Wake the Snake

In my story “Kopfwurmkundalini”, I imagined the square root of 2 as an infinitely long worm or snake whose endlessly varying digit-segments contained all stories ever (and never) written:

• √2 = 1·414213562373095048801688724209698078569671875376948073…

But there’s another way to get all stories ever written from the number 2. You don’t look at the root(s) of 2, but at the powers of 2:

• 2 = 2^1 = 2
• 4 = 2^2 = 2*2
• 8 = 2^3 = 2*2*2
• 16 = 2^4 = 2*2*2*2
• 32 = 2^5 = 2*2*2*2*2
• 64 = 2^6 = 2*2*2*2*2*2
• 128 = 2^7 = 2*2*2*2*2*2*2
• 256 = 2^8 = 2*2*2*2*2*2*2*2
• 512 = 2^9 = 2*2*2*2*2*2*2*2*2
• 1024 = 2^10
• 2048 = 2^11
• 4096 = 2^12
• 8192 = 2^13
• 16384 = 2^14
• 32768 = 2^15
• 65536 = 2^16
• 131072 = 2^17
• 262144 = 2^18
• 524288 = 2^19
• 1048576 = 2^20
• 2097152 = 2^21
• 4194304 = 2^22
• 8388608 = 2^23
• 16777216 = 2^24
• 33554432 = 2^25
• 67108864 = 2^26
• 134217728 = 2^27
• 268435456 = 2^28
• 536870912 = 2^29
• 1073741824 = 2^30

The powers of 2 are like an ever-lengthening snake swimming across a pool. The snake has an endlessly mutating head and a rhythmically waving tail with a regular but ever-more complex wake. That is, the leading digits of 2^p don’t repeat but the trailing digits do. Look at the single final digit of 2^p, for example:

• 02 = 2^1
• 04 = 2^2
• 08 = 2^3
• 16 = 2^4
• 32 = 2^5
• 64 = 2^6
• 128 = 2^7
• 256 = 2^8
• 512 = 2^9
• 1024 = 2^10
• 2048 = 2^11
• 4096 = 2^12
• 8192 = 2^13
• 16384 = 2^14
• 32768 = 2^15
• 65536 = 2^16
• 131072 = 2^17
• 262144 = 2^18
• 524288 = 2^19
• 1048576 = 2^20
• 2097152 = 2^21
• 4194304 = 2^22

The final digit of 2^p falls into a loop: 2 → 4 → 8 → 6 → 2 → 4→ 8…

Now try the final two digits of 2^p:

02 = 2^1
04 = 2^2
08 = 2^3
16 = 2^4
32 = 2^5
64 = 2^6
• 128 = 2^7
• 256 = 2^8
• 512 = 2^9
• 1024 = 2^10
• 2048 = 2^11
• 4096 = 2^12
• 8192 = 2^13
• 16384 = 2^14
• 32768 = 2^15
• 65536 = 2^16
• 131072 = 2^17
• 262144 = 2^18
• 524288 = 2^19
• 1048576 = 2^20
• 2097152 = 2^21
• 4194304 = 2^22
• 8388608 = 2^23
• 16777216 = 2^24
• 33554432 = 2^25
• 67108864 = 2^26
• 134217728 = 2^27
• 268435456 = 2^28
• 536870912 = 2^29
• 1073741824 = 2^30

Now there’s a longer loop: 02 → 04 → 08 → 16 → 32 → 64 → 28 → 56 → 12 → 24 → 48 → 96 → 92 → 84 → 68 → 36 → 72 → 44 → 88 → 76 → 52 → 04 → 08 → 16 → 32 → 64 → 28… Any number of trailing digits, 1 or 2 or one trillion, falls into a loop. It just takes longer as the number of trailing digits increases.

That’s the tail of the snake. At the other end, the head of the snake, the digits don’t fall into a loop (because of the carries from the lower digits). So, while you can get only 2, 4, 8 and 6 as the final digits of 2^p, you can get any digit but 0 as the first digit of 2^p. Indeed, I conjecture (but can’t prove) that not only will all integers eventually appear as the leading digits of 2^p, but they will do so infinitely often. Think of a number and it will appear as the leading digits of 2^p. Let’s try the numbers 1, 12, 123, 1234, 12345…:

16 = 2^4
128 = 2^7
12379400392853802748... = 2^90
12340799625835686853... = 2^1545
12345257952011458590... = 2^34555
12345695478410965346... = 2^63293
12345673811591269861... = 2^4869721
12345678260232358911... = 2^5194868
12345678999199154389... = 2^62759188

