He Say, He Sigh, He Sow #48

• « S’il est un homme tourmenté par la maudite ambition de mettre tout un livre dans une page, toute une page dans une phrase, et tout une phrase dans un mot, c’est moi. » — Joseph Jourbet (1754-1824)

• “If there is a man tormented by the cursed ambition to compress an entire book into a page, an entire page into a phrase, and that phrase into a word, it is I.” — Joseph Jourbet

Get Your Ox Off

Boustrophedon (pronounced “bough-stra-FEE-dun” or “boo-stra-FEE-dun”) is an ancient Greek word literally meaning “as the ox turns (in ploughing)”, that is, moving left-right, right-left, and so on. The word is used of writing that runs down the page in the same way. To see what that means, examine two versions of the first paragraph of Clark Ashton Smith’s story “The Demon of the Flower” (1933). The first is written in the usual way, the second is written boustrophedon:

Not as the plants and flowers of Earth, growing peacefully beneath a simple sun, were the blossoms of the planet Lophai. Coiling and uncoiling in double dawns; tossing tumultuously under vast suns of jade green and balas-ruby orange; swaying and weltering in rich twilights, in aurora-curtained nights, they resembled fields of rooted servants that dance eternally to an other-worldly music.

Not as the plants and flowers of Earth, growing peacefully
.iahpoL tenalp eht fo smossolb eht erew ,nus elpmis a htaeneb
Coiling and uncoiling in double dawns; tossing tumultuously
;egnaro ybur-salab dna neerg edaj fo snus tsav rednu
swaying and weltering in rich twilights, in aurora-curtained
ecnad taht stnavres detoor fo sdleif delbmeser yeht ,sthgin
eternally to an other-worldly music.

Boustrophedon writing was once common and sometimes the left-right lines would also be mirror-reversed, like this:

You could also use the term “boustrophedon” to describe the way this table of numbers is filled:


The table begins with “1” in the top left-hand corner, then moves right for “2”, then down for “3”, then right-and-up for “4”, “5” and “6”, then right for “7”, then left-and-down for “8”, “9” and “10”, and so on. You could also say that the numbers snake through the table. I’ve marked the primes among them, because I was interested in the patterns made by the primes when the numbers were represented as blocks on a grid, like this:


Primes are in solid white (compare the Ulam spiral). Here’s the boustrophedon prime-grid on a finer scale:


(click for full image)

And what about other number-tests? Here are the even numbers marked on the grid (i.e. n mod 2 = 0):


n mod 2 = 0

And here are some more examples of a modulus test:


n mod 3 = 0


n mod 5 = 0


n mod 9 = 0


n mod 15 = 0


n mod various = 0 (animated gif)

Next I looked at reciprocals (numbers divided into 1) marked on the grid, with the digits of a reciprocal marking the number of blank squares before a square is filled in (if the digit is “0”, the square is filled immediately). For example, in base ten 1/7 = 0.142857142857142857…, where the block “142857” repeats for ever. When represented on the grid, 1/7 has 1 blank square, then a filled square, then 4 blank squares, then a filled square, then 2 blank squares, then a filled square, and so on:


1/7 in base 10

And here are some more reciprocals (click for full images):


1/9 in base 2


1/13 in base 10


1/27 in base 10


1/41 in base 10


1/63 in base 10


1/82 in base 10


1/101 in base 10


1/104 in base 10


1/124 in base 10


1/143 in base 10


1/175 in base 10


1/604 in base 8


1/n in various bases (animated gif)

Sward and Sorcery

Watership Down by Richard Adams with cover by Pauline BaynesWatership Down, Richard Adams (1972)

A book is a magical thing. Black marks on white paper create words; words conjure worlds. But the sorcery of Watership Down is remarkable even by literary standards. The world conjured here defies expectation and suspends disbelief. Richard Adams took a seemingly ludicrous subject – the adventures of a group of rabbits – and made it something that could grip the imagination and stir the emotions of readers at any age.

He did this by combining two distinct traditions of writing about animals: the realism of Jack London’s Call of the Wild (1903) and White Fang (1906) with the fantasy of Wind in the Willows (1908) and Beatrix Potter (1866-1943). Jack London’s animals are real and don’t speak, but Grahame and Potter turned animals into miniature humans, bringing them into our world, taming and civilizing them. Adams does the reverse: he takes us into the world of animals. He kept his rabbits wild and on all fours, sworn to the sward that they create with their teeth, but he used one piece of anthropomorphism. Hazel, Fiver, Bigwig and the other rabbits can all talk. They have a language, Lapine, and communicate with other animals using a “very simple, limited lingua franca of the hedgerow and woodland” (Part II, ch. 20).

