of human intelligence in a system of nonintelligent entities, namely the neurons. Interestingly, the second law of thermodynamics is itself an emergent law. The motion of a molecule is governed by classical or ‘Newtonian’ mechanics, which has time-reversal symmetry, meaning that if you could somehow reverse the direction of time, the Newtonian equations of motion would still hold. And yet, when you put a large number of these molecules together, there are interactions among them and there emerges a direction of time: Time increases in the direction in which overall disorder increases. As I shall discuss later in the book, even the causality principle is an emergent principle.

10. The dynamics of evolution of a complex system of interacting entities is mostly through the operation of ‘local rules’. Chua (1998) has introduced the important notion of cellular nonlinear networks (CNNs), and enunciated a local-activity dogma. According to it, in order for a ‘nonconservative’ system or model to exhibit any form of complexity, the associated CNN parameters must be such that that either the cells or their couplings are locally active.

11. The most adaptable are the most likely to survive and propagate. Any species, if it is not to become extinct, must be able to survive and propagate, in an environment in which there is always some intra-species and/or inter-species competition because different individuals may all have to fight for the same limited resources like food or space. The fittest individuals or groups for this task (i.e., the most adaptable ones) stand a greater chance of winning this game and, as a result, the population gets better and better (more adapted) at survival and propagation in the prevailing conditions: the more adaptable or ‘fitter’ ones are not only more likely to survive, but also stand a greater chance to pass on their genes to the next generation.

It is remarkable that an enormous number and variety of natural phenomena can be understood in terms of just these few ‘commonsense’ facts, by adopting the complexity-science approach. Complexity science helps us understand, to a small or large extent, even those natural phenomena which fall outside the scope of conventional reductionistic science.

What is complexity science, and how is its operational space different from that of conventional science? Let us begin by answering the question: What does the phrase ‘system under investigation’ mean in conventional science? Strictly speaking, since everything interacts with everything else, the entire cosmos is one big single system. But such an approach cannot take us very far because it is neither tractable nor useful. So, depending on our interest, we define a subsystem which is a ‘quasi-isolated system’. A quasi-isolated system is an imaginary construct, such that what is outside it can be, to a good approximation, treated as an unchanging (usually large) ‘background’, or ‘heat bath’ etc. This approach is so common in conventional science that we just say ‘system’ when what we really mean is a carefully identified quasi-isolated system. An example from rocket science will illustrate the point. For predicting the initial trajectory of a rocket, we can assume safely that a truck moving an adequate distance away from the launching site will not affect the trajectory significantly. Conventional science deals mostly with such ‘simple’ or ‘simplifiable’ systems. Complexity science, by contrast, deals with systems which must be treated in their totality; for them it is mostly not possible to identify a ‘quasi-isolated subpart’.

By definition, a complex system is one which comprises of a large number of ‘members’, ‘elements’ or ‘agents’, which interact substantially with one another and with the environment, and which have the potential to generate qualitatively new collective behaviour. That is, there can be an emergence of new (unexpected) spatial, temporal, or functional structures or patterns. Different complex systems have different ‘degrees of complexity’, and the amount of information needed to describe the structure and function of a system is one of the measures of that degree of complexity (Wadhawan 2010).

‘Complexity’ is something we associate with a complex system (defined above). It is a technical term, and does not  mean the same thing as ‘complicatedness’.

The idea of writing this book took shape when I was working on my book Smart Structures: Blurring the Distinction between the Living and the Nonliving (Wadhawan 2007). Naturally, there was extensive exposure to concepts from complexity science. Like the subject of smart structures, complexity science also cuts across various disciplines, and highlights the basic unity of all science. The uneasy feeling grew in me that, in spite of the fact that complexity is so pervasive and important, it is not introduced as a well-defined subject even to science students. They are all taught, say, thermodynamics and quantum mechanics routinely, but not complexity science. Even among research workers, although a large number are working on one complex system or another (and not just in physics or chemistry, but also in biology, brain science, computational science, economics, etc.), not many have learnt about the basics of complexity science in a coherent manner at an early stage of their career. I have tried to write a book on complexity that takes this subject to the classroom at a fairly introductory but comprehensive level. There is no dumbing down of facts, even at the cost of appearing ‘too technical’ at times.

