Seven Clues to the Origin of Life by A.G. Cairns-Smith (1985)

The topic of the origin of life on the Earth is a branch of mineralogy. (p.99)

How did life begin? To be more precise, how did the inorganic chemicals formed in the early years of planet earth, on the molten rocks or in the salty sea or in the methane atmosphere, transform into ‘life’ – complex organisms which extract food from the environment and replicate, and from which all life forms today are ultimately descended? What, when and how was that first momentous step taken?

Thousands of biologists have devoted their careers to trying to answer this question, with the result that there are lots of speculative theories.

Alexander Graham Cairns-Smith (1931-2016) was an organic chemist and molecular biologist at the University of Glasgow, and this 120-page book was his attempt to answer the Big Question.

In a nutshell he suggested that life derived from self-replicating clay crystals. To use Wikipedia’s summary:

Clay minerals form naturally from silicates in solution. Clay crystals, like other crystals, preserve their external formal arrangement as they grow, snap, and grow further.

Clay crystal masses of a particular external form may happen to affect their environment in ways that affect their chances of further replication. For example, a ‘stickier’ clay crystal is more likely to silt up a stream bed, creating an environment conducive to further sedimentation.

It is conceivable that such effects could extend to the creation of flat areas likely to be exposed to air, dry, and turn to wind-borne dust, which could fall randomly in other streams.

Thus – by simple, inorganic, physical processes – a selection environment might exist for the reproduction of clay crystals of the ‘stickier’ shape.

Cairns-Smith’s book is densely argued, each chapter like a lecture or seminar packed with suggestive evidence about what we know about current life forms, a summary of the principles underlying Darwin’s theory of evolution, and about how we can slowly move backwards along the tree of life, speculating about how it developed.

But, as you can see from the summary above, in the end, it is just another educated guess.

Detective story

The blurb on the back and the introduction both claim the book is written in the style of a detective story. Oh no it isn’t. It is written in the style of a biology book – more precisely, a biology book which is looking at the underlying principles of life, the kind of abstract engineering principles underlying life – and all of these take quite some explaining, drawing in examples from molecular biology where required.

Sometimes (as in chapter 4 where he explains in detail how DNA and RNA and amino acids and proteins interact within a living cell) it becomes quite a demanding biology book.

What the author and publisher presumably mean is that, in attempt to sweeten the pill of a whole load of stuff about DNA and ribosomes, Cairns-Smith starts every chapter with a quote from a Sherlock Holmes story and from time to time claims to be pursuing his goal with Holmesian deduction.

You see Holmes, far from going for the easy bits first, would positively seek out those features in a case that were seemingly incomprehensible – ‘singular’ features he would call them… I think that the origin of life is a Holmesian problem. (p.ix)

Towards the very end, he remembers this metaphor and talks about ‘tracking down the suspect’ and ‘making an arrest’ (i.e. of the first gene machine, the origin of life). But this light dusting of Holmesiana doesn’t do much to conceal the sometimes quite demanding science, and the relentlessly pedagogical tone of the book.

Broad outline

1. Panspermia

First off, Cairns-Smith dismisses some of the other theories about the origin of life. He makes short work of the theories of Fred Hoyle and Francis Crick that organic life might have arrived on earth from outer space, carried in dust clouds or on meteors etc (Crick’s version of this was named ‘Panspermia’ – . I agree with him that this just relocates the problem somewhere else, but doesn’t solve it.

Cairns-Smith states the problem in three really fundamental facts:

  1. There is life on earth
  2. All known living things are at root the same (using the same carbon-based energy-gathering and DAN-replicating biochemistry)
  3. All known living things are very complicated

2. The theory of chemical evolution

In his day (the 1970s and 80s) the theory of ‘chemical evolution’ was widely thought to address the origin of life problem. This stated that lot of the basic amino acids and sugars which we find in organisms are relatively simple and so might well have been created by accident in the great sloshing oceans and lakes of pre-life earth, and that they then – somehow – came together to make more complex molecules which – somehow – learned how to replicate.

But it’s precisely on the vagueness of that ‘somehow’ that Cairns-Smith jumps. The leap from a random soup of semi-amino acids washing round in a lake and the immensely detailed and complex machinery of life demonstrated by even a tiny living organism – he selects the bacterium Escherichia coli – is just too vast a cliff face to have been climbed at random, by accident. It’s like saying if you left a bunch of wires and bits of metal sloshing around in a lake long enough they would eventually make a MacBook Air.

Cairns-Smith zeroes in on four keys aspects of life on earth which help to disprove the ‘chemical evolution’ theory.

  1. Life forms are complex systems. It is the whole machine which makes sense of its components.
  2. The systems are highly interlocked: catalysts are needed to make proteins, but proteins are needed to make catalysts; nucleic acids are needed to make proteins, yet proteins are needed to make nucleic acids;
  3. Life forms are very very complex.
  4. The system is governed by rules and conventions: the exact choice of the amino acid alphabet and the set of assignments of amino acid letters to nucleic acid words are examples.

3. The Miller-Urey experiments

Cairns-Smith then critiques the theory derived from the Miller-Urey experiments.

In 1953 a graduate student, Stanley Miller, and his professor, Harold Urey, performed an experiment that demonstrated how organic molecules could have spontaneously formed from inorganic precursors, under conditions like those posited by the Oparin-Haldane Hypothesis. The now-famous ‘Miller–Urey experiment’ used a highly reduced mixture of gases – methane, ammonia and hydrogen – to form basic organic monomers, such as amino acids. (Wikipedia)

Cairns-Smith spends four pages comprehensively demolishing this approach by showing that:

  1. the ultraviolet light its exponents claim could have helped synthesise organic molecules is in fact known to break covalent bonds and so degrade more than construct complex molecules
  2. regardless of light, most organic molecules are in fact very fragile and degrade easily unless kept in optimum conditions (i.e. inside a living cell)
  3. even if some organic molecules were created, organic chemists know only too well that there are hundreds of thousands of ways in which carbon, hydrogen, nitrogen and oxygen can combine, and most of them result in sticky sludges and tars in which nothing could ‘live’

So that:

  1. Only some of the molecules of life can be made this way
  2. Most of the molecules that would be made this way are emphatically not the ‘molecules of life’
  3. The ‘molecules of life’ are usually better made under conditions far most favourable than those obtaining back in the primordial soup era

He then does some back-of-a-matchbox calculations to speculate about how long it would take a random collection of organic molecules to ‘happen’ to all tumble together and create a life form: longer than the life of the universe, is his conclusion. No, this random approach won’t work.

Preliminary principles

Instead, he suggests a couple of principles of his own:

  1. That some and maybe all of the chemicals we now associate with ‘life’ were not present in the first replicating organisms; they came later; their exquisitely delicate interactivity suggests that they are the result not the cause of evolution
  2. Therefore, all lines of investigation which seek to account for the presence of the molecules of life are putting the cart before the horse: it isn’t the molecules which are important – it is the mechanism of replication with errors

Cairns-Smith thinks we should put the molecules of life question completely to one side, and instead seek for entirely inorganic systems which would replicate, with errors, so that the errors would be culled and more efficient ways of replicating tend to thrive on the available source material, beginning to create that dynamism and ‘sense of purpose’ which is one of life’s characteristics.