How else could there be a proper story? But that one piece of anthropomorphism is actually an umbrella sheltering many other things: intelligence, memory, planning, persuasion, story-telling, the ability to lie, and so on. With language, the rabbits become like a tribe of primitive humans, pre-literate, almost innumerate:

Rabbits can count up to four. Any number above that is Hrair – ‘a lot’ or ‘a thousand’. Thus they say U Hrair – ‘The Thousand’ – to mean, collectively, all the enemies (or elil, as they call them) of rabbits – fox, stoat, weasel, cat, owl, man, etc. There were probably more than five rabbits in the litter where Fiver was born, but his name, Hrairoo, means ‘Little thousand’, i.e. the little one of a lot, or, as they say of pigs, ‘the runt’. (Part I, “The Journey”, ch. 1, “The Notice Board”)

At the beginning of the book, Fiver is the unacknowledged shaman of Sandleford Warren and foresees the doom that approaches it. Unfortunately, few rabbits believe him, which is why Adams heads the first chapter with a quote from Aeschylus, Cassandra’s warning that “The house reeks of death and dripping blood.” Every other chapter has its apposite quote, ancient or modern, poetry or prose, whimsical or serious: Aeschylus, Shakespeare, Tennyson, Napoleon, W.H. Auden, Dr Johnson, Clausewitz, Walter de la Mare and so on. The quotes stitch Watership Down deftly into the literary canon and into history, because the book is, in part, a celebration of story-telling and the possibilities of language.

That celebration is echoed inside the book, because the narrative is broken up by stories of El-ahrairah, the rabbits’ trickster-prince and protector. He’s like Odysseus and Watership Down is like the Odyssey. It’s a cycle of folk-tales in the making. Like Odysseus, the rabbits have to rely on their cunning and their speed, tricking monsters, not directly confronting them. Their own adventures will, in time, be attributed to El-ahrairah. Without writing, they have no history and sooner or later real events will melt into myth. But that’s the natural way: writing is a mysterious and evil thing to those rabbits who can intuit its purpose:

In the livid, foggy twilight, Fiver stared at the board. As he stared, the black sticks flickered on the white surface. They raised their sharp, wedge-shaped little heads and chattered together like a nestful of young weasels. The sound, mocking and cruel, came faintly to his ears, as though muffled by sand or sacking. ‘In memory of Hazel-rah! In memory of Hazel-rah! In memory of Hazel-rah! Ha ha ha ha ha ha!’ (Part II, ch. 26, “Fiver Beyond”)

Like Tolkien in The Hobbit (1937) and Lord of the Rings (1954-5), Adams is writing against the evils of technology and modernity; unlike Tolkien, he lists writing among those evils. A book that condemns writing is a paradox, but Adams is adopting a rabbit’s perspective. Tolkien’s books were, I’d suggest, a strong hidden influence on Watership Down. Rabbits are hole-dwellers like hobbits and the band of rabbits who set out from Sandleford Warren are rather like the Company of the Ring. Adams treats Lapine the way Tolkien treats his invented languages, using it to make us aware of the gulf across which the story comes to us:

With them was a third rabbit, Hlao – Pipkin – a friend of Fiver. (Hlao means any small concavity in the grass where moisture may collect, e.g. the dimple formed by a dandelion or thistle-cup.) (Part 1, ch. 4, “The Departure”)

Meriadoc was chosen to fit the fact that this character’s shortened name, Kali, meant in the Westron ‘jolly, gay’, though this was actually an abbreviation of the now unmeaning Buckland name Kalimac. (Lord of the Rings, Appendix F, “On Translation”)

But I think Adams is more linguistically creative and subtle than Tolkien, whose invented languages still seem like real ones: Welsh, Finnish, Old Norse and so on. Lapine isn’t reminiscent of anything familiar and some of its words – pfeffa, “cat”, and hrududu, “motor vehicle” – are cleverly simple, just the sort of onomatopoeias you can imagine a talking rabbit would use.

Cover of a recent edition of Watership Down

Cover of a recent edition of Watership Down

Lapine is also like Nadsat, the teen-speak invented by Anthony Burgess for A Clockwork Orange (1962). Adams leaves some words of Lapine untranslated at first, letting context give them meaning, sprinkling them through the text and allowing them to sink slowly into the reader’s mind. By the end of the book, you’ll find that you can understand basic Lapine: “Siflay hraka, u embleer rah,” says Bigwig to General Woundwort and the line doesn’t need translation.