But what about the numbers 9, 98, 987, 986, 98765… as leading digits of 2^p? They don’t appear as quickly:

9007199254740992 = 2^53
98079714615416886934... = 2^186
98726397006685494828... = 2^1548
98768356967522174395... = 2^21257
98765563827287722773... = 2^63296
98765426081858871289... = 2^5194871
98765430693066680199... = 2^11627034
98765432584491513519... = 2^260855656
98765432109571471006... = 2^1641098748

Why do fragments of 123456789 appear much sooner than fragments of 987654321? Well, even though all integers occur infinitely often as leading digits of 2^p, some integers occur more often than others, as it were. The leading digits of 2^p are actually governed by a fascinating mathematical phenomenon known as Benford’s law, which states, for example, that the single first digit, d, will occur with the frequency log10(1 + 1/d). Here are the actual frequencies of 1..9 for all powers of 2 up to 2^101000, compared with the estimate by Benford’s law:

1: 30% of leading digits ↔ 30.1% estimated
2: 17.55% ↔ 17.6%
3: 12.45% ↔ 12.49%
4: 09.65% ↔ 9.69%
5: 07.89% ↔ 7.92%
6: 06.67% ↔ 6.69%
7: 05.77% ↔ 5.79%
8: 05.09% ↔ 5.11%
9: 04.56% ↔ 4.57%

Because (inter alia) 1 appears as the first digit of 2^p far more often than 9 does, the fragments of 123456789 appear faster than the fragments of 987654321. Mutatis mutandis, the same applies in all other bases (apart from bases that are powers of 2, where there’s a single leading digit, 1, 2, 4, 8…, followed by 0s). But although a number like 123456789 occurs much frequently than 987654321 in 2^p expressed in base 10 (and higher), both integers occur infinitely often.

As do all other integers. And because stories can be expressed as numbers, all stories ever (and never) written appear in the powers of 2. Infinitely often. You’ll just have to trim the tail of the story-snake.



მე, ზამთრისაგან ჯაჭვაწყვეტილი,
ნაცნობ ბაღისკენ მივემართები,
სად ფერად უცხო, ყნოსვად კეთილი,
ზამთარ და ზაფხულ ჰყვავის ვარდები.


Unchained from winter,
I walk to a long-known garden,
Where, sweet-scented and bright,
Roses grow winter and summer through.

ვარდები, გალაკტიონ ტაბიძე
“Roses”, Galaktion Tabidze — a translation into English

Performativizing Papyrocentricity #71

Papyrocentric Performativity Presents…

Clive DriveUnreliable Memoirs (1980) and Always Unreliable: The Memoirs (2001), Clive James

Nou’s WhoArt Nouveau, Camilla de la Bedoyere (Flame Tree Publishing 2005)

Hit and MistletoeThrough It All I’ve Always Laughed, Count Arthur Strong (Faber & Faber 2013)

Beauties and BeastsShardik, Richard Adams (1974)

Or Read a Review at Random: RaRaR

Genoa Ultramarina

«Il mare è la civiltà», disse [Franco Scoglio] una volta, «il sentimento, la passione, le tempeste, ma l’amore, gli sbarchi, le partenze, il mare è tutto. La follia va di pari passo con il mare». — Ultrà. Il volto nascosto delle tifoserie di calcio in Italia, Tobias Jones (2020)

• “The sea is civilization,” [Franco Scoglio] said once, “sentiment, passion, storms, love, landings, leavings, the sea is everything… madness walks with the sea.” — Ultra: The Underworld of Italian Football, Tobias Jones (2019)

Post-Performative Post-Scriptum

I’m not sure if the Italian is the original Italian or an Italian translation of Jones’s English translation of the original Italian. But it seems to be the former.

Elsewhere other-accessible…

Franco Scoglio en italiano
Franco Scoglio in English

Bash the Trash

From George Orwell’s “As I Please” for 11th February 1944, Tribune:

THE FOLLOWING lines are quoted in Anthony Trollope’s Autobiography:

When Payne-Knight’s Taste was issued on the town
A few Greek verses in the text set down
Were torn to pieces, mangled into hash,
Hurled to the flames as execrable trash;
In short, were butchered rather than dissected
And several false quantities detected;
Till, when the smoke had risen from the cinders
It was discovered that — the lines were Pindar’s!

