Wednesday, September 03, 2014

Upside down and inside out

Tomorrow a new exhibition by Peter Randall-Page opens at Pangolin London, called Upside Down & Inside Out. Peter has a long-standing interest in natural processes responsible for the appearance of pattern and form, inspired by the ideas of D’Arcy Thompson. It has been my privilege to write an essay for the catalogue of this exhibition, which is freely available online. Here’s the piece anyway.

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There are, in the crudest of terms, two approaches to understanding the world. Some seek to uncover general, universal principles behind the bewildering accumulation of particulars; others find more enlightenment in life’s variety than in the simplifying approximations demanded in a quest for unity. The former are Platonists, and in science they tend to be found in greater numbers among physicists. The latter are Aristotelians, and they are best represented in biology. The Platonists follow the tree to its trunk, the Aristotelians work in the other direction, towards branch and leaf.

The work of artist and sculptor Peter Randall-Page explores these opposing – or perhaps one should say complementary – tendencies. He sees them in terms of the musical notion of theme and variation: a single Platonic theme can give rise to countless Aristotelian variations. The theme alone risks being static, even monotonous; a little disorder, a dash of unpredictability, generates enriching diversity, but that random noise must be kept under control if the result is not to become incomprehensible chaos. It is perhaps precisely because this tension obtains in evolution, in music and language, in much of our experience of life and world, that its expression in art has the potential to elicit emotion and identification from abstract forms. This balance of order and chaos is one that we recognize instinctively.

This is why Peter’s works commonly come as a series: they are multiple expressions of a single underlying idea, and only when viewed together do they give us a sense both of the fundamental generating principle and its fecund creative potential. The diversity depends on chance, on happy accidents or unplanned contingencies that allow the generative laws to unfold across rock or paper in ways quite unforeseen and unforeseeable. Like Paul Klee, Peter takes lines for a walk – but they are never random walks, there are rules that they must respect. And as with Klee, this apparent constraint is ultimately liberating to the imagination: given the safety net of the basic principles, the artist’s mind is free to play.

It might seem odd to talk about creativity in what is essentially an algorithmic process, an unfolding of laws. But it is hard to think of a better or more appropriate term to describe the “endless forms most beautiful” that we find in nature, and not just in animate nature. We could hardly fail to marvel at the inventiveness of a mind that could conceive of the countless variations on a theme that we observe in snowflakes, and it seems unfair to deny nature here inventiveness merely because we can see no need to attribute to her a mind, just as Alan Turing insisted that we have no grounds for denying a machine “intelligence” if we cannot distinguish its responses from those of a human.

This emergence of variety from simplicity is an old notion. “Nature”, wrote Ralph Waldo Emerson, “is an endless combination and repetition of a very few laws. She hums the old well-known air through innumerable variations.” When Emerson attested that such “sublime laws play indifferently through atoms and galaxies”, it is surely the word “play” that speaks loudest: there is a gaiety and spontaneity here that seems far removed from the mechanical determinism of which physics is sometimes accused. For Charles Darwin, one can’t help feel that the Aristotelian diversity of nature – in barnacles, earthworms and orchids – held at least as much attraction as the Platonic principle of natural selection.

But one of Peter’s most inspirational figures was skeptical of an all-embracing Darwinism as the weaver of nature’s threads. The Scottish zoologist D’Arcy Thompson felt that natural selection was all too readily advanced as the agency of every wrinkle and rhythm of organic nature. The biologists of his time tended to claim that all shape, form and regularity was the way it was because of adaptation. If biology has a more nuanced view today, Thompson must take some of the credit. He argued that it was often physical and mechanical principles that governed nature’s forms and patterns, not some infinitely malleable Darwinian force. Yet at root, Thompson’s picture – presented in his encyclopaedic 1917 book On Growth and Form – was not so different from Darwin’s insofar as it posited some quite general principles that could give rise to a vast gallery of variations. Thompson simply said that those principles need not be Darwinian or selective, but could apply both to the living and the inorganic worlds. In this view, it should be no coincidence that the branching shapes of river networks resemble those of blood vessels or lung passages, or that a potato resembles a pebble, or that the filigree skeletal shell of a radiolarian echoes the junctions of soap films in a foam. Thompson was a pioneer of the field loosely termed morphogenesis: the formation of shape. In particular, he established the idea that the appearance of pattern and regularity in nature may be a spontaneous affair, arising from the interplay of conflicting tendencies. No genes specify where a zebra’s stripes are to go: if anything is genetically encoded, it is merely the biochemical machinery for covering an arbitrary form with stripes.


The exoskeleton of a radiolarian

It is a fascination with these ideas that gives nearly all of Peter’s works their characteristic and compelling feature: you can’t quite decide whether the impetus for these complex but curiously geometric forms came from biology or from elsewhere, from cracks and crystals and splashes. That ambiguity fixes the imagination, inviting us to decode the riddle. This dance between geometry and organism is immediately apparent in the monumental sculpture Seed commissioned by the Eden Project in Cornwall: an egg-shaped block of granite 13 feet high and weighing 70 tonnes, the surface of which is covered in bumps that you quickly discern to be as apparently orderly as atoms packed together in a crystal. But are they? These bumps adapt their size to the curvature of the surface, and you soon notice that they progress around the ovoid in spirals, recalling the arrangements of leaflets on a pine-cone or florets on a sunflower head. Can living nature really be so geometric? Certainly it can, for both of those plant structures, like the compartments on a pineapple, obey mathematical laws that have puzzled botanists (including Darwin) for centuries. These plant patterns are called phyllotaxis, and the reason for them is still being debated. Some argue that they are ordered by the constraints on the buckling and wrinkling of new stem tissue, others that there is a biochemical process – not unlike that responsible for the zebra’s stripes and the leopard’s spots – that generates order among the successively sprouting buds.



Seed, by Peter Randall-Page, at the Eden Project, Cornwall, and the inspiration provided by pine cones.

The bulbous, raspberry-like surface of Seed was carved out of the pristine rock. But in nature such structures are typically grown from the inside outwards, the cells and compartments budding and swelling under the expansive pressures of biological proliferation. “Everything is what it is”, D’Arcy Thompson wrote, “because it got that way” – a seemingly obvious statement, but one that brings the focus to how it got that way: to the process of growth that created it. With this in mind, the bronze casts that Peter has created for this exhibition are also made “from the inside”. They are cast from natural boulders shaped by erosion, but Peter has worked the inner surfaces of the moulds using a special tool to scoop out hemispherical impressions packed like the cells of a honeycomb, so that the shapes cast from them follow the basic contours of the boulders while acquiring these new frogspawn-like cellular patterns on their surface. By subtracting material from the mould, the cast object is itself “grown”, emerging transformed and hitherto unseen from its chrysalis.