We keep coming to this idea that at some earlier phase of evolution, before life as we know it, there were other kinds of evolving system, other organisms that, in effect, invented our system. (p.61)

This seems, intuitively, like a more satisfying approach. Random forces will never make a MacBook Air and, as he has shown in chapter 4, even an entity like Escherichia coli is so staggeringly complex and amazingly finely-tuned as to be inconceivable as the product of chance.

Trying to show that complex molecules like ribosomes or RNA or amino acids – which rely on each other to be made and maintained, which cannot exist deprived of the intricately complicated interplay within each living cell – came about by chance is approaching the problem the wrong way. All these complex organic molecules must be the result of evolution. Evolution itself must have started with something much, much simpler – with the ‘invention’ of the basic engine, motor, the fundamental principle – and this is replication with errors. In other words:

Evolution started with ‘low-tech’ organisms that did not have to be, and probably were not made from, ‘the molecules of life’. (p.65)

Crystals

And it is at this point that Cairns-Smith introduces his Big Idea – the central role of clay crystals – in a chapter titled, unsurprisingly, ‘Crystals’ (pp.75-79).

he now explains in some detail the surprisingly complicated and varied world of clay crystals. These naturally form in various solutions and, if splashed up onto surfaces like rocks or stones, crystallise out into lattices, but with the crystallisation process also commonly involves errors and mutations.

His description of the different types of crystals and their properties is fascinating – who knew there were so many types, shapes, patterns and processes, starting with an introduction to the processes of saturation and super-saturation. The point is that crystals naturally occur and naturally mutate. He lists the ways they can vary or diverge from their ‘pure’ forms: twinning, stacking errors, cation substitutions, growth in preferred directions, break-up along preferred planes (p.97).

There follows a chapter about the prevalence of crystals in mud and clay and, therefore, their widespread presence in the conditions of the early planet earth.

And then, finally, he explains the big leap whereby replicating crystals may have attracted to themselves other molecules.

There follows a process of natural selection for clay crystals that trap certain forms of molecules to their surfaces that may enhance their replication potential. Complex proto-organic molecules can be catalysed by the surface properties of silicates.

Genetic takeover of the crystals

It is at this point that he introduces the idea of a ‘genetic takeover’.

When complex molecules perform a ‘genetic takeover’ from their clay ‘vehicle’, they become an independent locus of replication – an evolutionary moment that might be understood as the first exaptation.

(Exaptation = ‘the process by which features acquire functions for which they were not originally adapted or selected’)

Cairns-Smith had already described this process – the ‘genetic takeover’ of an initial, non-organic process by more complex, potentially organic molecules – in his earlier, longer and far more technical book, Genetic Takeover: And the Mineral Origins of Life, published in 1982.

This book – the Seven Clues – is a much shorter, non-technical and more accessible popularisation of the earlier tome. Hence the frivolous references to Sherlock Holmes.

Proliferating crystals form the scaffold for molecules which learn to replicate without them

The final chapter explains how these very common and proliferating entities – clay crystals – might have formed into structures and arrangements which attracted – for purely chemical reasons – various elementary organic molecules to them.

Certain repeating structures might attract molecules which then build up into more complex molecules, into molecules which are more efficient at converting the energy of the sun into further molecular combinations. And thus the principle of replication with variation, and competition for resources among the various types of replicating molecule, would have been established.

Thoughts

At this point the book ends, his case presented. It has been a fascinating journey because a) it is interesting to learn about all the different shapes and types of clay crystal b) he forces the reader to think about the fundamental engineering and logistical aspects of life forms, to consider the underlying principles which must inform all life forms, which is challenging and rewarding.

But, even in his own terms, Cairns-Smith’s notion of more and more complex potentially organic molecules being haphazardly replicated on a framework of proliferating clay crystals is still a long, long, long way from even the most primitive life forms known to us, with their vastly complex structure of cell membrane, nucleus and internal sea awash with DNA-controlled biochemical processes.


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The Double Helix by James Watson (1968)

The short paper by James Watson and Francis Crick establishing the helical structure of the DNA molecule was published in the science journal, Nature, on April 25, 1953. The blurb of this book describes it as the scientific breakthrough of the 20th century. Quite probably, although it was a busy century – the discovery of antibiotics was quite important, too, not to mention the atom bomb.

James Watson and Francis Crick with their DNA model at the Cavendish Laboratories in 1953

Anyway, what makes this first-person account of the events leading up to the discovery such fun is Watson’s prose style and mentality. He is fearless. He takes no prisoners. He is brutally honest about his own shortcomings and everyone else’s and, in doing so, sheds extraordinarily candid light on how science is actually done. He tells us that foreign conferences where nobody speaks English are often pointless. Many scientists are just plain stupid. Some colleagues are useless, some make vital contributions at just the right moment.

Watson has no hesitation in telling us that, when he arrived in Cambridge in 1951, aged just 23, he was unqualified in almost every way – although he had a degree from the University of Chicago, he had done his best to avoid learning any physics or chemistry, and as a graduate student at Indiana he had also avoided learning any chemistry. In fact the book keeps referring to his astonishing ignorance of almost all the key aspects of the field he was meant to be studying.

The one thing he did have was a determination to solve the problem which had been becoming ever-more prominent in the world of biology, what is a gene? Watson says he was inspired by Erwin Schrödinger’s 1946 book, What Is Life? which pointed out that ‘genes’ were the key component of living cells and that, to understand what life is, we must understand what genes are and how they work. The bacteriologist O.T. Avery had already shown that hereditary traits were passed from one bacterium to another by purified DNA molecules, so this much was common knowledge in the scientific community.

DNA was probably the agent of hereditary traits, but what did it look like and how did it work?

Our hero gets a U.S. government research grant to go to Copenhagen to study with biochemist Herman Kalckar, his PhD supervisor Salvador Luria hoping the Dane would teach him something but… no. Watson’s interest wasn’t sparked, partly because Kalckar was working on the structure of nucleotides, which young Jim didn’t think were immediately relevant to his quest, also because Herman was hard to understand –

At times I stood about nervously while Herman went through the motions of a biochemist, and on several days I even understood what he said. (p.34)

A situation compounded when Herman began to undergo a painful divorce and his mind wandered from his work altogether.

It was a chance encounter at a conference in Naples that motivated Watson to seek out the conducive-sounding environment of Cambridge (despite the reluctance of his funding authorities back in the States to let him go so easily). John Kendrew, the British biochemist and crystallographer, at that point studying the structure of myoglobin, helped smooth his passage to the fens.