Here are some examples of complex systems: beehives; ant colonies; self-organized supramolecular assemblies; ecosystems; spin-glasses and other complex materials; stock markets; economies of nations; the world economy; the global weather pattern. The origin and evolution of life on Earth was itself a series of emergent phenomena that occurred in highly complex systems. Evolution of complexity is generally a one-way traffic: The new emergent features may (in principle) be deducible from, but are not reducible to, those operating at the next lower level of complexity. Reductionism stands discounted.

As I said earlier, emergent behaviour is a hallmark of complex systems. Human intelligence is also an emergent property: Thoughts, feelings, and purpose result from the interactions among the neurons. Similarly, even memories are emergent phenomena, arising out of the interactions among the large number of ‘unmemory-like’ fragments of information stored in the brain.

What goes on in a complex system is essentially as follows: There is a large number of interacting agents, which may be viewed as forming a network. In the network-theory jargon, the agents are the ‘nodes’ of the network, and a line joining any two nodes (i.e., an ‘edge’) represents the interaction between that pair of agents. Any interaction amounts to communication or exchange of information. The action or behaviour of each agent is determined by what it ‘sees’ others doing, and its actions, in turn, determine what the other agents may do. Further, the term game-playing is used for this mutual interaction in the case of those complex systems in which the agents are ‘thinking’ organisms (particularly humans). Therefore a partial list of topics covered in this book is: information theory; network theory; cellular automata; game theory.

Exchange of information in complex systems, controlled like other macroscopic phenomena by the second law of thermodynamics, leads to self-organization and emergence. In particular, biological evolution is a natural and inevitable consequence of such ongoing processes, an additional factor for them being the cumulative effects of mutations and natural selection. This book has chapters on evolution of complexity of all types: cosmic, chemical, biological, artificial, cultural.

Networked or ‘webbed’ systems have the all-important nonlinearity feature. In fact, nonlinear response, in conjunction with substantial departure from equilibrium, is the crux of complex behaviour. There are many types of nonlinear systems. The most important for our purposes in this book are those in which, although the output (y) is indeed proportional to the input (x), the proportionality factor (m) is not independent of the input; i.e., m is not a constant factor, but rather varies with what x is. For a linear system we have y = m x, with m having a fixed value, not varying with x. But for a nonlinear system, the equation becomes   y = m(x) x; now m is not a constant.  This has far-reaching consequences for the (always networked) complex system. In particular, its future progression of events is very sensitive to conditions at any particular point of time (the so-called ‘initial conditions’). This sensitivity to initial conditions is also the hallmark of chaotic systems. In fact, there is a well-justified viewpoint that it is impossible to discuss several types of complex systems without bringing in concepts from chaos theory. And, what is more, complex systems tend to evolve to a configuration wherein they can operate near the so-called edge of chaos (neither too much order, nor too much chaos). There is a chapter on chaos which elaborates on these things.

Inanimate systems can also be complex. Whirlpools and whirlwinds are familiar examples of dynamic nonbiological complex systems. Even static physical systems like some nanocomposites may exhibit properties that cannot always be deduced from those of the constituents of the composite. A particularly fascinating class of complex materials are the so-called multiferroics. A multiferroic is actually a ferroic crystalline material (a ‘natural’ composite) which just refuses to be homogeneous over macroscopic length scales, so that the same crystal may be, say, ferroelectric in some part, and ferromagnetic in another. In a multiferroic, two or all three of the electric, magnetic and elastic interactions compete in a delicately balanced manner, and even a very minor local factor can tilt the balance in favour of one or the other. This class of materials offers great scope for basic research and for device applications, particularly in smart structures.

The current concern about ecological conservation and global warming points to the need for a good understanding of complex systems, particularly their holistic nature. Mother Earth is a single, highly complex, system, now increasingly referred to as the System Earth.