A new work by Peter Randall-Page (on the right) being cast at the foundry.

The organic and unfolding character of Peter’s work is nowhere more evident than in his “drawings” of branching, tree-like networks: Blood Tree, Sap River and Source Seed. These are made by allowing ink or wet pigment to flow under gravity across the paper in a quasi-controlled manner, so that not only does the flow generate repeated bifurcations but the branches acquire perfect mirror symmetry by folding the absorbent paper, just like the bilateral symmetry of the human body. The results are ordered, but punctuated and decorated with unique accidents. The final images are inverted so that the rivulets seem to stream upwards in increasingly fine filaments, defying gravity: a process of division without end, arbitrarily truncated and all emanating from a single seed. The inversion suggests growth and vitality, a reaching towards the infinite, although of course in real plants we know that these branches are echoed downwards in the traceries of the roots. There is irony too in the fact that, while sap does indeed rise from trunk to tip, driven by the evaporation of water from the leaf, water in a river network flows the other way, being gathered into the tributaries and converging into the central channel. Nature indeed makes varied use of these branching networks – and often for the same reason, that they are particular efficient at distributing fluid and dissipating the energy of flow. But we must be vigilant in making distinctions as well as analogies in how they are used.



Peter Randall-Page, Blood Tree and Sap River V.

Were real trees ever quite so regular, however? Some of these look more like genealogies, a mathematically precise doubling of branch density by bifurcation in each generation – until, perhaps, the individual branches blur into a continuum. We could almost be looking at a circuit diagram or technical chart – and yet the splodgy irregularities of the channels warn us that there is still something unpredictable here, as though these are computer networks grown from bacteria (as indeed some researchers are attempting to do). If there can be said to be beauty in the images, it depends on this uncertainty: as Ernst Gombrich put it, the aesthetic sense is awakened by “a struggle between two opponents of equal power, the formless chaos, on which we impose our ideas, and the all-too-formed monotony, which we brighten up by new accents”.

The vision of the world offered by Peter Randall-Page is therefore neither Platonic nor Aristotelian. We might better describe it as Neoplatonic: as asserting analogies and correspondences between apparently unrelated things. This tendency, which thrived in the Renaissance and can be discerned in the parallels that Leonardo da Vinci drew between the circulation of blood and of natural waters in rivers, later came to seem disreputable: like so much of the occult philosophy, it attempted to connect the unconnected, relying on mere visual puns and resemblances without regard to causative mechanisms (or perhaps, mistaking those analogies for a kind of mechanism itself). But thanks to the work of D’Arcy Thompson, and now modern scientific theories of complexity and pattern formation, a contemporary Neoplatonism has re-emerged as a valid way to understand the natural world. There are indeed real, quantifiable and verifiable reasons why zebra stripes look like the ripples of windblown sand, or why both the Giant’s Causeway and the tortoise shell are divided into polygonal networks. When we experience these objects and structures, we experience what art historian Martin Kemp has called “structural intuitions”, which are surely what the Neoplatonists were responding to. And these intuitions are what Peter’s work, with all its intricate balance of order and randomness, awakens in us.

To find out more: see Peter Randall-Page, “On theme and variation”, Interdisciplinary Science Reviews 38, 52-62 (2013) [here].

Saturday, August 30, 2014

When and why does biology go quantum?

Here is my latest Crucible column for Chemistry World. Do look out for Jim and Johnjoe’s book Life of the Edge, which very nicely rounds up where quantum biology stands right now – and Jim has just started filming a two-parter on this (for BBC4, I believe).

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“Quantum biology” was always going to be a winning formula. What could be more irresistible than the idea that two of the most mysterious subjects in science – quantum physics and the existence of life – are connected? Indeed, you get the third big mystery – consciousness – thrown in for good measure, if you accept the highly controversial suggestion by Roger Penrose and Stuart Hameroff that quantum behaviour of protein filaments called microtubules are responsible for the computational capability of the human mind [1].

Chemists might sigh that once again those two attention-grabbers, physics and biology, are appropriating what essentially belongs to chemistry. For the fact is that all of the facets of quantum biology that are so far reasonably established or at least well grounded in experiment and theory are chemical ones. The most arguably mundane, but at the same time the least disputable, area in which quantum effects make their presence felt in a biological context is enzyme catalysis, where quantum tunneling processes operate during reactions involving proton and electron transfer [2]. It also appears beyond dispute that photosynthesis involves transfer of energy from the excited chromophore to the reaction centre in an excitonic wavefunction that maintains a state of quantum coherence [3,4]. It still seems rather staggering to find in the warm, messy environment of the cell a quantum phenomenon that physicists and engineers are still struggling to harness at cryogenic conditions for quantum computing. The riskier reaches of quantum biology also address chemical problems: the mechanism of olfaction (proposed to happen by sensing of odorant vibrational spectra using electron tunneling [5]) and of magnetic direction-sensing in birds (which might involve quantum entanglement of electron spins on free radicals [6]).

Yet it is no quirk of fate that these phenomena are sold as a union of physics and biology, bypassing chemistry. For as Jim Al-Khalili and Johnjoe McFadden explain in a forthcoming comprehensive overview of the field, Life On the Edge (Doubleday), the first quantum biologists were pioneers of quantum theory: Pascual Jordan, Niels Bohr and Erwin Schrödinger. Bohr was never shy of pushing his view of quantum theory – the Copenhagen interpretation – into fields beyond physics, and his 1932 lecture “Light and Life” seems to have been influential in persuading Max Delbrück to turn from physics to genetics, on which his work later won him a Nobel Prize.

But it is Schrödinger’s contribution that is probably best known, for the notes from his lectures at Trinity College Dublin that he collected into his little 1944 book What Is Life? remain remarkable for their prescience and influence. Most famously, Schrödinger here formulated the idea that life somehow opposes the entropic tendency towards dissolution – it feeds on negative entropy, as he put it – and he also argued that genetic information might be transmitted by an arrangement of atoms that he called an “aperiodic crystal” – a description of DNA, whose structure was decoded nine years later (partly by another former physicist, Francis Crick), that still looks entirely apt.

One of the most puzzling of biological facts for Schrödinger was that genetic mutations, which were fundamentally probabilistic quantum events on a single-atom scale, could become fixed into the genome and effect macroscopic changes of phenotype. By the same token, replication of genes (which was understood before Crick and Watson revealed the mechanism) happened with far greater fidelity than one should expect from the statistical nature of molecular interactions. Schrödinger reconciled these facts by arguing that it was the very discreteness of quantum events that gave them an accuracy and stability not amenable to classical continuum states.