Head of the Cavendish Laboratory in Cambridge where Watson now found himself was Sir Lawrence Bragg, Nobel Prize winner and one of the founders of crystallography. The unit collecting X-ray diffraction photographs of haemoglobin was headed up by the Austrian Max Perutz, and included Francis Crick, at this stage (in 1951) 35-years-old and definitely an acquired taste. Indeed the famous opening sentence of the book is:

I have never seen Francis Crick in a modest mood.

followed by the observation that:

he talked louder and faster than anybody else, and when he laughed, his location within the Cavendish was obvious.

So he had found a home of sorts and, in Francis Crick, a motormouth accomplice who was also obsessed by DNA – but there were two problems.

  1. The powers that be didn’t like Crick, who was constantly getting into trouble and nearly got thrown out when he accused the head of the lab, Bragg, of stealing one of his ideas in a research paper.
  2. Most of the work on the crystallography of DNA was being done at King’s College, London, where Maurice Wilkins had patiently been acquiring X-rays of the molecule for nearly ten years.

There was a sub-problem here which was that Wilkins was being forced to work alongside Rosalind Franklin, an expert in X-ray crystallography, who was an independent-minded 31-year-old woman (b.1920) and under the impression that she had been invited in to lead the NA project. The very young Watson and the not-very-securely-based Crick both felt daunted at having to ask to borrow and interpret Wilkins’s material, not least because he himself would have to extract it from the sometimes obstreperous Franklin.

And in fact there was a third big problem, which was that Linus Pauling, probably the world’s leading chemist and based at Cal Tech in the States, was himself becoming interested in the structure of DNA and the possibility that it was the basis of the much-vaunted hereditary material.

Pauling’s twinkling eyes and dramatic flair when making presentations is vividly described (pp.37-8). Along the same lines, Watson later gives a deliberately comical account of how he is scoffed and ignored by the eminent biochemist Erwin Chargaff after making some (typically) elementary mistakes in basic chemical bonding.

It is fascinating to read the insights scattered throughout the book about the relative reputations of the different areas of science – physics, biology, biochemistry, crystallography and so on. Typical comments are:

  • ‘the witchcraft-like techniques of the biochemist’, p.63
  • ‘In England, if not everywhere, most botanists and zoologists were a muddled lot.’ p.63

In a typical anecdote, after attending a lecture in London given by Franklin about her work, Watson goes for a Chinese meal in Soho with Maurice Wilkins who is worried that he made a mistake moving into biology, compared to the exciting and well-funded world of physics.

The physics of the time was dominated by the aftershock of the massive wartime atom bomb project, and with ongoing work to develop both the H-bomb and peacetime projects for nuclear power.

During the war Wilkins had helped to develop improved radar screens at Birmingham, then worked on isotope separation at the Manhattan Project at the University of California, Berkeley. Now he was stuck in a dingy lab in King’s College arguing with Franklin almost every day about who should use the best samples of DNA and the X-ray equipment and so on. (Later on, Watson tells us Wilkins’ and Franklin’s relationship deteriorated so badly that he (Watson) was worried about lending the London team the Cambridge team’s wire models in case Franklin strangled Wilkins with them. At one point, when Watson walks in on Franklin conducting an experiment, she becomes so angry at him he is scared she’s going to attack him. Wilkins confirms there have been occasions when he has run away in fear of her assaulting him.)

It’s in this respect – the insights into the way the lives of scientists are as plagued by uncertainty, professional rivalry, and doubts about whether they’re in the right job, or researching the right subject, gnawing envy of more glamorous, better-funded labs and so on – that the book bursts with insight and human interest.

Deoxyribonucleic acid

By about page 50 Watson has painted vivid thumbnail portraits of all the players involved in the story, the state of contemporary scientific knowledge, and the way different groups or individuals (Wilkins, Franklin, Pauling, Crick and various crystallographer associates at the Cavendish) are all throwing around ideas and speculations about the structure of DNA, on bus trips, in their freezing cold digs, or over gooseberry pie at their local pub, the Eagle in Cambridge (p.75).

For the outsider, I think the real revelation is learning how very small the final achievement of Crick and Watson seems. Avery had shown that DNA was the molecule of heredity. Chergaff had shown it contained equal parts of the four bases. Wilkins and Franklin had produced X-ray photos which strongly hinted at the structure and the famous photo 51 from their lab put it almost beyond doubt that DNA had a helix structure. Pauling, in America, had worked out the helical structure of other long proteins and had now began to speculate about DNA (although Watson conveys his and Crick’s immense relief that Pauling’s paper on the subject, published in early 1953, betrayed some surprisingly elementary mistakes in its chemistry.) But the clock was definitely ticking very loudly, rivals were closing in on the answer, and the pages leading up to the breakthrough are genuinely gripping.

In other words, the final deduction of the double helix structure doesn’t come out of the blue; the precise opposite; from Watson’s account it seems like it would have only been a matter of time before one or other of these groups had stumbled across the correct structure.

But it is very exciting when Watson comes into work one day, clears all the clutter from his desk and starts playing around with pretty basic cardboard cutouts of the four molecules which, by now, had become strongly associated with DNA, adenine and guanine, cytosine and thymine.

Suddenly, in a flash, he sees how they assemble the molecules naturally arrange themselves into pairs linked by hydrogen bonds – adenine with thymine and cytosine with guanine.

For a long time they’d been thinking the helix had one strand at the core and that the bases stuck out from it, like quills on a porcupine. Now, in a flash, Watson realises that the the base pairs, which join together so naturally, form a kind of zip, and the bands of sugar-phosphates holding the thing together run along the outside – creating a double helix shape.

The structure of the DNA double helix. The atoms in the structure are colour-coded by element and the detailed structures of two base pairs are shown in the bottom right. (Source: Wikipedia)

Conclusion

I am not qualified to summarise the impact of the discovery of DNA has had on the world. Maybe it would take books to do so adequately. I’ll quote the book’s blurb:

By elucidating the structure of DNA, the molecule underlying all life, Francis Crick and James Watson revolutionised biochemistry. At the time, Watson was only 24. His uncompromisingly honest account of those heady days lifts the lid on the real world of great scientists, with their very human faults and foibles, their petty rivalries and driving ambition. Above all, he captures the extraordinary excitement of their desperate efforts to beat their rivals at King’s College to the solution to one of the great enigmas of the life sciences.

The science is interesting, but has been overtaken and superseded generations ago. It’s the characters and the atmosphere of the time (the dingy English rooms with no heating, the appalling English food), the dramatic reality of scientific competition, and then the genuinely exciting pages leading up to the breakthrough which makes Watson’s book such a readable classic.

Rosalind Franklin

I marked all the places in the text where a feminist might explode with anger. Both Watson, but even more Crick, assume pretty young girls are made for their entertainment. They are referred to throughout as ‘popsies’ and Crick in particular, although married, betrays an endless interest in the pretty little secretaries and au pairs which adorn Cambridge parties.