A better understanding of complexity may well become a matter of life and death for the human race. And the subject of complexity science is still at the periphery of science. It has not yet become mainstream, in the sense that it is not taught routinely even at the college level. That cannot go on.

There are already a substantial number of great books on complexity science, and I have drawn on them. But I believe that this book is student-friendly and teacher-friendly, and it brings home the all-pervasive nature of the subject. Here are its salient features:

1. It provides a comprehensive update on the subject.

2. It can serve as introductory or supplementary reading for an undergraduate or graduate course on any branch of complexity science.

3. Practically all the mathematical treatment of the subject has been pushed to the appendices at the end of the book, so the main text can be comprehended even by those who are not too comfortable with equations. This is important because a large fraction of the educated public must get the hang of the nature of complexity, so that we can successfully meet the challenges posed to our very survival as a species.

4. Both among scientists and nonscientists there is a large proportion of people who are insufficiently trained about the explaining power of complexity science when it comes to some of the deepest puzzles of Nature and, hopefully, this book would help remedy the situation to some extent.

5. The book has a certain all-under-one-roof character. The topics covered are so many and so diverse that it would be well-nigh impossible for a reader, specializing in a particular branch of complexity science, not to get exposed to what is going on in the rest of complexity science! This is important, because using the insights gained in one complex system for trying to understand another complex system is the hallmark of complexity science.

6. A proper understanding of what complexity science has already achieved will also help discredit many of the claims of mystics, supernaturalists, and pseudoscientists.


September, 2017

Preface to the Second Edition

A number of minor corrections and other improvements have been incorporated. The font size has been reduced by 10%. New information has been added, and some less relevant material has been removed.

Vinod Wadhawan
September 2017

Preface to the Second Print

This print includes a Foreword by Prof. A. M. Glazer of the University of Oxford. A former Vice President of the International Union of Crystallography, he is a veteran crystallographer and a great teacher. I am grateful to him for his kind words and also for many other inputs.