But this doesn’t sound right today. For the fact is that Schrödinger was underestimating biology. Far from being at the mercy of replication errors incurred by thermal fluctuations, cells have proof-reading mechanisms to check for and correct these mistakes.

There is an equal danger that quantum biologists may overestimate biology. For it’s all too tempting, when a quantum effect such as tunneling is discovered in a biological process, to assume that evolution has put it there, or at least found a way to capitalize on it. Tunnelling is nigh inevitable in proton transfer; but if we want to argue that biology exploits quantum physics here, we need to ask if its occurrence is enhanced by adaptation. Nobel laureate biochemist Arieh Warshel has rejected that idea, calling it a “red herring” [7].

Similarly in photosynthesis, it’s not yet clear if quantum coherence is adaptive. It does seem to help the efficiency of energy transfer, but that might be a happy accident – Graham Fleming, one of the pioneers in this area, says that it may be simply “a byproduct of the dense packing of chromophores required to optimize solar absorption” [8].

These are the kind of questions that may determine what becomes of quantum biology. For its appeal lies largely with the implication that biology and quantum physics collaborate, rather than being mere fellow travellers. We have yet to see how far that is true.

1. R. Penrose, Shadows of the Mind (Oxford University Press, 1994).
2. A. Kohen & J. P. Klinman, Acc. Chem. Res. 31, 397 (1998).
3. G. S. Engel et al., Nature 446, 782 (2007).
4. H. Lee, Y.-C. Cheng & G. R. Fleming, Science 316, 1462 (2007).
5. L. Turin, Chem. Senses 21, 773 (1996).
6. E. M. Gauger, E. Rieper, J. J. L. Morton, S. C. Benjamin & V. Vedral, Phys. Rev. Lett. 106, 040503 (2011).
7. P. Ball, Nature 431, 396 (2004).
8. P. Ball, Nature 474, 272 (2011).

Thursday, August 07, 2014

Calvino's culturomics

Italo Calvino’s If On a Winter’s Night a Traveller is one of the finest and funniest meditations on writing that I’ve ever read. It also contains a glorious pre-emptive critique on what began as Zipf’s law and is now called culturomics: the statistical mining of vast bodies of text for word frequencies, trends and stylistic features. What is so nice about it (apart from the wit) is that Calvino seems to recognize that this approach is not without validity (and I certainly think it is not), while at the same time commenting on the gulf that separates this clinical enumeration from the true craft of writing – and for that matter, of reading. I am going to quote the passage in full – I don’t know what copyright law might have to say about that, but I am trusting to the fact that anyone familiar with Calvino’s book would be deterred from trying to enforce ownership of the text by the baroque level of irony that would entail.

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[From Vintage edition 1998, translated by William Weaver]

I asked Lotaria if she has already read some books of mine that I lent her. She said no, because here she doesn’t have a computer at her disposal.

She explained to me that a suitably programmed computer can read a novel in a few minutes and record the list of all the words contained in the text, in order of frequency. ‘That way I can have an already completed reading at hand,” Lotaria says, “with an incalculable saving of time. What is the reading of a text, in fact, except the recording of certain thematic recurrences, certain insistences of forms and meanings? An electronic reading supplies me with a list of the frequencies, which I have only to glance at to form an idea of the problems the book suggests to my critical study. Naturally, at the highest frequencies the list records countless articles, pronouns, particles, but I don’t pay them any attention. I head straight for the words richest in meaning; they can give me a fairly precise notion of the book.”

Lotaria brought me some novels electronically transcribed, in the form of words listed in the order of their frequency. “In a novel of fifty to a hundred thousand words,” she said to me, “I advise you to observe immediately the words that are repeated about twenty times. Look here. Words that appear nineteen times:
“blood, cartridge belt, commander, do, have, immediately, it, life, seen, sentry, shots, spider, teeth, together, your…”
“Words that appear eighteen times:
“boys, cap, come, dead, eat, enough, evening, French, go, handsome, new, passes, period, potatoes, those, until…”

“Don’t you already have a clear idea what it’s about?” Lotaria says. “There’s no question: it’s a war novel, all actions, brisk writing, with a certain underlying violence. The narration is entirely on the surface, I would say; but to make sure, it’s always a good idea to take a look at the list of words used only once, though no less important for that. Take this sequence, for example:
“underarm, underbrush, undercover, underdog, underfed, underfoot, undergo, undergraduate, underground, undergrowth, underhand, underprivileged, undershirt, underwear, underweight…”

“No, the book isn’t completely superficial, as it seemed. There must be something hidden; I can direct my research along these lines.”

Lotaria shows me another series of lists. “This is an entirely different novel. It’s immediately obvious. Look at the words that recur about fifty times:
“had, his, husband, little, Riccardo (51) answered, been, before, has, station, what (48) all, barely, bedroom, Mario, some, Times (47) morning, seemed, went, whom (46) should (45) hand, listen, until, were (43) Cecilia, Delaia, evening, girl, hands, six, who, years (42) almost, alone, could, man returned, window (41) me, wanted (40) life (39)"

“What do you think of that? An intimatist narration, subtle feelings, understated, a humble setting, everyday life in the provinces … As a confirmation, we’ll take a sample of words used a single time:
“chilled, deceived, downward, engineer, enlargement, fattening, ingenious, ingenious, injustice, jealous, kneeling, swallow, swallowed, swallowing…"

“So we already have an idea of the atmosphere, the moods, the social background… We can go on to a third book:
“according, account, body, especially, God, hair, money, times, went (29) evening, flour, food, rain, reason, somebody, stay, Vincenzo, wine (38) death, eggs, green, hers, legs, sweet, therefore (36) black, bosom, children, day, even, ha, head, machine, make, remained, stays, stuffs, white, would (35)"

“Here I would say we’re dealing with a full-blooded story, violent, everything concrete, a bit brusque, with a direct sensuality, no refinement, popular eroticism. But here again, let’s go on to the list of words with a frequency of one. Look, for example:
“ashamed, shame, shamed, shameful, shameless, shames, shaming, vegetables, verify, vermouth, virgins…"

“You see? A guilt complex, pure and simple! A valuable indication: the critical inquiry can start with that, establish some working hypothesis…What did I tell you? Isn’t this a quick, effective system?”