It is through this patronising and sexist prism that the pair judged the efforts of Franklin who was, reasonably enough, a hard-working scientist not at all interested in her appearance or inclined to conform to gender stereotypes of the day. She felt marginalised and bullied at the King’s College lab, and irritated by the ignorance and superficiality of most of Watson and Crick’s ideas, untainted as they were by any genuine understanding of the difficult art of X-ray crystallography – an ignorance which Watson, to his credit, openly admits.

Eventually, Franklin found working with Wilkins so intolerable that she left to take up a position at Birkbeck College and then, tragically, discovered she had incurable cancer, although she worked right up to her death in April 1958.

Franklin has become a feminist heroine, a classic example of a woman struggling to make it in a man’s world, patronised by everyone around her. But if you forget her gender and just think of her as the scientist called Franklin, it is still a story of misunderstandings and poisonous professional relations such as I’ve encountered in numerous workplaces. Watson and Crick’s patronising tone must have exacerbated the situation, but the fundamental problem was that she was given clear written instructions that she would be in charge of the X-ray crystallography at King’s College but then discovered that Wilkins thought he had full control of the project. This was a management screw-up more than anything else.

It does seem unfair that she wasn’t cited in the Nobel Prize which was awarded to Crick, Watson and Wilkins in 1962, but then she had died in 1958, and the Swedish Academy had a simple rule of not awarding the prize to dead people.

Still, it’s not like her name has disappeared from the annals of history. Quite the reverse:

Impressive list, don’t you think?

And anyone who hasn’t read the book might be easily persuaded that she was an unjustly victimised, patronised and ignored figure. But just to set the record straight, Watson chooses to end the entire book not with swank about his and Crick’s later careers, but with a tribute to Franklin’s character and scientific achievement.

In 1958, Rosalind Franklin died at the early age of thirty-seven. Since my initial impressions of her, both scientific and personal (as recorded in the early pages of this book), were often wrong, I want to say something here about her achievements. The X-ray work she did at King’s is increasingly regarded as superb. The sorting out of the A and B forms [of DNA], by itself, would have made her reputation; even better was her 1952 demonstration, using Patterson superposition methods, that the phosphate groups must be on the outside of the DNA molecule. Later, when she moved to Bernal’s lab, she took up work on tobacco mosaic virus and quickly extended our qualitative ideas about helical construction into a precise quantitative picture, definitely establishing the essential helical parameters and locating the ribonucleic chain halfway out from the central axis.

Because I was then teaching in the States, I did not see her as often as did Francis, to whom she frequently came for advice or when she had done something very pretty, to be sure he agreed with her reasoning. By then all traces of our early bickering were forgotten, and we both came to appreciate greatly her personal honesty and generosity, realising years too late the struggles that the intelligent woman faces to be accepted by a scientific world which often regards women as mere diversions from serious thinking. Rosalind’s exemplary courage and integrity were apparent to all when, knowing she was mortally ill, she did not complain but continued working on a high level until a few weeks before her death. (p.175)

That is a fine, generous and moving tribute, don’t you think? And as candid and honest as the rest of the book in admitting his and Crick’s complete misreading of her situation and character.

In a literal sense the entire book leads up to this final page [these are the last words of the book] and this book became a surprise bestseller and the standard source to begin understanding the events surrounding the discovery. So it’s hard to claim that her achievement was suppressed or ignored when this is the climax of the best-selling account of the story.


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  • What Is Life? How Chemistry Becomes Biology by Addy Pross (2012)
  • The Diversity of Life by Edward O. Wilson (1992)
  • Seven Clues to the Origin of Life by A.G. Cairns-Smith (1985)
  • The Double Helix by James Watson (1968)

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The Periodic Kingdom: A Journey Into the Land of the Chemical Elements by Peter Atkins (1995)

Chemistry is the science of changes in matter. (p.37)

At just under 150 pages long, A Journey Into the Land of the Chemical Elements is intended as a novel and imaginative introduction to the 118 or so chemical elements which are the basic components of chemistry, and which, for the past 100 years or so, have been laid out in the grid arrangement known as the periodic table.

The periodic table explained

Just to refresh your memory, it’s called the periodic table because it is arranged into rows called ‘periods’. These are numbered 1 to 7 down the left-hand side.

What is a period? The ‘period number’ of an element signifies ‘the highest energy level an electron in that element occupies (in the unexcited state)’. To put it another way, the ‘period number’ of an element is its number of atomic orbitals. An orbital is the number of orbital positions an electron can take around the nucleus. Think of it like the orbit of the earth round the sun.

For each element there is a limited number of these ‘orbits’ which electrons can take up. Hydrogen, in row one, can only have one electron because it only has one possible orbital for an electron to take up around its nucleus. All the elements in row 2 have two orbitals for their electrons, and so on.

Sodium, for instance, sits in the third period, which means a sodium atom typically has electrons in the first three energy levels. Moving down the table, periods are longer because it takes more electrons to fill the larger and more complex outer levels.

The columns of the table are arranged into ‘groups’ from 1 to 18 along the top. Elements that occupy the same column or group have the same number of electrons in their outer orbital. These outer electrons are called ‘valence electrons’. The electrons in the outer orbital are the first ones to be involved in chemical bonds with other elements; they are relatively easy to dislodge, the ones in the lower orbitals progressively harder.

Elements with identical ‘valance electron configurations’ tend to behave in a similar fashion chemically. For example, all the elements in group or column 18 are gases which are slow to interact with other chemicals and so are known as the inert gases – helium, neon etc. Atkins describes the amazing achievement of the Scottish chemist William Ramsey in discovering almost all the inert gases in the 1890s.

Although there are 18 columns, the actual number of electrons in the outer orbital only goes up to 8. Take nitrogen in row 2 column 15. Nitrogen has the atomic number seven. The atomic number means there are seven electrons in a neutral atom of nitrogen. How many electrons are in its outer orbital? Although nitrogen is in the fifteenth column, that column is actually labelled ‘5A’. 5 represents the number of electrons in the outer orbital. So all this tells you that nitrogen has seven electrons in two orbitals around the nucleus, two in the first orbital and five in the second (2-5).

 

The Periodic Table. Karl Tate © LiveScience.com

Note that each element has two numbers in its cell. The one at the top is the atomic number. This is the number of protons in the nucleus of the element. Note how the atomic number increases in a regular, linear manner, from 1 for hydrogen at the top left, to 118 for Oganesson at the bottom right. After number 83, bismuth, all the elements are radioactive.

(N.B. When Atkins’s book was published in 1995 the table stopped at number 109, Meitnerium. As I write this, 24 years later, it has been extended to number 118, Oganesson. These later elements have been created in minute quantities in laboratories and some of them only exist for a few moments.)

Beneath the element name is the atomic weight. This is the mass of a given atom, measured on a scale in which the hydrogen atom has the weight of one. Because most of the mass in an atom is in the nucleus, and each proton and neutron has an atomic weight near one, the atomic weight is very nearly equal to the number of protons and neutrons in the nucleus.