General Woundwort is the Polyphemus or Sauron of Watership Down: a rabbit almost as big as a hare, the cunning and vicious megalomaniac who leads the slave-warren Efrafra. His wickedness is on a much smaller scale than Sauron’s, of course, but that makes it more credible and so more powerful. Lord of the Rings is more ambitious than The Hobbit, which is admirable, but also less successful, which was inevitable. Bilbo sets out to slay a dragon, not save the world. The rabbits in Watership Down are unwilling refugees who want to found a permanent warren of their own. It’s a small thing within the wider world, where humans rear giant metal pylons, span rivers with bridges, and speed to and fro in hrududim, but then human affairs are small within the wider universe.

It doesn’t matter: significance is not determined by size, purpose doesn’t have to be blunted by futility. The rabbits’ instincts drive them on and their ambitions are big enough for their abilities. They don’t need more. It’s General Woundwort’s desire to be great that prevents him from being so. He’s the most human of the rabbits and so the most evil: “All other elil do what they have to do and Frith moves them as he moves us. They live on the earth and they need food. Men will never rest till they’ve spoiled the earth and destroyed the animals.” (Part II, ch. 21, “For El-ahrairah to Cry”)

Man’s restlessness and meddling are a theme Adams took up again in The Plague Dogs (1977), a novel about two dogs that escape from a research laboratory in the Lake District. It’s a weak book set beside Watership Down, written more self-consciously and less coherently. Adams doesn’t stitch literary allusions into the story: he nails them in like corrugated iron. But his sympathy for animals is still there and so is his ability to describe the world through their sharper and subtler senses. The rabbits of Watership Down are like a primitive tribe of humans, but you never forget that they aren’t actually human:

A robin on a low branch twittered a phrase and listened for another that answered to him from beyond the farmhouse. A chaffinch gave its little falling song and farther off, high in an elm, a chiff-chaff began to call. Hazel stopped and then sat up, the better to scent the air. Powerful smells of straw and cow-dung mingled with those of elm-leaves, ashes and cattle-feed. Fainter traces came to his nose as the overtones of a bell sound in a trained ear. Tobacco, naturally: a good deal of cat and rather less dog and then, suddenly and beyond doubt, rabbit. He looked at Pipkin and saw that he too had caught it. (Part II, ch. 24, “Nuthanger Farm”)

That’s describing a raid on a farm that keeps pet rabbits. Hazel wants to find some does for the warren at Watership Down, where he and his fellow hlessil – “wanderers, scratchers, vagabonds” – seem to have finally found sanctuary. They’ve come a long way through strange country, but they’ll go further and see stranger before the end of the book. Watership Down is first and foremost an adventure story, but it’s also a celebration of the English countryside: its flowers, trees, birds, streams and rivers; its sounds, scents, shapes; its delights and dangers. The rabbits have their place there, naming themselves from nature, and unlike man, with his stinks and cacophonies, they don’t desire dominion over it.

The raucous gull Kehaar, their ally in their struggle with General Woundwort and Efrafra, brings word of far-off places and the mysterious sea, but their world is room enough. It fills their senses, challenges their cunning and ingenuity, sustains them, in the end will slay them. The countryside is the biggest character, as the title suggests, and rabbits were the best way to bring that character into a book. They’re social animals, mostly warren-dwelling, occasionally wandering, and if Adams could suspend disbelief and give them language, he could conjure a world of wonders through their eyes, ears, noses and mouths.

He could and did exactly that with the help of R.M. Lockley, who wrote The Private Life of the Rabbit, the “remarkable book” on which he drew for a “knowledge of rabbits and their ways” (“Acknowledgments”). Rabbits are in fact remarkable animals, but most people won’t realize that until they read the remarkable book called Watership Down. It’s a microcosm that mirrors the macrocosm, both reflecting man and reflecting on our ways. Rabbits “don’t name the stars”, Adams tells us, but in truth they don’t name anything, because Lapine doesn’t exist. It was his great achievement to make that impossibility plausible, turning sward-munchers into adventurers, mystics and dynasts with the sorcery of words:

A few minutes later there was not a rabbit to be seen on the down. The sun sank beneath Ladle Hill and the autumn stars began to shine in the darkening east – Perseus and the Pleiades, Cassiopeia, faint Pisces and the great square of Pegasus. The wind freshened, and soon myriads of dry beech leaves were filling the ditches and hollows and blowing in gusts across the dark miles of open grass. Underground, the story continued. (Part IV, ch. 50, “And Last”)

Stories and Stars

A story is stranger than a star. Stronger too. What do I mean? I mean that the story has more secrets than a star and holds its secrets more tightly. A full scientific description of a star is easier than a full scientific description of a story. Stars are much more primitive, much closer to the fundamentals of the universe. They’re huge and impressive, but they’re relatively simple things: giant spheres of flaming gas. Mathematically speaking, they’re more compressible: you have to put fewer numbers into fewer formulae to model their behaviour. A universe with just stars in it isn’t very complex, as you would expect from the evolution of our own universe. There were stars in it long before there were stories.