The idea that Lotaria reads my books in this way creates some problems for me. Now, every time I write a word, I see it spun around by the electronic brain, ranked according to its frequency, next to other words whose identity I cannot know, and so I wonder how many times I have used it, I feel the whole responsibility of writing weigh on those isolated syllables, I try to imagine what conclusions can be drawn from the fact that I have used this word once or fifty times. Maybe it would be better for me to erase it…But whatever other word I try to use seems unable to withstand the test…Perhaps instead of a book I could write lists of words, in alphabetical order, an avalanche of isolated words which expresses that truth I still do not know, and from which the computer, reversing its program, could construct the book, my book.

On the side of the angels


Here’s my take on Dürer’s Melencolia I on its 500th anniversary, published in Nature this week.

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Albrecht Dürer’s engraving Melencholia I, produced 500 years ago, seems an open invitation to the cryptologist. Packed with occult symbolism from alchemy, astrology, mathematics and medicine, it promises hidden messages and recondite meanings. What it really tells us, however, is that Dürer was a philosopher-artist of the same stamp as Leonardo da Vinci, immersed in the intellectual currents of his time. In the words of art historian John Gage, Melencolia I is “almost an anthology of alchemical ideas about the structure of matter and the role of time” [1].

Dürer’s brooding angel is surrounded by, the instruments of the proto-scientist: a balance, an hourglass, measuring calipers, a crucible on a blazing fire. Here too is numerological symbolism in the “magic square” of the integers 1-16, the rows, columns and main diagonals of which all add up to 34: a common emblem of both folk and philosophical magic. Here is the astrological portent of a comet, streaming across a sky in which an improbable rainbow arches, a symbol of the colour-changing processes of the alchemical route to the philosopher’s stone. And here is the title itself: melancholy, associated in ancient medicine with black bile, the same colour of the material with which the alchemist’s Great Work to make gold was supposed to begin.

But why the tools of the craftsman – the woodworking implements in the foreground, the polygonal block of stone awaiting the sculptor’s hammer and chisel? Why the tormented, introspective eyes of the androgynous angel?

Melencolia I is part of a trio of complex etchings on copper plate that Dürer made in 1513-14. Known as the Master Engravings, they are considered collectively to raise this new art to an unprecedented standard of technical skill and psychological depth. This cluttered, virtuosic image is widely thought often said to represent a portrait of Dürer’s own artistic spirit. Melancholy, often considered the least desirable of the four classical humours then believed to govern health and medicine, was traditionally associated with insanity. But during the Renaissance it was ‘reinvented’ as the humour of the artistic temperament, originating the link popularly asserted between madness and creative genius. The German physician and writer Cornelius Agrippa, whose influential Occult Philosophy (widely circulated in manuscript form from 1510) Dürer is almost certain to have read, claimed that “celestial spirits” were apt to possess the melancholy man and imbue him with the imagination required of an “excellent painter”. For it took imagination to be an image-maker – but also to be a magician.

The connection to Agrippa was first made by the art historian Erwin Panofsky, a doyen of symbolism in art, in 1943. He argued that what leaves Dürer’s art-angel so vexed is the artist’s constant sense of failure: an inability to fly, to exceed the bounds of the human imagination and create the truly wondrous. Her tools, in consequence, lie abandoned. Why astronomy, geometry, meteorology and chemistry should have any relation to the artistic temperament is not obvious today, but in the early sixteenth century the connection would have been taken for granted by anyone familiar with the Neoplatonic idea of correspondences in nature. This notion, which pervades Agrippa’s writing, held that, which joined all natural phenomena, including the predispositions of humankind, are joined into a web of hidden forces and symbols. Melancholy, for instance, is the humour governed by the planet Saturn, whence “saturnine.” That blend of ideas was still present in Robert Burton’s The Anatomy of Melancholy, published a century later, which called melancholics “dull, sad, sour, lumpish, ill-disposed, solitary, any way moved, or displeased.” A harsh description perhaps, but Burton reminds us that “from these melancholy dispositions no man living is free” – for melancholy is in the end “the character of Mortality.” But some are more prone than others: Agrippa reminded his readers of Aristotle’s opinion “that all men that were excellent in any Science, were for the most part melancholy.”

So there would have been nothing obscure about this picture for its intended audience of intellectual connoisseurs. It was precisely because Dürer mastered and exploited the new technologies of printmaking that he could distribute these works widely, and he indicated in his diaries that he sold many on his travels, as well as giving others as gifts to friends and humanist scholars such as Erasmus of Rotterdam. Unlike paintings, you needed only moderate wealth to afford a print. Ferdinand Columbus, son of Christopher, collected over 3,000, 390 of which were by Dürer and his workshop [2].

But even if the alchemical imagery of Melencolia I was part of the ‘occult parcel’ that this engraving presents, Besides all this, it would be wrong to imagine that alchemy was, to Dürer and his contemporaries, purely an esoteric art associated with gold-making. As Lawrence Principe has recently argued (The Secrets of Alchemy, University of Chicago Press, 2013), this precursor to chemistry was not just or even primarily about furtive and futile experimentation to make gold from base metals. It was also a practical craft, not least in providing artists with their pigments. For this reason, artists commonly knew something of its techniques; Dürer’s friend, the German artist Lucas Cranach the Elder, was a pharmacist on the side, which may explain why he was almost unique in Northern Europe in using the rare and poisonous yellow pigment orpiment, an arsenic sulphide. The extent of Dürer’s chemical knowledge is not known, but he was one of the first artists to use acids for etching metal, a technique developed only at the start of the sixteenth century. The process required specialist knowledge: it typically used nitric acid, made from saltpetre, alum and ferrous sulphate, mixed with dilute hydrochloric acid and potassium chlorate (“Dutch mordant”).

Humility should perhaps compel us to concur with art historian Keith Moxey that “the significance of Melencolia I is ultimately and necessarily beyond our capacity to define” [3] – we are too removed from it now for its themes to resonate. But what surely endures in this image is a reminder that for the Renaissance artist there was continuity between theories about the world, matter and human nature, the practical skills of the artisan, and the business of making art.

References
1. Gage, J. Colour and Culture, p.149. Thames & Hudson, London, 1993.
2. McDonald, M. in Albrecht Dürer and his Legacy, ed. G. Bartrum. British Museum, London, 2003.
3. Moxey, K. The Practice of Theory, p.93. Cornell University Press, Ithaca, 1994.