Vinod Wadhawan
January 2018




I. Complexity Basics

1. Overview 1

1.1 Preamble 3

1.2 A whirlpool as an example of self-organization 5

1.3 Spontaneous pattern formation: the Bénard instability 6

1.4 Recent history of investigations in complexity science 8

1.5 Organization of the book 8

2. The Philosophical and Computational Underpinnings of Complexity Science 9

2.1 The scientific method for understanding natural phenomena 9

2.2 Reductionism and its inadequacy for dealing with complexity 12

2.3 The Laplace demon 13

2.4 Holism 15

2.5 Emergence 16

2.6 Scientific determinism, effective theories 17

2.7 Free will 18

2.8 Actions, reactions, interactions, causality 21

2.9 The nature of reality 23

3. The Second Law of Thermodynamics 25

3.1 The second law for isolated systems 25

3.2 Entropy 26

3.3 The second law for open systems 27

3.4 Nucleation and growth of a crystal 29

3.5 The second law is an emergent law 32

3.6 Emergence, weak and strong 33

3.7 Nature abhors gradients 33

3.8 Systems not in equilibrium 34

3.9 Thermodynamics of small systems 35

4. Dynamical Evolution 37

4.1 Dynamical systems 37

4.2 Phase-space trajectories 37

4.3 Attractors in phase space 38

4.4 Nonlinear dynamical systems 40

4.5 Equilibrium, stable and unstable 40

4.6 Dissipative structures and processes 42

4.7 Bifurcations in phase space 43

4.8 Self-organization and order in dissipative structures 44

5.Relativity Theory and Quantum Mechanics 47

5.1.Special theory of relativity 47

5.2 General theory of relativity 49

5.3 Quantum mechanics 52

5.4 Summing over multiple histories 55

6.The Nature of Information 57

6.1 Russell’s paradox 57

6.2 Hilbert’s formal axiomatic approach to mathematics 58

6.3 Gödel’s incompleteness theorem 59

6.4 Turing’s halting problem 60

6.5 Elementary information theory 63

6.6 Entropy means unavailable or missing information 65

6.7 Algorithmic information theory 66

6.8 Algorithmic probability and Ockham’s razor 69

6.9 Algorithmic information content and effective complexity 70

6.10 Classification of problems in terms of computational complexity70

6.11 ‘Irreducible complexity’ deconstructed 71

7.Darwinian Evolution, Complex Adaptive Systems, Sociobiology 75

7.1 Darwinian evolution 75

7.2 Complex adaptive systems 77

7.3 The inevitability of emergence of life on Earth 79

7.4 Sociobiology, altruism, morality, group selection 81

8. Symmetry is Supreme 83

8.1 Of socks and shoes 83

8.2 Connection between symmetry and conservation laws 83

8.3 Why so much symmetry? 84

8.4 Growth of a crystal as an ordering process 85

8.5 Broken symmetry 86

8.6 Symmetry aspects of phase transitions 88

8.7 Latent symmetry 89

8.8 Latent symmetry and the phenomenon of emergence in complex systems 90

8.9 Broken symmetry and complexity 91

8.10 Symmetry of complex networks 92

9. The Standard Model of Particle Physics 95

9.1 The four fundamental interactions 95

9.2 Bosons and fermions 96

9.3 The standard model and the Higgs mechanism 98

10. Cosmology Basics 101

10.1 The ultimate causes of all cosmic order and structure 101

10.2 The Big Bang and its aftermath 102

10.3 Dark matter and dark energy 105

10.4 Cosmic inflation 108

10.5 Supersymmetry, string theories, M-theory 109

10.6 Has modern cosmology got it all wrong? 111

11. Uncertainty, Complexity, and the Arrow of Time 117

11.1 Irreversible processes, and not entropy, determine the arrow of time 117

11.2 Irreversible processes can lead to order         117

11.3 The arrow of time and the early universe 118

11.4 When did time begin? 119

11.5 Uncertainty and complex adaptive systems 120

12. The Cosmic Evolution of Complexity 123

12.1 Our cosmic history 123

12.2 We are star stuff 124

13. Why Are the Laws of Nature What They Are? 127

13.1 The laws of Nature in our universe 127

13.2 The anthropic principle 128

14. The Universe is a Quantum Computer 131

14.1 Quantum computation 131

14.2 Quantum entanglement 132

14.3 The universe regarded as a quantum computer 133

15. Chaos, Fractals, and Complexity 135

15.1 Nonlinear dynamics 135

15.2 Extreme sensitivity to initial conditions 136

15.3 Chaotic rhythms of population sizes 137

15.4 Fractal nature of the strange attractor 139

15.5 Chaos and complexity 141

16. Cellular Automata as Models of Complex Systems 143

16.1 Cellular automata 143

16.2 Conway’s Game of Life 143