Trollope does not make clear who is the author of these lines, and I should be very glad if any reader could let me know. But I also quote them for their own sake — that is, for the terrible warning to literary critics that they contain — and for the sake of drawing attention to Trollope’s Autobiography, which is a most fascinating book, although or because it is largely concerned with money.

Elsewhere Other-Accessible…

Pindar (c. 518-438 BC) at Wikipedia
An Analytical Inquiry Into the Principles of Taste (1806) by Richard Payne-Knight at Archive.org
An Autobiography and Other Writings (1869) by Anthony Trollope at Gutenberg

Beauties and Beasts

Shardik, Richard Adams (1974)

Is it thirty years since I last read Shardik? No, it think it’s nearer forty. But as I read the book in March this year I began remembering small things before I came to them again. And I realized how deep the characters and story had sunk into my mind on those early readings long ago. Indeed, I felt that coming across the book again in a second-hand shop had been important-with-a-capital-I, as though I’d been meant to meet it again now.

Maybe it wasn’t and maybe I hadn’t. But the opening chapters, in which the simple hunter Kelderek finds and helps to capture the giant bear Shardik, have been some of the most vivid and enjoyable literature I’ve ever read. Adams conjures the forest fire that drives Shardik, burned and near-dead, across the great river Telthearna; brings Kelderek and other characters to life with something like Dickensian vividness and depth; gives them a solid and scented world to inhabit; and evokes a genuine sense of matriarchal mystery and magic around the island of Quiso, where the Tuginda and her priestesses have awaited the return of Shardik for centuries. And Shardik himself is a huge and dangerous presence, slapping a leopard aside like a twig before he collapses and begins to die of his burns. He’s awesome even in his distress:

The bear was still lying among the scarlet trepsis, but already it looked less foul and wretched. Its great wounds had been dressed with some kind of yellow ointment. One girl was keeping the flies from its eyes and ears with a fan of fern-fronds, while another, with a jar of ointment, was working along its back and as much as she could reach of the flank on which it was lying. Two others had brought sand to cover patches of soiled ground which they had already cleaned and hoed with pointed sticks. The Tuginda was holding a soaked cloth to the bear’s mouth, as [Kelderek] himself had done, but was dipping it not in the pool but in a water-jar at her feet. The unhurried bearing of the girls contrasted strangely with the gashed and monstrous body of the creature they were tending. Kelderek watched them pause in their work, waiting as the bear stirred restlessly. Its mouth gaped open and one hind leg kicked weakly before coming to rest once more among the trepsis. – end of chapter 10 in Book I, “Ortelga”

If Shardik continued like that, I think it would be much better-known today. But it doesn’t. It turns not just grimmer, but less well-written and less psychologically plausible. The simple hunter Kelderek, friend of children and awestruck acolyte of Shardik, turns into a ruthless priest-king who cages his bear-god and oversees a trade in child-slaves to finance a war of attrition against the enemies of his tribe. And that small and impoverished tribe, from the half-forgotten river-island of Ortelga in the far north, has overthrown an empire by then. Shardik has given them victory, becoming a literal deus ex machina in a crucial early battle. Or perhaps that should be deus in machina:

Suddenly a snarling roar, louder even than the surrounding din of battle, filled the tunnel-like roadway under the trees. There followed a clanging and clattering of iron, sharp cracks of snapped wood, panic cries and a noise of dragging and scraping. Baltis’ voice shouted, “Let go, you fools!” Then again broke out the snarling, full of savagery and ferocious rage. Kelderek leapt to his feet.

The cage had broken loose and was rushing down the hill, swaying and jumping as the crude wheels ploughed ruts in the mud and struck against protruding stones. The roof had split apart and the bars were hanging outwards, some trailing along the ground, others lashing sideways like a giant’s flails. Shardik was standing upright, surrounded by long, white splinters of wood. Blood was running down one shoulder and he foamed at the mouth, beating the iron bars around him as Baltis’ hammers had never beaten them.