Note the freestanding pair of rows at the bottom, coloured in purple and orange. These are the lanthanides and actinides. We’ll come to them in a moment.

Not only are the elements arranged into periods and groups but they are also categorised into groupings according to their qualities. In this diagram (taken from LiveScience.com) the different groupings are colour-coded. The groupings are, moving from left to right:

Alkali metals The alkali metals make up most of Group 1, the table’s first column. Shiny and soft enough to cut with a knife, these metals start with lithium (Li) and end with francium (Fr), among the rarest elements on earth: Atkins tells us that at any one moment there are only seventeen atoms of francium on the entire planet. The alkali metals are extremely reactive and burst into flame or even explode on contact with water, so chemists store them in oils or inert gases. Hydrogen, with its single electron, also lives in Group 1, but is considered a non-metal.

Alkaline-earth metals The alkaline-earth metals make up Group 2 of the periodic table, from beryllium (Be) through radium (Ra). Each of these elements has two electrons in its outermost energy level, which makes the alkaline earths reactive enough that they’re rarely found in pure form in nature. But they’re not as reactive as the alkali metals. Their chemical reactions typically occur more slowly and produce less heat compared to the alkali metals.

Lanthanides The third group is much too long to fit into the third column, so it is broken out and flipped sideways to become the top row of what Atkins calls ‘the Southern Island’ that floats at the bottom of the table. This is the lanthanides, elements 57 through 71, lanthanum (La) to lutetium (Lu). The elements in this group have a silvery white color and tarnish on contact with air.

Actinides The actinides line forms the bottom row of the Southern Island and comprise elements 89, actinium (Ac) to 103, lawrencium (Lr). Of these elements, only thorium (Th) and uranium (U) occur naturally on earth in substantial amounts. All are radioactive. The actinides and the lanthanides together form a group called the inner transition metals.

Transition metals Returning to the main body of the table, the remainder of Groups 3 through 12 represent the rest of the transition metals. Hard but malleable, shiny, and possessing good conductivity, these elements are what you normally associate with the word metal. This is the location of many of the best known metals, including gold, silver, iron and platinum.

Post-transition metals Ahead of the jump into the non-metal world, shared characteristics aren’t neatly divided along vertical group lines. The post-transition metals are aluminum (Al), gallium (Ga), indium (In), thallium (Tl), tin (Sn), lead (Pb) and bismuth (Bi), and they span Group 13 to Group 17. These elements have some of the classic characteristics of the transition metals, but they tend to be softer and conduct more poorly than other transition metals. Many periodic tables will feature a highlighted ‘staircase’ line below the diagonal connecting boron with astatine. The post-transition metals cluster to the lower left of this line. Atkins points out that all the elements beyond bismuth (row 6, column 15) are radioactive. Here be skull-and-crossbones warning signs.

Metalloids The metalloids are boron (B), silicon (Si), germanium (Ge), arsenic (As), antimony (Sb), tellurium (Te) and polonium (Po). They form the staircase that represents the gradual transition from metals to non-metals. These elements sometimes behave as semiconductors (B, Si, Ge) rather than as conductors. Metalloids are also called ‘semi-metals’ or ‘poor metals’.

Non-metals Everything else to the upper right of the staircase (plus hydrogen (H), stranded way back in Group 1) is a non-metal. These include the crucial elements for life on earth, carbon (C), nitrogen (N), phosphorus (P), oxygen (O), sulfur (S) and selenium (Se).

Halogens The top four elements of Group 17, from fluorine (F) through astatine (At), represent one of two subsets of the non-metals. The halogens are quite chemically reactive and tend to pair up with alkali metals to produce various types of salt. Common salt is a marriage between the alkali metal sodium and the halogen chlorine.

Noble gases Colorless, odourless and almost completely non-reactive, the inert, or noble gases round out the table in Group 18. The low boiling point of helium makes it a useful refrigerant when exceptionally low temperatures are required; most of them give off a colourful display when electric current is passed through them, hence the generic name of neon lights, invented in 1910 by Georges Claude.

The metaphor of the Periodic Kingdom

In fact the summary I’ve given above isn’t at all how Atkins’s book sounds. It is the way I have had to make notes to myself to understand the table.

Atkins’ book is far from being so clear and straightforward. The Periodic Kingdom is dominated by the central conceit that Atkins treats the periodic table as if it were an actual country. His book is not a comprehensive encyclopedia of biochemistry, mineralogy and industrial chemistry; it is a light-hearted ‘traveller’s guide’ (p.27) to the table which he never refers to as a table, but as a kingdom, complete with its own geography, layout, mountain peaks and ravines, and surrounded by a sea of nothingness.

Hence, from start to finish of the book, Atkins uses metaphors from landscape and exploration to describe the kingdom, talking about ‘the Western desert’, ‘the Southern Shore’ and so on. Here’s a characteristic sentence:

The general disposition of the land is one of metals in the west, giving way, as you travel eastward, to a varied landscape of nonmetals, which terminates in largely inert elements at the eastern shoreline. (p.9)

I guess the idea is to help us memorise the table by describing its characteristics and the changes in atomic weight, physical character, alkalinity, reactivity and so on of the various elements, in terms of geography. Presumably he thinks it’s easier to remember geography than raw information. His approach certainly gives rise to striking analogies:

North of the mainland, situated rather like Iceland off the northwestern edge of Europe, lies a single, isolated region – hydrogen. This simple but gifted element is an essential outpost of the kingdom, for despite its simplicity it is rich in chemical personality. It is also the most abundant element in the universe and the fuel of the stars. (p.9)

Above all the extended metaphor (the periodic table imagined as a country) frees Atkins not to have to lay out the subject in either a technical nor a chronological order but to take a pleasant stroll across the landscape, pointing out interesting features and making a wide variety of linkages, pointing out the secret patterns and subterranean connections between elements in the same ‘regions’ of the table.

There are quite a few of these, for example the way iron can easily form alliances with the metals close to it such as cobalt, nickel and manganese to produce steel. Or the way the march of civilisation progressed from ‘east’ to ‘west’ through the metals, i.e. moving from copper, to iron and steel, each representing a new level of culture and technology.

The kingdom metaphor also allows him to get straight to core facts about each element without getting tangled in pedantic introductions: thus we learn there would be no life without nitrogen which is a key building block of all proteins, not to mention the DNA molecule; or that sodium and potassium (both alkali metals) are vital in the functioning of brain and nervous system cells.

And hence the generally light-hearted, whimsical tone allows him to make fanciful connections: calcium is a key ingredient in the bones of endoskeletons and the shells of exoskeletons, compacted dead shells made chalk, but in another format made the limestone which the Romans and others ground up to make the mortar which held their houses together.