A universe with stories in it, by contrast, is definitely complex. This is because stories depend on language and language is the scientific mother-lode, the most difficult and important problem of all. Or rather, the human brain is. The human brain understands a lot about stars, despite their distance, but relatively little about itself, despite brains being right on the spot. Consciousness is a tough nut to crack, for example. Perhaps it’s uncrackable. Language looks easier, but linguistics is still more like stamp-collecting than science. We can describe the structure of language in detail – use labels like “pluperfect subjunctive”, “synecdoche”, “bilabial fricative” and so on – but we don’t know how that structure is instantiated in the brain or where language came from. How did it evolve? How is it coded in the human genome? How does meaning get into and out of sounds and shapes, into and out of speech and writing? These are big, important and very interesting questions, but we’ve barely begun to answer them.

Distribution of dental fricatives and the O blood-group in Europe (from David Crystal's )

Distribution of dental fricatives and the O blood-group in Europe (from David Crystal’s Cambridge Encyclopedia of Language)

But certain things seem clear already. Language-genes must differ in important ways between different groups, influencing their linguistic skills and their preferences in phonetics and grammar. For example, there are some interesting correlations between blood-groups and use of dental fricatives in Europe. The invention of writing has exerted evolutionary pressures in Europe and Asia in ways it hasn’t in Africa, Australasia and the Americas. Glossogenetics, or the study of language and genes, will find important differences between races and within them, running parallel with differences in psychology and physiology. Language is a human universal, but that doesn’t mean one set of identical genes underlies the linguistic behaviour of all human groups. Skin, bones and blood are human universals too, but they differ between groups for genetic reasons.

Understanding the evolution and effects of these genetic differences is ultimately a mathematical exercise, and understanding language will be too. So will understanding the brain. For one thing, the brain must, at bottom, be a maths-engine or math-engine: a mechanism receiving, processing and sending information according to rules. But that’s a bit like saying fish are wet. Fish can’t escape water and human beings can’t escape mathematics. Nothing can: to exist is to stand in relation to other entities, to influence and be influenced by them, and mathematics is about that inter-play of entities. Or rather, that inter-play is Mathematics, with a big “M”, and nothing escapes it. Human beings have invented a way of modelling that fundamental micro- and macroscopic inter-play, which is mathematics with a small “m”. When they use this model, human beings can make mistakes. But when they do go wrong, they can do so in ways detectable to other human beings using the same model:

In 1853 William Shanks published his calculations of π to 707 decimal places. He used the same formula as [John] Machin and calculated in the process several logarithms to 137 decimal places, and the exact value of 2^721. A Victorian commentator asserted: “These tremendous stretches of calculation… prove more than the capacity of this or that computer for labor and accuracy; they show that there is in the community an increase in skill and courage…”

Augustus de Morgan thought he saw something else in Shanks’s labours. The digit 7 appeared suspiciously less often than the other digits, only 44 times against an average expected frequency of 61 for each digit. De Morgan calculated that the odds against such a low frequency were 45 to 1. De Morgan, or rather William Shanks, was wrong. In 1945, using a desk calculator, Ferguson found that Shanks had made an error; his calculation was wrong from place 528 onwards. Shanks, fortunately, was long dead. (The Penguin Dictionary of Curious and Interesting Numbers, 1986, David Wells, entry for π, pg. 51)

Unlike theology or politics, mathematics is not merely self-correcting, but multiply so: there are different routes to the same truths and different ways of testing a result. Science too is self-correcting and can test its results by different means, partly because science is a mathematical activity and partly because it is studying a mathematical artifact: the gigantic structure of space, matter and energy known as the Universe. Some scientists and philosophers have puzzled over what the physicist Eugene Wigner (1902-95) called “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”. In his essay on the topic, Wigner tried to make two points:

The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it. Second, it is just this uncanny usefulness of mathematical concepts that raises the question of the uniqueness of our physical theories. (Op. cit., in Communications in Pure and Applied Mathematics, vol. 13, No. I, February 1960)

I disagree with Wigner: it is not mysterious or uncanny and there is a rational explanation for it. The “effectiveness” of small-m maths for scientists is just as reasonable as the effectiveness of fins for fish or of wings for birds. The sea is water and the sky is air. The universe contains both sea and sky: and the universe is maths. Fins and wings are mechanisms that allow fish and birds to operate effectively in their water- and air-filled environments. Maths is a mechanism that allows scientists to operate effectively in their maths-filled environment. Scientists have, in a sense, evolved towards using maths just as fish and birds have evolved towards using fins and wings. Men have always used language to model the universe, but language is not “unreasonably effective” for understanding the universe. It isn’t effective at all.