The point of a sharp, splintered stake had pierced his neck and as it swayed up and down, levering itself in the wound, he roared with pain and anger. Red-eyed, frothing and bloody, his head smashing through the flimsy lower branches of the trees overhanging the track, he rode down upon the battle like some beast-god of apocalypse. – Book I, ch. 22, “The Cage”

I don’t like that “splintered stake … levering itself in the wound.” It seems gratuitous. And that kind of thing doesn’t stop. Shardik suffers from beginning to end of the book and at times I felt as though he’d become little more than a punch-bag for the plot. Although many readers will come to this book as young fans of Watership Down (1972), I don’t think it’s a good book for children. There are cruelty and ruthlessness in Watership Down, but they don’t overwhelm the story as they come to do in Shardik. And the characters who suffer in Watership Down are rabbits; in Shardik, they’re children and a giant bear. There was one act of cruelty that struck me with horror when I read it as a teenager, because it suddenly and ruthlessly smashed the hope I had invested in a character.

I barely noticed the incident this time, because I knew it was coming and because I wasn’t captivated by Adams’ prose any more. He starts the book well, but his best here isn’t as good as his best in Watership Down. And his prose gets much less good after Book I. Plus, I could see his influences more clearly: classical myth and history, the Bible, Dickens. The book begins with these lines from Homer:

οἴκτιστον δὴ κεῖνο ἐμοῖς ἴδον ὀφθαλμοῖσι
πάντων, ὅσσ᾽ ἐμόγησα πόρους ἁλὸς ἐξερεείνων.

They’re not translated, but they mean:

It was the most pitiable sight of all I saw exploring the pathways of the sea. – Odyssey XII, 258

Homer’s influence hovers below the surface everywhere in Book I, sometimes bursting through in long and elaborate similes that don’t always work very well. But I think that something else that doesn’t always work very well is part of Adams’ linguistic cleverness rather than his clumsiness. Shardik is set in a fantasy universe with simple technology and some kind of magic. Like many writers before him and after, Adams creates new languages to go with his new world. The hunter Kelderek is nicknamed Zenzuata, meaning “Play-with-children”. Later he becomes Crendrik, the “Eye of God” and high-priest of Shardik, the “Power of God.” When he’s still a simple hunter he hears a song with the refrain “Senandril na kora, senandril na ro”; at another time he marvels at the beauty of a gold-and-purple bird called a kynat; at another he eats the ripe fruit of a tendriona on the island of Quiso, where the high-priestess is called the Tuginda and addressed with the honorific säiyett.

The strange names and words transport you from the here-and-now of reality to the elsewhere-and-elsewhen of fantasy. But what about Kabin, one of the cities of the Beklan Empire, and Deelguy, one of the lands bordering the Empire? Kabin echoes English “cabin” and Deelguy echoes English “deal” and “guy”. They don’t look or sound right (though perhaps Deelguy is meant to be pronounced “deel-goo-ee”). But that’s linguistic cleverness, I think. The paradox is that it’s not right if all the words and names of an invented language sound right to the ears of Anglophones. If they all sound right, that is, if they’re all exotic and alien, it means that they’ve been created with English in mind. So they’re a kind of un-English or anti-English, rather than something existing without any regard to English. In Shardik, it’s as though Kabin echoes English by chance, which is just what you might expect of a truly exotic and alien language. So that’s linguistic cleverness, I think.

And it’s also linguistically clever of Adams to invent an accent within the story for native speakers of Deelguy who are talking Beklan or Ortelgan. Here’s the slimy slave-trader Lalloc speaking to the chief villain of the story, the evil slave-trader Genshed: “I was in Kabin, Gensh, when the Ikats come north. Thought I had plonty of time to gotting back to Bekla, but left it too late – you ever know soldiers go so fost, Gensh, you ever know? Cot off, couldn’t gotting to Bekla […] no governor in Kabin – new governor, man called Mollo, been killed in Bekla, they were saying – the king kill him with his own honds – no one would take money to protect me.” (Book VI, ch. 51, “The Gap of Linsho”) The diminutive “Gensh” used by Lalloc is clever too. Genshed is a monster, but Lalloc thinks that the two of them are friends. His accent works as a kind of fantastic realism: yes, when someone from Deelguy spoke Beklan, he would speak in a strange way. And Adams captures that in English.

However, he puts words into the mouth of another character that are clumsy rather than clever: “the resources of this splendid establishment” (used of an inn); “riparian witch-doctor” (used of Kelderek); “bruin-boys [who] burst on an astonished world” (used of the followers of Shardik); “bear-bemused river-boys” (ditto); “some nice, lonely place with no propinquitous walls or boulders”; and so on. Those are the words of Elleroth, Ban of Sarkid, a “dandified” aristocrat who is secretly working against Kelderek and the Ortelgans. He’s an important character, central to the plot, so it’s a pity that, in part, he’s also a cliché out of old-fashioned boys’ literature. He’s a fop who’s also a fighter and whose languid, drawling irony covers serious purpose and emotion. It’s as though an Old Etonian or Harrovian has suddenly appeared. The way he’s presented is out of place in the fantasy universe of Shardik: “propinquitous” would work in one of Clark Ashton Smith’s Hyperborea stories. But it doesn’t work here as dialogue.