Wednesday, August 06, 2014

All hail the man who makes the bangs


The nerd with the safety specs who is always cropping up on TV doing crazy experiments for Jim Al-Khalili or Mark Miodownik or Michael Mosley, while threatening to upstage them with his patter? That’s Andrea Sella of UCL, who has just been awarded the Michael Faraday Prize by the Royal Society. And this is a very splendid thing. With previous recipients including Peter Medawar, Richard Dawkins, David Attenborough, Robert Winston and Brian Cox, it is clear what a prestigious award this is. But whereas those folks have on the whole found themselves celebrated and supported for their science-communication work, Andrea has sometimes been under a lot of pressure to justify doing this stuff instead of concentrating on his research (on lanthanides). I hope very much that this recognition will help to underline the value of what we now call “outreach activities” when conducted by people in regular research positions, rather than just by those who have managed to establish science communication as a central component of their work. Being able to talk about science (and in Andrea’s case, show it in spectacular fashion) is a rare skill, the challenge of which is sometimes under-estimated and under-valued, and so it is very heartening to see it recognized here.

Monday, August 04, 2014

Dreams of invisibility

Here’s my Point of View piece from the Guardian Review a week ago. It’s fair to say that my new book Invisible is now out, and I’m delighted that John Carey seemed to like it (although I’m afraid you can’t fully see why without a subscription).

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H. G. Wells claimed in his autobiography that he and Joseph Conrad had “never really ‘got on’ together”, but you’d never suspect that from the gushing fan letter Conrad sent to Wells, 8 years his junior but far more established as a writer, in 1897. Before their friendship soured Conrad was a great admirer of Wells, and in that letter he rhapsodized the author of scientific romances as the “Realist of the Fantastic”. It’s a perceptive formulation of the way Wells blended speculative invention with social realism: tea and cakes and time machines. That aspect is nowhere more evident than in the book that stimulated Conrad to write to his idol: The Invisible Man.

To judge from Wells’ own account of his aims, Conrad had divined them perfectly. “For the writer of fantastic stories to help the reader to play the game properly”, he wrote in 1934, “he must help him in every possible unobtrusive way to domesticate the impossible hypothesis… instead of the usual interview with the devil or a magician, an ingenious use of scientific patter might with advantage be substituted. I simply brought the fetish stuff up to date, and made it as near actual theory as possible.”

In other words, Wells wanted to turn myth into science, or at least something that would pass for it. This is why The Invisible Man is a touchstone for interpreting the claims of modern physicists and engineers to be making what they call “invisibility cloaks”: physical structures that try to hide from sight what lies beneath. The temptation is to suggest that, as with atomic bombs, Wells’ fertile imagination was anticipating what science would later realise. But the light that his invisible man sheds on today’s technological magic is much more revealing.

It’s likely Wells was explicitly updating myth. One of the earliest stories about invisibility appears near the start of Plato’s Republic, a book that had impressed Wells in his youth. Plato’s narrator Glaucon tells of a Lydian shepherd named Gyges who discovered a ring of invisibility in the bowels of the earth. Without further ado, Gyges used the power to seduce the queen, kill the king and establish a new dynasty of Lydian rulers. In a single sentence Plato tells us what many subsequent stories of invisibility would reiterate about the desires that the dream of invisibility feeds: they are about sex, power and death.

Evidently this power corrupts – which is one reason why Tolkien made much more mythologically valid use of invisibility magic than did J. K. Rowling. But Glaucon’s point has nothing to do with invisibility itself; it is about moral responsibility. Given this power to pass unseen, he says, no one “would be so incorruptible that he would stay on the path of justice, when he could with impunity take whatever he wanted from the market, go into houses and have sexual relations with anyone he wanted, kill anyone, and do the other things which would make him like a god among men.” The challenge is how to keep rulers just if they can keep their injustices hidden.

The point about Gyges’ ring is that it doesn’t need to be explained, because it is metaphorical. The same is true of this and other magic effects in fairy tales: they just happen, because they are not about the doing but the consequences. Fairy-tale invisibility often functions as an agent of seduction and voyeurism (see the Grimms’ “The Twelve Dancing Princesses”), or a gateway to Faerie and other liminal realms. It’s precisely because children don’t ask “how is that possible?” that we shouldn’t fret about filling them with false beliefs.

But it seems to be a peculiarity of our age that we focus on the means of making magic and not the motive. The value of The Invisible Man is precisely that it highlights the messy outcome of this collision between science and myth. True, Wells makes some attempt to convince us that his anti-hero Griffin is corrupted by discovering the “secret of invisibility” – but it is one of the central weaknesses of the tale that Griffin scarcely has any distance to fall, since he is thoroughly obnoxious from the outset, driving his poor father to suicide by swindling him out of money he doesn’t possess in order to fund his lone research. If we are meant to laugh at the superstitions of the bucolic villages of Iping as the invisible Griffin rains blows on them, I for one root for the bumpkins.

No, where the book both impresses and exposes is in its description of how Griffin becomes invisible. A plausible account of that trick had been attempted before, for example in Edward Page Mitchell’s 1881 short story “The Crystal Man”, but Wells had enough scientific nous to make it convincing. While Mitchell’s scientist simply makes his body transparent, Wells knew that it was necessary not just to eliminate pigmentation (which Griffin achieves chemically) but to eliminate refraction too: the bending of light that we see through glass or water. There was no known way of doing that, and Wells was forced to resort to the kind of “jiggery-pokery magic” he had criticized in Mary Shelley’s Frankenstein. He exploited the very recent discovery of X-rays by saying that Griffin had discovered another related form of “ethereal vibration” that gives materials the same refractive strength as air.

Despite this, Griffin finds that invisibility is more a burden than a liberation. He dreams of world domination but, forgetting to vanish his clothes too, has to wander naked in the winter streets of London, bruised by unseeing crowds and frightened that he will be betrayed by the snow that threatens to settle on his body and record his footsteps. His eventual demise has no real tragedy in it but is like the lynching of a common criminal, betrayed by sneezes, sore feet and his digestive tract (in which food visibly lingers for a time). In all this, Wells shows us what it means to domesticate the impossible, and what we should expect when science tries to do magic.

That same gap between principle and practice hangs over today’s “invisibility cloaks”. They work in a different, and technologically marvelous, way: not by transparency, but by guiding light around the object they hide. But when the first of them was unveiled in 2006, it was perplexing: for there it sat, several concentric rings of printed circuits, as visible as you or me. It was, the scientists explained, invisible to microwaves, not to visible light. What had this to do with Gyges, or even with Griffin?

Some scientists argue that, for all their technical brilliance (which is considerable, and improving steadily), these constructs should be regarded as clever optical devices, not as invisibility cloaks. It’s hard to imagine how they could ever conceal a person walking around in daylight. This “magic” is cumbersome and compromised: it is not the way to seduce the queen, kill the king and become a tyrant.

This isn’t to disparage the invention and imagination that today’s “invisibility cloaks” embody. But it’s a reminder that myth is not a technical challenge, not a blueprint for the engineer. It’s about us, with all our desires, flaws, and dreams.