16.3 Self-reproducing automata 145

16.4 The four Wolfram classes of cellular automata 146

16.5 Universal cellular automata 147

17. Wolfram’s ‘New Kind of Science’ 149

17.1 Introduction 149

17.2 Wolfram’s principle of computational equivalence (PCE) 150

17.3 The PCE and the rampant occurrence of complexity 151

17.4 Why does the universe run the way it does? 152

17.5 Criticism of Wolfram’s NKS 153

18. Swarm Intelligence 157

18.1 Emergence of swarm intelligence in a beehive 157

18.2 Ant logic 159

18.3 Positive and negative feedback in complex systems 160

19. Nonadaptive Complex Systems 163

19.1 Composite materials 163

19.2 Ferroic materials 163

19.3 Multiferroics 164

19.4 Spin glasses 165

19.5 Relaxor ferroelectrics 166

19.6 Relaxor ferroelectrics as vivisystems 167

20. Self-Organized Criticality, Power Laws 169

20.1 The sandpile experiment 169

20.2 Power-law behaviour and complexity 170

20.3 Robust and nonrobust criticality 173

21. Characteristics of Complex Systems 175

II. Pre-Human Evolution of Complexity

22. Evolution of Structure and Order in the Cosmos 183

22.1 The three eras in the cosmic evolution of complexity 183

22.2 Chaisson’s parameter for quantifying the degree of complexity 183

22.3 Cosmic evolution of information 184

22.4 Why so much terrestrial complexity? 186

23. The Primary and Secondary Chemical Bonds 187

23.1 The primary chemical bonds 187

23.2 The secondary chemical bonds 189

23.3 The hydrogen bond and the hydrophobic interaction 190

24. Cell Biology Basics 193

25. Evolution of Chemical Complexity 197

25.1 Of locks and keys in the world of molecular self-assembly 197

25.2 Self-organization of matter 199

25.3 Emergence of autocatalytic sets of molecules 202

25.4 Positive feedback, pattern formation, emergent phenomena 204

25.5 Pattern formation: the BZ reaction 205

26. What is Life? 207

26.1 Schrödinger and life 207

26.2 Koshland’s ‘seven pillars of life’ 209

27. Models for the Origins of Life 211

27.1 The early work 211

27.2 The RNA-world model for the origin of life 213

27.3 Dyson’s proteins-first model for the origins of life 215

27.4 Why was evolution extremely fast for the earliest life? 218

28. Genetic Regulatory Networks and Cell Differentiation 219

28.1 Circuits in genetic networks 220

28.2 Kauffman’s work on genetic regulatory networks 221

29. Ideas on the Origins of Species: From Darwin to Margulis 223

29.1 Darwinism and neo-Darwinism 223

29.2 Biological symbiosis and evolution 225

29.3 What is a species 227

29.4 Horizontal gene transfer in the earliest life forms 228

29.5 Epigenetics 229

30. Coevolution of Species 231

30.1 Punctuated equilibrium in the coevolution of species 231

30.2 Evolutionarily stable strategies 232

30.3 Of hawks and doves in the logic of animal conflicts 234

30.4 Evolutionary arms races and the life-dinner principle 236

31. The Various Energy Regimes in the Evolution of Our Ecosphere 241

31.1 The thermophilic energy regime 242

31.2 The phototrophic energy regime 244

31.3 The aerobic energy regime 245

III. Humans and the Evolution of Complexity

32. Evolution of Niele’s Energy Staircase After the Emergence of Humans   249

32.1 The pyrocultural energy regime 249

32.2 The agrocultural energy regime 251

32.3 The carbocultural energy regime 252

32.4 The green-valley approach to System Earth 253

32.5 The imperial approach to System Earth         254

32.6 A nucleocultural energy regime? 256

32.7 A possible ‘heliocultural’ energy regime 258

33. Computational Intelligence 261

33.1 Introduction 261

33.2 Fuzzy logic 262

33.3 Neural networks, real and artificial 263

33.4 Genetic algorithms 265

33.5 Genetic programming: Evolution of computer programs 267

33.6 Artificial life 271

34. Adaptation and Learning in Complex Adaptive Systems 273

34.1 Holland’s model for adaptation and learning 273

34.2 The bucket brigade in Holland’s algorithm 274

34.3 Langton’s work on adaptive computation 276

34.4 The edge-of-chaos existence of complex adaptive systems 278

35. Smart Structures 281

35.1 The three main components of a smart structure 281

35.2 Reconfigurable computers and machines that can evolve 283

36. Robots and Their Dependence on Computer Power 287

36.1 Behaviour-based robotics 287

36.2 Evolutionary robotics 288

36.3 Evolution of computer power per unit cost 290

37. Machine Intelligence 295