Then there is magnesium. I didn’t think magnesium was particularly special, but learned from Atkins that a single magnesium atom is at the heart of the chlorophyll molecule, and:

Without chlorophyll, the world would be a damp warm rock instead of the softly green haven of life that we know, for chlorophyll holds its magnesium eye to the sun and captures the energy of sunlight, in the first step of photosynthesis. (p.16)

You see how the writing is aspiring to an evocative, poetic quality- a deliberate antidote to the dry and factual way chemistry was taught to us at school. He means to convey the sense of wonder, the strange patterns and secret linkages underlying these wonderful entities. I liked it when he tells us that life is about capturing, storing and deploying energy.

Life is a controlled unwinding of energy.

Or about how phosphorus, in the form of adenosine triphosphate (ATP) is a perfect vector for the deployment of energy, common to all living cells. Hence the importance of phosphates as fertiliser to grow the plants we need to survive. Arsenic is such an effective poison because it is a neighbour of phosphorus, shares some of its qualities, and so inserts itself into chemical reactions usually carried out by phosphorus but blocking them, nulling them, killing the host organism.

All the facts I explained in the first half of this post (mostly cribbed from the LiveScience.com website) are not reached or explained until about page 100 of this 150-page-long book. Personally, I felt I needed them earlier. As soon as I looked at the big diagram of the table he gives right at the end of the book I became intrigued by the layout and the numbers and couldn’t wait for him to get round to explaining them, which is why I went on the internet to find out more, more quickly, and why Istarted my review with a factual summary.

And eventually, the very extended conceit of ‘the kingdom’ gets rather tiresome. Whether intentional or not, the continual references to ‘the kingdom’ begin to sound Biblical and pretentious.

Now the kingdom is virtually fully formed. It rises above the sea of nonbeing and will remain substantially the same almost forever. The kingdom was formed in and among the stars.. (p.75)

The chapter on the scientists who first isolated the elements and began sketching out the table continues the metaphor by referring to them as ‘cartographers’, and the kingdom as made of islands and archipelagos.

As an assistant professor of chemistry at the University of Jena, [Johann Döbereiner] noticed that reports of some of the kingdom’s islands – reports brought back by their chemical explorers – suggested a brotherhood of sorts between the regions. (p.79)

For me, the obsessive use of the geographical metaphor teeters on the border between being useful, and becoming irritating. He introduces me to the names of the great pioneers – I was particularly interested in Dalton, Michael Faraday, Humphrey Davy (who isolated a bunch of elements in the early 1800s) and then William Ramsey – but I had to go to Wikipedia to really understand their achievements.

Atkins speculates that some day we might find another bunch or set of elements, which might even form an entire new ‘continent’, though it is unlikely. This use of a metaphor is sort of useful for spatially imagining how this might happen, but I quickly got bored of him calling this possible set of new discoveries ‘Atlantis’, and of the poetic language as a whole.

Is the kingdom eternal, or will it slip beneath the waves? There is a good chance that one day – in a few years, or a few hundred years at most – Atlantis will be found, which will be an intellectual achievement but probably not one of great practical significance…

A likely (but not certain) scenario is that in that distant time, perhaps 10100 years into the future, all matter will have decayed into radiation, it is even possible to imagine the process. Gradually the peaks and dales of the kingdom will slip away and Mount Iron will rise higher, as elements collapse into its lazy, low-energy form. Provided that matter does not decay into radiation first (which is one possibility), the kingdom will become a lonely pinnacle, with iron the only protuberance from the sea of nonbeing… (p.77)

And I felt the tone sometimes bordered on the patronising.

The second chemical squabble is in the far North, and concerns the location of the offshore Northern Island of hydrogen. To those who do not like offshore islands, there is the problem of where to put it on the mainland. This is the war of the Big-Endians versus the Little-Endians. Big-Endians want to tow the island ashore to form a new Northwestern Cape, immediately north of lithium and beryllium and across from the Northeastern Cape of helium… (p.90)

Hard core chemistry

Unfortunately, none of these imaginative metaphors can help when you come to chapter 9, an unexpectedly brutal bombardment of uncompromising hard core information about the quantum mechanics underlying the structure of the elements.

In quick succession this introduces us to a blizzard of ideas: orbitals, energy levels, Pauli’s law of exclusion, and then the three imaginary lobes of orbitals.

As I understood it, the Pauli exclusion principle states that no two electrons can inhabit a particular orbital or ‘layer’ or shell. But what complicates the picture is that these orbitals come in three lobes conceived as lying along imaginary x, y and z axes. This overlapped with the information that there are four types of orbitals – s, p, d and f orbitals. In addition, there are three p-orbitals, five d-orbitals, seven f-orbitals. And the two lobes of a p-orbital are on either side of an imaginary plane cutting through the nucleus, there are two such planes in a d-orbital and three in an f-orbital.

After pages of amiable waffle about kingdoms and Atlantis, this was like being smacked in the face with a wet towel. Even rereading the chapter three times, I still found it impossible to process and understand this information.

I understand Atkins when he says it is the nature of the orbitals, and which lobes they lie along, which dictates an element’s place in the table, but he lost me when he said a number of electrons lie inside the nucleus – which is the opposite of everything I was ever taught – and then when described the way electrons fly across or through the nucleus, something to do with the processes of ‘shielding’ and ‘penetration’.

The conspiracy of shielding and penetration ensure that the 2s-orbital is somewhat lower in energy than the p-orbitals of the same rank. By extension, where other types of orbitals are possible, ns- and np-orbitals both lie lower in energy than nd-orbitals, and nd-orbitals in turn have lower energy than nf-orbitals. An s-orbital has no nodal plane, and electrons can be found at the nucleus. A p-orbital has one plane, and the electron is excluded from the nucleus. A d-orbital has two intersecting planes, and the exclusion of the electron is greater. An f-orbital has three planes, and the exclusion is correspondingly greater still. (p.118)

Note how all the chummy metaphors of kingdoms and deserts and mountains have disappeared. This is the hard-core quantum mechanical basis of the elements, and at least part of the reason it is so difficult to understand is because he has made the weird decision to throw half a dozen complex ideas at the reader at the same time. I read the chapter three times, still didn’t get it, and eventually wanted to cry with frustration.

This online lecture gives you a flavour of the subject, although it doesn’t mention ‘lobes’ or penetration or shielding.

In the next chapter, Atkins, briskly assuming  his readers have processed and understood all of this information, goes on to combine the stuff about lobes and orbitals with a passage from earlier in the book, where he had introduced the concept of ions, cations, and anions:

  • ion an atom or molecule with a net electric charge due to the loss or gain of one or more electrons
  • cation a positively charged ion
  • anion a negatively charged ion

He had also explained the concept of electron affinity

The electron affinity (Eea) of an atom or molecule is defined as the amount of energy released or spent when an electron is added to a neutral atom or molecule in the gaseous state to form a negative ion.

Isn’t ‘affinity’ a really bad word to describe this? ‘Affinity’ usually means ‘a natural liking for and understanding of someone or something’. If it is the amount of energy released, why don’t they call it something useful like the ‘energy release’? I felt the same about the terms ‘cation’ and ‘anion’ – that they had been deliberately coined to mystify and confuse. I kept having to stop and look up what they meant since the name is absolutely no use whatsoever.