Cutting-edge metallurgy


This is my Materials Witness column for the August issue of Nature Materials. I normally figure these columns are a bit too specialized to put up here, but this subject is just lovely: there is evidently so much more to the "sword culture" of the so-called Dark Ages, the Viking era and the early medieval period than a bunch of blokes running amok with big blades. As Snorri Sturluson surely said, you can't beat a good sword.

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There can be few more mythologized ancient materials technologies than sword-making. The common view – that ancient metalsmiths had an extraordinary empirical grasp of how to manipulate alloy microstructure to make the finest-quality blades – contains a fair amount of truth. Perhaps the most remarkable example of this was discovered several years ago: the near-legendary Damascus blades used by Islamic warriors, which were flexible yet strong and hard enough to cleave the armour of Crusaders, contained carbon nanotubes [1]. Formation of the nanotubes was apparently catalysed by impurities such as vanadium in the steel, and these nanostructures assisted the growth of cementite (Fe3C) fibres that thread through the unusually high-carbon steel known as wootz, making it hard without paying the price of brittleness.

Yet it seems that the skill of the swordsmith wasn’t directed purely at making swords mechanically superior. Thiele et al. report that the practice called pattern-welding, well established in swords from the second century AD to the early medieval period, was primarily used for decorative rather than mechanical purposes and, unless used with care, could even have compromised the quality of the blades [2].

Pattern-welding involved the lamination and folding of two materials – high-phosphorus iron and low-phosphorus mild steel or iron – to produce a surface that could be polished and etched to striking decorative effect. After twisting and grinding, the metal surface could acquire striped, chevron and sinuous patterns that were highly prized. A letter to a Germanic tribe in the sixth century AD, complimenting them for the swords they gave to the Ostrogothic king Theodoric, conqueror of Italy, praised the interplay of shadows and colours in the blades, comparing the pattern to tiny snakes.


This and the image above are modern pattern-welded swords made by Patrick Barta using traditional methods.

But was it all about appearance? Surely what mattered most to a warrior was that his sword could be relied on to slice, stab and maim without breaking? It seems not. Thiele et al. commissioned internationally renowned swordsmith Patrick Barta to make pattern-welded rods for them using traditional techniques and re-smelted medieval iron. In these samples the high-phosphorus component was iron and not, as some earlier studies have mistakenly assumed, steel.

They subjected the samples to mechanical tests that probed the stresses typically experienced by a sword: impact, bending and buckling. In no cases did the pattern-welded samples perform any better than hardened and tempered steel. This is not so surprising, given that phosphoric iron itself has rather poor toughness, no matter how it is laminated with other materials.

The prettiness of pattern welding didn’t, however, have to compromise the sword’s strength, since – at least in later examples – the patterned section was confined to panels in the central “fuller” of the blade, while the cutting edge was steel. All the same, here’s an example of how materials use may be determined as much by social as by technical and mechanical considerations. From the Early to the High Middle Ages, swords weren’t just or even primarily for killing people with. For the Frankish warrior, the spear and axe were the main weapons; swords were largely symbols of power and status, carried by chieftains, jarls and princes but used only rarely. Judging by the modern reproductions, they looked almost too gorgeous to stain with blood.

References
1. Reibold, M. et al., Nature 444, 286 (2006).
2. Thiele, A., Hosek, J., Kucypera, P. & Dévényi, L. Archaeometry online publication doi:10.1111/arcm.12114 (2014).

Thursday, July 24, 2014

Science books you (and I) should read


La Recherche asked me to recommend my favourite science book for a special issue of the magazine. I had to go for Richard Holmes’ The Age of Wonder (Harper Press, London, 2008). Lots of science books have interested me, and many have captivated me with their wonderful writing. But this is the only one that left me feeling quite so excited.

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Who should write books about science? A Nobel laureate once made his views on that plain enough to me, saying “I have a healthy disregard for anybody and everybody who has not made advances in the field in which they are pontificating.” And in compiling The Oxford Book of Modern Science Writing, Richard Dawkins proclaimed that “This is a collection of good writing by professional scientists, not excursions into science by professional writers” – implying not only that those two groups are distinct but that true science writing embraces only the former.

There are plenty of good examples that demonstrate the wisdom of the academic impulse never to stray outside your own field – in which you have perhaps painstakingly accumulated expertise over decades. The results of forays into foreign intellectual territory have sometimes been disastrous. But the idea that non-scientists have nothing to say about science that could possibly be useful or interesting to scientists, or that scientists from one discipline are unlikely to say anything valuable about another, is one that I find not just dismaying but terrifying.

I don’t think Dawkins or my Nobel colleague actually doubts for a moment that science can be effectively popularized by non-experts. After all, as many people have read and been informed by Bill Bryson’s A Short History of Nearly Everything (2005) as they have many of Dawkins’ splendid and profoundly erudite expositions on evolution. But can outsiders actually bring anything new to the table?

My answer to that is Richard Holmes’ book The Age of Wonder. It tells of the period in the late eighteenth and early nineteenth centuries when scientists sat down with artists and poets – Humphry Davy and Samuel Coleridge exchanged mutually admiring correspondence, for example. They all shared a common view of nature as a source of sublime wonder, the exploration of which was a voyage of romantic discovery that needed the cadences of poetry as much as the precision of scientific experiment and observation. This material could have become a formulaic lament about the “two cultures” that have allegedly arisen since, but Holmes does something much subtler, richer and more fulfilling.

Beginning with James Cook’s expedition to Tahiti in 1769, on which the botanist was the future president of the Royal Society Joseph Banks, Holmes takes in episodes of “romantic science” that include William and Caroline’s telescopic investigations of the moon and stars, early balloon flights and Davy’s experiments with laughing gas and the miner’s lamp. Holmes’ sights are trained firmly on the cultural setting and reception of these studies, and the mindset that informed them. “For many Romantic scientists”, he writes,

there was no immediate contradiction between religion and science: rather the opposite. Science was a gift of God or Providence to mankind, and its purpose was to reveal the wonders of His design.

Holmes has insisted that he knows rather little about science. This is excessively modest, but not falsely so. One doesn’t doubt, from the confident tone of the book, either that he spared any pains to find out what he needed to know or that he knew what to do with that information. Precisely because Holmes is an expert on the lives and intellectual milieu of Coleridge, Percy Shelley and the British and French Romantics generally, he was able to draw out themes and ideas that historians of science would not have seen.