37.1 Artificial distributed intelligence 295

37.2 Evolution of machine intelligence 296

37.3 The future of intelligence and the status of humans 298

38. Evolution of Language 303

39. Memes and Their Evolution 307

40. Evolution of the Human Brain, and the Nature of Our Neocortex 311

40.1 Evolution of the brain 312

40.2 The human neocortex 313

40.3 The history of intelligence 315

41. Minsky’s and Hawkins’ Models for how Our Brain Functions 319

41.1 Marvin Minsky’s ‘Society of Mind’ 319

41.2 Can we make decisions without involving emotions? 320

41.3 Hawkins’ model for intelligence and consciousness 323

42. Inside the Human Brain 325

42.1 Probing the human Brain 325

42.2 Peering into the human brain 327

43. Kurzweil’s Pattern-Recognition Theory of Mind 331

44. The Knowledge Era and Complexity Science 337

44.1 The wide-ranging applications of complexity science 337

44.2 Econophysics 338

44.3 Application of complexity-science ideas in management science 341

44.4 Cultural evolution and complexity transitions 343

44.5 Complexity leadership theory 345

44.6 Complexity science in everyday life 345

45. Epilogue 347

IV. Appendices

A1. Equilibrium Thermodynamics and Statistical Mechanics 357

A1.1 Equilibrium thermodynamics 357

A1.2 Statistical mechanics 360

A1.3 The ergodicity hypothesis 360

A1.4 The partition function 361

A1.5 Tsallis thermodynamics of small systems 361

A2. Probability Theory 365

A2.1 The notion of probability 365

A2.2 Multivariate probabilities 365

A2.3 Determinism and predictability 367

A3. Information and Uncertainty 369

A3.1 Information theory 369

A3.2 Shannon’s formula for a numerical measure of information 370

A3.3 Shannon entropy and thermodynamic entropy 371

A3.4 Uncertainty 372

A3.5 Algorithmic information theory 373

A4. Thermodynamics and Information 375

A4.1 Entropy and information 375

A4.2 Kolmogorov-Sinai entropy 376

A4.3 Mutual information and redundancy of information 377

A5. Systems Far from Equilibrium 379

A5.1 Emergence of complexity in systems far from equilibrium 379

A5.2 Nonequilibrium classical dynamics 380

A5.3 When does the Newtonian description break down? 383

A5.4 Generalization of Newtonian dynamics         384

A5.5 Pitchfork bifurcation 386

A5.6 Extension of Newton’s laws 386

A6. Quantum Theory and Particle Physics 389

A6.1 Introduction 389

A6.2 The Heisenberg uncertainty principle 389

A6.3 The Schrödinger equation 390

A6.4 The Copenhagen interpretation 391

A6.5 Time asymmetry 391

A6.6 Multiple universes 391

A6.7 Feynman’s sum-over-histories formulation 392

A6.8 Quantum Darwinism 393

A6.9 Gell-Mann’s coarse-graining interpretation 393

A6.10 Poincaré resonances and quantum theory 394

A6.11 Model-dependent realism, intelligence, existence 396

A6.12 The principle of conservation of quantum information 397

A6.13 Particle physics 398

A7. Theory of Phase Transitions and Critical Phenomena 401

A7.1 A typical phase transition 401

A7.2 Liberal definitions of phase transitions 401

A7.3 Instabilities can cause phase transitions 402

A7.4 Order parameter of a phase transition 403

A7.5 The response function corresponding to the order parameter 404

A7.6 Phase transitions near thermodynamic equilibrium 404

A7.7 The Landau theory of phase transitions 405

A7.8 Spontaneous breaking of symmetry 407

A7.9 Field-induced phase transitions 407

A7.10 Ferroic phase transitions 408

A7.11 Prototype symmetry 409

A7.12 Critical phenomena 409

A7.13 Universality classes and critical exponents 410

A8. Chaos Theory 413

A8.1 The logistic equation 413

A8.2 Lyapunov exponents 416

A8.3 Divergence of neighbouring trajectories 417

A8.4 Chaotic attractors 419

A9. Network Theory and Complexity 421

A9.1 Graphs 421

A9.2 Networks 425

A9.3 The travelling-salesman problem 426

A9.4 Random networks 427

A9.5 Percolation transitions in random networks 428

A9.6 Small-world networks 429

A9.7 Scale-free networks 431

A9.8 Evolution of complex networks 432

A9.9 Emergence of symmetry in complex networks 433

A9.10 Chua’s cellular nonlinear networks as a paradigm for emergence and complexity 435

A10. Game Theory 439

A10.1 Introduction 439

A10.2 Dual or two-player games 442

A10.3 Noncooperative games 449

A10.4 Nash equilibrium 450

A10.5 Cooperative games 450

Bibliography 453

Index 481

Acknowledgements 491

About the Author 492