And the electronvolt – ‘An electronvolt (eV) is the amount of kinetic energy gained or lost by a single electron accelerating from rest through an electric potential difference of one volt in vacuum.’

Combining the not-very-easily understandable material about electron volts with the incomprehensible stuff about orbitals means that the final 30 pages or so of The Periodic Kingdom is thirty pages of this sort of thing:

Take sodium: it has a single electron outside a compact, noble-gaslike core (its structure is [Ne]3s¹). The first electron is quite easy to remove (its removal requires an investment of 5.1 eV), but removal of the second, which has come from the core that lies close to the nucleus, requires an enormous energy – nearly ten times as much, in fact (47.3 eV). (p.130)

This reminds me of the comparable moment in John Allen Paulos’s book Innumeracy where I ceased to follow the argument. After rereading the passage where I stumbled and fell I eventually realised it was because Paulos had introduced three or so important facts about probability theory very, very quickly, without fully explaining them or letting them bed in – and then had spun a fancy variation on them…. leaving me standing gaping on the shore.

Same thing happens here. I almost but don’t quite understand what [Ne]3s¹ means, and almost but don’t quite grasp the scale of electronvolts, so when he goes on to say that releasing the second electron requires ten times as much energy, of course I understand the words, but I cannot quite grasp why it should be so because I have not understood the first two premises.

As with Paulos, the author has gone too fast. These are not simple ideas you can whistle through and expect your readers to lap up. These are very, very difficult ideas most readers will be completely unused to.

I felt the sub-atomic structure chapter should almost have been written twice, approached from entirely different points of view. Even the diagrams were no use because I didn’t understand what they were illustrating because I didn’t understand his swift introduction of half a dozen impenetrable concepts in half a page.

Once through, briskly, is simply not enough. The more I tried to reread the chapter, the more the words started to float in front of my eyes and my brain began to hurt. It is packed with sentences like these:

Now imagine a 2 p-electron… (an electron that occupies a 2 p-orbital). Such an electron is banished from the nucleus on account of the existence of the nodal plane. This electron is more completely shielded from the pull of the nucleus, and so it is not gripped as tightly.In other words, because of the interplay of shielding and penetration, a 2 s-orbital has a lower energy (an electron in it is gripped more tightly) than a 2 p-orbital… Thus the third and final electron of lithium enters the 2 s-orbital, and its overall structure is 1s²2s¹. (p.118)

I very nearly understand what some of these words meant, but the cumulative impact of sentences like these was like being punched to the ground and then given a good kicking. And when the last thirty pages went on to add the subtleties of electronvoltages and micro-electric charges into the mix, to produce ever-more complex explanations for the sub-atomic interactivity of different elements, I gave up.

Summary

The first 90 or so pages of The Periodic Kingdom do manage to give you a feel for the size and shape and underlying patterns of the periodic table. Although it eventually becomes irritating, the ruling metaphor of seeing the whole place as a country with different regions and terrains works – up to a point – to explain or suggest the patterns of size, weight, reactivity and so on underlying the elements.

When he introduced ions was when he first lost me, but I stumbled on through the entertaining trivia and titbits surrounding the chemistry pioneers who first isolated and named many of the elements and the first tentative attempts to create a table for another thirty pages or so.

But the chapter about the sub-atomic structure of chemical elements comprehensively lost me. I was already staggering, and this finished me off.

If Atkins’s aim was to explain the basics of chemistry to an educated layman, then the book was, for me, a complete failure. I sort of quarter understood the orbitals, lobes, nodes section but anything less than 100% understanding means you won’t be able to follow him to the next level of complexity.

As with the Paulos book, I don’t think I failed because I am stupid – I think that, on both occasions, the author failed to understand how challenging his subject matter is, and introduced a flurry of concepts far too quickly, at far too advanced a level.

Looking really closely I realise it is on the same page (page 111) that Atkins introduces the concepts of energy levels, orbitals, the fact that there are three two-lobed orbitals, and the vital existence of nodal planes. On the same page! Why the rush?

An interesting and seemingly trivial feature of a p-orbital, but a feature on which the structure of the kingdom will later be seen to hinge, is that the electron will never be found on the imaginary plane passing through the nucleus and dividing the two lobes of the orbital. This plane is called a nodal plane. An s-orbital does not have such a nodal plane, and the electron it describes may be found at the nucleus. Every p-orbital has a nodal plane of this kind, and therefore an electron that occupies a p-orbital will never be found at the nucleus. (p.111)

Do you understand that? Because if you don’t, you won’t understand the last 40 or so pages of the book, because this is the ‘feature on which the structure of the kingdom will later be seen to hinge’.

I struggled through the final 40 pages weeping tears of frustration, and flushed with anger at having the thing explained to me so badly. Exactly how I felt during my chemistry lessons at school forty years ago.


Related links

Reviews of other science books

Chemistry

Cosmology

The environment

Human evolution

Genetics and life

  • What Is Life? How Chemistry Becomes Biology by Addy Pross (2012)
  • The Diversity of Life by Edward O. Wilson (1992)
  • The Double Helix by James Watson (1968)

Maths

Particle physics

Psychology

Nature’s Numbers by Ian Stewart (1995)

Stewart is a mathematician and prolific author, having written over 40 books on all aspects of maths, as well as publishing guides to the maths used in Terry Pratchett’s Discworld books, authoring half a dozen textbooks for students and co-authoring a couple of science fiction novels.

He writes in a marvellously clear style, but more importantly, he is interesting: he sees the world in an interesting way and manages to convey the wonder and strangeness and powerful insights which seeing the world in terms of patterns and shapes, numbers and maths, gives you.

He wants to help us see the world as a mathematician sees it, full of clues and information which can lead us to deeper and deeper appreciation of the patterns and harmonies all around us.

1. The Natural Order

Thus he begins the book by describing just some of nature’s patterns: the regular movements of the stars in the night sky; the sixfold symmetry of snowflakes; the stripes of tigers and zebras; the recurring patterns of sand dunes; rainbows; the spiral of a snail’s shell; why nearly all flowers have petals arranged in one of the following numbers 5, 8, 13, 21, 34, 55, 89; the regular patterns or ‘rhythms’ made by animals scuttling, walking, flying and swimming.

2. What Mathematics is For

Mathematics is brilliant at helping us to solve puzzles. it is a more or less systematic way of digging out the rules and structures that lie behind some observed pattern or regularity, and then using those rules and structures to explain what’s going on. (p.16)

Kepler discovers the planets move in ellipses. The nature of acceleration, ‘not a fundamental quality, a rate of change’. Newton and Leibniz invent calculus to help us work out rates of change.

Two of the main things that maths are for are 1. providing the tools which let scientists understand what nature is doing 2. providing new theoretical questions for mathematicians to explore further. Applied and pure mathematics.