But the book isn’t just to be celebrated for its fresh perspective. It is also a joy to read. Every page delivers something interesting, always with elegance and wryness. Even the footnotes (mark this, academics) are not to be missed. I have read a lot of science books – when I first read The Age of Wonder it was as a judge of the Royal Society Science Book Prize (which Holmes won), and so I was wading through literally stacks of them. But none has left me with such genuine exhilaration as this one. And none has better illuminated the case which Holmes makes at the end, and which surely all scientists would applaud:

Perhaps most important, right now, is a changing appreciation of how scientists themselves fit into society as a whole, and the nature of the particular creativity they bring to it. We need to consider how they are increasingly vital to any culture of progressive knowledge, to the education of young people (and the not so young), and to our understanding of the planet and its future.

More challenging to some, I suspect, is Holmes’ corollary:

The old rigid debates and boundaries – science versus religion, science versus the arts, science versus traditional ethics – are no longer enough. We should be impatient with them. We need a wider, more generous, more imaginative perspective.

Here already is the beginning of that perspective.

Wednesday, July 23, 2014

The hidden structure of liquids


Here’s a Commentary written for the collection of essays curated by Nature Materials for the International Year of Crystallography. It is worth also taking a look at Nature’s Crystallography Milestones collection.

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From its earliest days, crystallography has been viewed as a means to probe order in matter. J. D. Bernal’s work on the structure of water reframed it as a means of examining the extent to which matter can be regarded as orderly.

In 1953, J. Desmond Bernal wrote that, as a result of the development of X-ray crystallography,

“science was beginning to find explanations in terms of atoms and their combinations not only of the phenomena of physics and chemistry but of the behaviour of ordinary things. The beating out of metal under the hammer, the brittleness of glass and the cleavage of mica, the plasticity of clay, the lightness of ice, the greasiness of oil, the elasticity of rubber, the contraction of muscle, the waving of hair, and the hardening of a boiled egg are among the hundreds of phenomena that had already been completely or partially explained." [1]

What is striking here is how far beyond crystalline and ordered matter Bernal perceived the technique to have gone: into soft matter, amorphous solids, and viscous liquids. For biological polymers, Bernal himself had pioneered the study of globular proteins, while William Astbury, Bernal’s one-time colleague in William Bragg’s laboratory at the Royal Institution in London, had by mutual agreement focused on the fibrous proteins that constitute hair and muscle. Of course, in the year in which Bernal was writing, the most celebrated X-ray structure of a fibrous biological macromolecule, DNA, was solved by Crick and Watson under the somewhat sceptical auspices of William’s son Lawrence Bragg, head of the Cavendish Laboratory in Cambridge.

All those macromolecular materials do form crystals. But one of Bernal’s great insights (if not his alone) was to recognize that the lack of long-ranged order in a material was no obstacle to the use of X-rays for deducing its structure. That one could meaningfully talk about a structure for the liquid state was itself something of a revelation. What is sometimes overlooked is the good fortune that the natural first choice for such investigation of liquids – water, ubiquitous and central to life and the environment – happens to have an unusually high degree of structure. Indeed, Bernal first began his studies of liquid-state structure by regarding it as a kind of defective crystal.

The liquid state is notoriously problematic precisely because it bridges other states that can, at least in ideal terms, be considered as perfectly ordered (the crystal) and perfectly disordered (the gas). Is the liquid a dense gas or an imperfect solid? It has become clear today that neither view does full justice to the issue – not least because, in liquids, structure must be considered not only as a spatial but also as a temporal property. We are still coming to terms with that fact and how best to represent it, which is one reason why there is still no consensual “structure of water” in the same way as there is a structure of ice. What is more, it is also now recognized that there is a rich middle ground between crystal and gas, of which the liquid occupies only a part: this discussion must also encompass the quasi-order or partial order of liquid crystals and quasicrystals, the ‘frozen disorder’ of glasses, and the delicate interplay of kinetic and thermodynamic stability. X-ray diffraction has been central to all of these ideas, and it offered Bernal and others the first inkling of how we might meaningfully talk about the elusive liquid state.

Mixed metaphors

One of the first attempts to provide a molecular picture of liquid water came from the discoverer of X-rays themselves, Wilhelm Röntgen. In 1891 Röntgen suggested that the liquid might be a mixture of freely diffusing water molecules and what he termed “ice molecules” – something akin to ice-like clusters dispersed in the fluid state. He suggested that such a ‘mixture model’, as it has become known, could account for many of water’s anomalous properties, such as the decrease in viscosity at high pressure. Mixture models are still proposed today [2,3], attesting to the tenacity of the idea that there is something crystal-like in water structure.

X-ray scattering was already being applied to liquids, in particular to water, by Peter Debye and others in the late 1920s. These experiments showed that there was structural information in the pattern: a few broad but clearly identifiable peaks, which Debye interpreted as coming from both intra- and intermolecular interference. In 1933 Bernal and his colleague Ralph Fowler set out to devise a structural model that might explain the diffraction pattern measured from water. It had been found only the previous year that the water molecule has a V shape, and Bernal and Fowler argued from quantum-chemical considerations that it should have positive charges at the hydrogen atoms, balanced by two lobes of negative charge on the oxygen to produce a tetrahedral motif. On electrostatic grounds, each molecule should then form hydrogen bonds with four others in a tetrahedral arrangement. Noting the similarity with the tetrahedral structure in silicates, Bernal and Fowler developed a model in which water was regarded as a kind of distorted quartz. Their calculations produced fair agreement with the X-ray data: the peaks were in the right places, even if their intensities did not match so well [4].

This work established some of the core ideas of water structure, in particular the tetrahedral coordination. It set the scene for other models that started from a crystalline viewpoint. Notably, Henry Eyring and colleagues at the University of Utah devised a general picture of the liquid state consisting of an essentially crystalline close-packing threaded with many dislocations [5]. Molecules that escape from this close-packing can, in Eyring’s picture, wander almost gas-like between the dense clusters, making it a descendent of Röntgen’s mixture model.

Building liquids by hand

But Bernal was not happy with this view of the liquid as a defective solid, saying that it postulates “a greater degree of order…in the liquid than actually exists there” [6]. In the 1950s he started again, this time by considering a ‘simple liquid’ in which the molecules are spheres that clump together in an unstructured (and presumably dynamic) heap. Bernal needed physical models to guide his intuition, and during this period he constructed many of them, some now sadly lost. He used ball bearings to build dense random packings, or to see the internal structure better he would prop apart ping-pong balls or rubber balls with wires or rods, sometimes trying to turn himself into the required randomizing influence by selecting rods of different length without thinking. He was able to construct models of water that respected the local tetrahedral arrangement while producing no long- or medium-range order among molecules: a random hydrogen-bonded network in which the molecules are connected in rings with between four and seven members, as opposed to the uniformly six-membered rings of ordinary ice. Not only did this structure produce a good fit to the X-ray data (he counted out the interatomic distances by hand and plotted them as histograms), but the model liquid proved to have a higher density than ice, just as is the case for water [7].