It is effective, however, in manipulating and controlling other human beings, which explains its importance in politics and theology. In politics, language is used to manipulate; in science, language is used to explain. That is why mathematics is so important in science and so carefully avoided in politics. And in certain academic disciplines. But the paradox is that physics is much more intellectually demanding than, say, literary theory because the raw stuff of physics is actually much simpler than literature. To understand the paradox, imagine that two kinds of boulder are strewn on a plain. One kind is huge and made of black granite. The other kind is relatively small and made of chalk. Two tribes of academic live on the plain, one devoted to studying the black granite boulders, the other devoted to studying the chalk boulders.

The granite academics, being unable to lift or cut into their boulders, will have no need of physical strength or tool-making ability. Instead, they will justify their existence by sitting on their boulders and telling stories about them or describing their bumps and contours in minute detail. The chalk academics, by contrast, will be lifting and cutting into their boulders and will know far more about them. So the chalk academics will need physical strength and tool-making ability. In other words, physics, being inherently simpler than literature, is within the grasp of a sufficiently powerful human intellect in a way literature is not. Appreciating literature depends on intuition rather than intellect. And so strong intellects are able to lift and cut into the problems of physics as they aren’t able to lift and cut into the problems of literature, because the problems of literature depend on consciousness and on the hugely complex mechanisms of language, society and psychology.

Intuition is extremely powerful, but isn’t under conscious control like intellect and isn’t transparent to consciousness in the same way. In the fullest sense, it includes the senses, but who can control his own vision and hearing or understand how they turn the raw stuff of the sense-organs into the magic tapestry of conscious experience? Flickering nerve impulses create a world of sight, sound, scent, taste and touch and human beings are able to turn that world into the symbols of language, then extract it again from the symbols. This linguifaction is a far more complex process than the ignifaction that drives a star. At present it’s beyond the grasp of our intellects, so the people who study it don’t need and don’t build intellectual muscle in the way that physicists do.

Or one could say that literature is at a higher level of physics. In theory, it is ultimately and entirely reducible to physics, but the mathematics governing its emergence from physics are complex and not well-understood. It’s like the difference between a caterpillar and a butterfly. They are two aspects of one creature, but it’s difficult to understand how one becomes the other, as a caterpillar dissolves into chemical soup inside a chrysalis and turns into something entirely different in appearance and behaviour. Modelling the behaviour of a caterpillar is simpler than modelling the behaviour of a butterfly. A caterpillar’s brain has less to cope with than a butterfly’s. Caterpillars crawl and butterflies fly. Caterpillars eat and butterflies mate. And so on.

Stars can be compared to caterpillars, stories to butterflies. It’s easier to explain stars than to explain stories. And one of the things we don’t understand about stories is how we understand stories.

2:1 Now when Jesus was born in Bethlehem of Judaea in the days of Herod the king, behold, there came wise men from the east to Jerusalem, 2:2 Saying, Where is he that is born King of the Jews? for we have seen his star in the east, and are come to worship him. 2:3 When Herod the king had heard these things, he was troubled, and all Jerusalem with him. 2:4 And when he had gathered all the chief priests and scribes of the people together, he demanded of them where Christ should be born. 2:5 And they said unto him, In Bethlehem of Judaea: for thus it is written by the prophet, 2:6 And thou Bethlehem, in the land of Juda, art not the least among the princes of Juda: for out of thee shall come a Governor, that shall rule my people Israel. 2:7 Then Herod, when he had privily called the wise men, enquired of them diligently what time the star appeared. 2:8 And he sent them to Bethlehem, and said, Go and search diligently for the young child; and when ye have found him, bring me word again, that I may come and worship him also. 2:9 When they had heard the king, they departed; and, lo, the star, which they saw in the east, went before them, till it came and stood over where the young child was. 2:10 When they saw the star, they rejoiced with exceeding great joy. 2:11 And when they were come into the house, they saw the young child with Mary his mother, and fell down, and worshipped him: and when they had opened their treasures, they presented unto him gifts; gold, and frankincense and myrrh. – From The Gospel According to Saint Matthew.