He mentions one of the oddities, paradoxes or thought-provoking things that comes up in many science books which is the eerie way that good mathematics, whatever its source, eventually turns out to be useful, to be applicable to the real world, to explain some aspect of nature. Many philosophers have wondered why. Is there a deep congruence between the human mind and the structure of the universe? Did God make the universe mathematically and implant an understanding of maths in us? Is the universe made of maths?

Stewart’s answer is simple and elegant: he thinks that nature exploits every pattern that there is, which is why we keep discovering patterns everywhere. We humans express these patterns in numbers, but it isn’t the numbers nature uses – it’s the patterns and shapes and possibilities which the numbers express, or define.

Mendel noticing the numerical relationships with which characteristics of peas are expressed when they are crossbred. The double helix structure of DNA. The computer simulation of the evolution of the eye from an initial mutation providing for skin cells sensitive to light, published by Daniel Nilsson and Susanne Pelger in 1994.

Resonance = the relationship between periodically moving bodies in which their cycles lock together so that they take up the same relative positions at regular intervals. The cycle time is the period of the system. The individual bodies have different periods. The moon’s rotational period is the same as its revolution around the earth, so there is a 1:1 resonance of its orbital and rotational period.

Mathematics doesn’t just analyse, it can predict, predict how all kinds of systems will work, from the aerodynamics which keep planes flying to the amount of fertiliser required to increase crop yield to the complicated calculations which keep communications satellites in orbit round the earth and therefore sustain the internet and mobile phone networks.

Time lags: the gap between a new mathematical idea being developed and its practical implementation can be a century or more: it was 17th century interest in the vibration of a violin string which led, three hundred years later, to the invention of radio, radar and TV.

3. What Mathematics is About

The word ‘number’ does not have any immutable, God-given meaning. (p.42)

Numbers are the most prominent part of mathematics and everyone is taught arithmetic at school, but numbers are just one type of object that mathematics is interested in.

The invention of numbers. Fractions. Some time in the Dark Ages the invention of 0. The invention of negative numbers, then of square roots. Irrational numbers. ‘Real’ numbers.

Whole numbers 1, 2, 3… are known as the natural numbers. If you include negative whole number, the series is known as integers. Positive and negative numbers are known as rational numbers. Then there are real numbers and complex numbers. Five systems in total.

But maths is also about operations such as addition, subtraction, multiplication and division. And functions, also known as transformations, rules for transforming one mathematical object into another. Many of these processes can be thought of as things which help to create data structures.

Maths is like a landscape with similar proofs and theories clustered together to create peaks and troughs.

4. The Constants of Change

Newton’s basic insight was that changes in nature can be described by mathematical processes. Stewart explains how detailed consideration of what happens to a cannonball fired out of a cannon helps us towards Newton’s fundamental law, that force = mass x acceleration.

Newton invented calculus to help work out solutions to moving bodies. Its two basic operations – integration and differentiation – mean that, given one element – force, mass or acceleration – you can work out the other two. Differentiation is the technique for finding rates of change; integration is the technique for ‘undoing’ the effect of differentiation.

Calculating rates of change is a crucial aspect of maths, engineering, cosmology and many other areas.

5. From Violins to Videos

A fascinating historical recap of how initial investigations into the way a violin string vibrates gave rise to formulae and equations which turned out to be useful in mapping electricity and magnetism, which turned out to be aspects of the same fundamental force, understanding which underpinned the invention of radio, radar, TV etc, taking in contributions from Michael Faraday, James Clerk Maxwell, Heinrich Hertz and Giulielmo Marconi.

Stewart makes the point that mathematical theory tends to start with the simple and immediate and grow ever-more complicated. This is because you have to start somewhere.

6. Broken Symmetry

A symmetry of an object or system is any transformation that leaves it invariant. (p.87)

There are many types of symmetry. The most important ones are reflections, rotations and translations.

7. The Rhythm of Life

The nature of oscillation and Hopf bifurcation (if a simplified system wobbles, then so must the complex system it is derived from) leads into a discussion of how animals, specifically animals with legs, move, which is by staggered or syncopated oscillations, oscillations of muscles triggered by neural circuits in the brain.

This is a subject Stewart has written about elsewhere and is something of an expert on. The seven types of quadrupedal gait are: the trot, pace, bound, walk, rotary gallop, transverse gallop, and canter.

8. Do Dice Play God?

Stewart’s take on chaos theory.

Chaotic behaviour obeys deterministic laws, but is so irregular that to the untrained eye it looks pretty much random. Chaos is not complicated, patternless behaviour; it is much more subtle. Chaos is apparently complicated, apparently patternless behaviour that actually has a simple, deterministic explanation. (p.130)

19th century scientists thought that, if you knew the starting conditions, and then the rules governing any system, you could completely predict the outcomes. In the 1970s and 80s it became increasingly unclear that this was wrong. It is impossible because you can never define the starting conditions with complete certainty.

Thus all real world behaviours are subject to ‘sensitivity to initial conditions’. From minuscule divergences at the start point, cataclysmic differences may eventually emerge.

He then explains the concept of phase space developed by Henri Poincaré: this is an imaginary mathematical space that represents all possible motions in a given dynamic system. The phase space is the 3-D place in which you plot the behaviour in order to create the phase portrait. Instead of having to define a formula and worrying about identifying every number of the behaviour, the general shape can be determined.

Much use of phase portraits has shown that dynamic systems tend to have set shapes which emerge and which systems move towards. These are called attractors.

9. Drops, Dynamics and Daisies

The book ends by drawing a kind of philosophical conclusion.

Chaos theory has all sorts of implications but the one he closes on is this: the world is not chaotic; if anything, it is boringly predictable. And at the level of basic physics and maths, the laws which seem to underpin it are also schematic and simple. And yet what we are only really beginning to appreciate is how complicated things are in the middle.

It is as if nature can only get from simple laws (like Newton’s incredibly simple law of thermodynamics) to fairly simple outcomes (the orbit of the planets) via almost incomprehensibly complex processes. To end, Stewart gives us three examples of the way apparently ‘simple’ phenomena in nature derive from stupefying complexity:

  • what exactly happens when a drop of water falls off a tap
  • computer modelling of the growth of fox and rabbit populations
  • why petals on flowers are arranged in numbers derived from the Fibonacci sequence

In all three cases the underlying principles seem to be resolvable into laws and functions – and we see water dropping off taps or flowerheads all the time – and yet the intermediate steps are bogglingly complex.

Coda: Morphomatics

He ends the book with an epilogue speculating, hoping and wishing for a new kind of mathematics which incorporates chaos theory and the other elements he’s discussed – a theory and study of form, which takes everything we already know about mathematics and seeks to work out how the almost incomprehensible complexity we are discovering in nature gives rise to all the ‘simple’ patterns which we see around us.


Related links

Reviews of other science books

Cosmology

Environment / human impact

Genetics

  • The Double Helix by James Watson (1968)

Maths

Particle physics

Psychology

  • Irrationality: The Enemy Within by Stuart Sutherland (1992)
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