This ‘mixed-ring’ random network supplies the basis for most subsequent molecular models of water [8, 9], although it is now clear that the network is highly dynamic – hydrogen bonds have a lifetime of typically 1 ps – and permeated with defects such as bifurcated and distorted hydrogen bonds [9, 10].

But although the tetrahedron seems to fit the local structure of water, that liquid is unusual in this regard, having a local geometry that is dictated by the high directionality of the hydrogen bonds. At the same time as Bernal was developing these ideas in the 1950s, Charles Frank at the University of Bristol proposed that for simple liquids, such as monatomic liquids and molten metals, a very common motif of short-ranged structure is instead the icosahedron [11]. This structure, Frank argued, provides the closest packing for a small number of atoms. But as one adds successive layers to an icosahedral cluster, the close-packing breaks down. What is more, the clusters have fivefold symmetry, which is incompatible with any crystalline arrangement. It is because of this incommensurability, Frank said, that liquid metals can be deeply supercooled without nucleating the solid phase. It was after hearing Frank speak on these ideas about polyhedral packings with local fivefold symmetry – very much within the context of the solid-state physics that was Frank’s speciality – that Bernal was prompted to revisit his model of the liquid state in the 1950s.

Forbidden order

Both X-ray scattering [12, 13] and X-ray spectroscopy [14] now offer some support for Frank’s picture of liquid metals, showing that something like icosahedral structures do form in metastable supercooled melts. Frank’s hypothesis was already recalled in 1984, however, when the first discovery was reported of a material that seemed to have a crystalline icosahedral order: a quasicrystal [15]. X-ray diffraction from an alloy of aluminium and manganese produced a pattern of sharp peaks with tenfold symmetry, which is now rationalized in terms of a solid structure that has local five- and tenfold symmetry but no perfect long-range translational order. Such materials are now recognized by the International Union of Crystallography as formally crystalline, according to the definition that they produce sharp, regular diffraction peaks. Frank’s icosahedral liquid clusters could provide the nuclei from which these quasicrystalline phases form, and indeed synchrotron X-ray crystallography of a supercooled melt of a Ti-Zr-Ni alloy shows that their formation does indeed precede the activated formation first of a metastable quasicrystalline phase and then of a stable crystal [12].

It seems fitting that Linus Pauling, whose work helped to explain the structures of water and ice [16], should have entered the early debate over the interpretation of quasicrystal diffraction. Pauling was on the wrong side, insisting dismissively that this was probably just a case of crystal twinning. But he, Bernal, Frank, and indeed William Bragg himself (a pioneer of X-ray studies of liquid crystals), all grappled with the question of determining how far ideas from crystalline matter can be imported into the study of the liquid state. Or to put it another way, they showed that X-ray crystallography is better viewed not as a method of probing order in matter, but as a means of examining the extent to which matter can be regarded as orderly. With the advent of high-intensity synchrotron sources that reduce the exposure times sufficiently to study ultrafast dynamic processes by X-ray diffraction [17, 18], it is now possible to explore that question as a function of the timescale being probed. It has been suggested that recent ddebate about the structure of water – a discussion that has oscillated between the poles of Bernal’s random tetrahedral network and mixture models – is itself all a matter of defining the notion of ‘structure’ on an appropriate timescale [19].

Such studies also seem to be confirming that Bernal asked the right question about the liquid state in 1959 (even if he phrased it as a statement): “it is not the fluidity of the liquid that gives rise to its irregularity. It is its irregularity that gives rise to its fluidity.” [7] Which is it, really? Do defects such as bifurcated hydrogen-bonds give water its fluidity [10]? Or is it dynamical making and breaking of hydrogen-bonds that undermines the clathrate-like regularity proposed by Pauling [20]? Whatever the case, there is little doubt now that Bernal’s perception that extending X-ray diffraction to biomolecules and liquids – and now to quasicrystals and all manner of soft matter – has led to a broader view of what crystallography is:

“And so there are no rules, or the old rules are enormously changed… We are seeing now a generalized crystallography, although it hasn’t been written up as such… [These materials] have their own inner logic, the same kind of logic but a different chapter of the logic that applies to the three-dimensional regular lattice crystals.” [21]

References
[1] A. L. Mackay, Journal of Physics: Conference Series 57, 1–16 (2007)
[2] C. H Cho, S. Singh & G. W. Robinson, Faraday Discuss. 103, 19-27 (1996)
[3] C. Huang et al., Proc. Natl Acad. Sci. 106, 15214-15218 (2009)
[4] J. D. Bernal and R. H. Fowler, J. Chem. Phys. 1, 515 (1933)
[5] H. Eyring, F. W. Cagle, Jr. and Carl J. Chritiansen, Proc. Natl Acad. Sci. 44 , 123-126 (1958)
[6] J. L. Finney, Journal of Physics: Conference Series 57, 40–52 (2007)
[7] J. D. Bernal, Proc. R. Inst. Great Britain 37, 355-393 (1959)
[8] H. Stillinger, Science 209, 451-457 (1980)
[9] J. L Finney, Philos. Trans. R. Soc. Lond. B 359 1145-1165 (2004)
[10] F. Sciortino, A. Geiger & H. E. Stanley, Nature 354, 218-221 (1991)
[11] F. C. Frank, Proc. R. Soc. London, Ser. A 215, 43-46 (1952)
[12] K. F. Kelton et al., Phys. Rev. Lett. 90, 195504 (2003)
[13] T. Schenk et al., Phys. Rev. Lett. 89, 075507 (2002)
[14] A. Filipponi, A. Di Cicco & S. De Panfilis, Phys. Rev. Lett. 83, 560 (1999).
[15] D. Shechtman, I. Blech, D. Gratias, & J. W. Cahn, Phys. Rev. Lett. 53, 1951–1953 (1984)
[16] Pauling, L., Nature 317, 512–514 (1985)
[17] J. Ihee et al., Science 309, 1223-1227 (2005)
[18] S. Bratos and M. Wulff, Adv. Chem. Phys. 137, 1-29 (2008)
[19] T. D. Kühne and R. Z. Khaliullin, Nat. Commun. 4, 1450 (2013)
[20] L. Pauling, L., in Hadzi, D. & Thompson, H. W. (eds), Hydrogen Bonding, 1-6 (Pergamon Press, New York, 1959).
[21] J. D. Bernal (1966). Opening remarks, in G. E. W. Wolstenholme & M. O’Connor (eds.), Principles of Biomolecular Organization. Little Brown & Co., Boston.