Another aspect of Elleroth’s character does work. Before he appears, we’ve seen Shardik through the eyes of his devoted followers, who swear “by the Bear” and see him triumph over all doubt and lead the Ortelgans to victory. After Elleroth appears, we suddenly see Shardik and his cult through the eyes of someone who despises “the bear” and his followers. To Elleroth, the Ortelgans are ursine swine. Later still, the perspective shifts in another way. The final chapters of the book are partly in the form of home-bound letters by an ambassador from Zakalon, a hitherto unknown land where they swear “by the Cat”. What is that about? What cult is practised in Zakalon? We never learn, but the glimpse of something beyond the story increases the power and reality of Shardik’s world.

And Shardik is, despite its frequent clumsiness, a powerful book. Sometimes its power is beautiful, sometimes it’s horrific, and new readers will remember both the beauty and the horror as I did in all the time that has passed since my last readings. Forty years on, I’m glad to have met it again, read it again, and re-acquainted myself with its power and its beauty. It isn’t as good as Watership Down, but it’s better than The Plague Dogs. And not many books are as good as Watership Down.

Elsewhere other-accessible…

Sward and Sorcery – a review of Watership Down (1972)
Paw is Less – a review of The Plague Dogs (1977)

Pascal’s Paradox

« Je n’ai fait celle-ci plus longue que parce que je n’ai pas eu le loisir de la faire plus courte. » — Blaise Pascal, Lettres provinciales (1657)

“I’ve made this [letter] longer only because I haven’t had time to make it shorter.” — Blaise Pascal


Basque is one. Etruscan is another. Sumerian is a third.

What are they? Well, they’re all languages, but they’re more than that. John Donne said that no man is an island, entire of itself. That isn’t true of languages. Basque, Etruscan and Sumerian are all what might be called language-islands, entire of themselves and unrelated to any other language in the world, alive or dead.

I find that a powerful idea in all sorts of ways. Living or dead, language isolates (as linguists call them) are weird and wonderful things. But in some ways they’re at their weirdest and most wonderful when they’re poised between life and death. Many people down the millennia have been the last living speaker of a once widely spoken language. Often that language will still have had far-thrown and flourishing relatives, so its imminent death won’t throw all its treasures of phonology and syntax and lexicon into oblivion.

But sometimes the last living speaker will be of a language isolate. And when the speaker dies, an entire linguistic world will die with them. That kind of tragedy reminds me of one of Clark Ashton Smith’s most memorable and moving stories: the brief but brain-ballistic “Sadastor”. It’s the weird tale of a huge planet “far-fissured with enormous chasms, and covered from pole to pole with the never-ebbing tides of the desert sand.” The planet is called Sadastor and is “without moon or satellite, an abomination and a token of doom to fairer and younger worlds.” I won’t describe what the demon Charnadis discovers on Sadastor, because you should read the story for yourself if you haven’t already.

If you do read it, or you already know it, you’ll understand why it reminds me of the last living speaker of a language isolate. And here is such a speaker and such a language:

Gyani Maiya Sen, a 75-year-old woman from western Nepal, can perhaps be forgiven for feeling that the weight of the world rests on her shoulders. She is the only person still alive in Nepal who fluently speaks the Kusunda language. The unknown origins and mysterious sentence structures of Kusunda have long baffled linguists.

As such, she has become a star attraction for campaigners eager to preserve her dying tongue. Madhav Prasad Pokharel, a professor of linguistics at Nepal’s Tribhuwan University, has spent a decade researching the vanishing Kusunda tribe.

Professor Pokharel describes Kusunda as a “language isolate”, not related to any common language of the world. “There are about 20 language families in the world,” he said, “among them are the Indo-European, Sino-Tibetan and Austro-Asiatic group of languages.

“Kusunda stands out because it is not phonologically, morphologically, syntactically and lexically related to any other languages of the world.” – Nepal’s mystery language on the verge of extinction, BBC News, 13v12

Wikitionary has a word-list for Kusunda and I looked there for something that also reminds me of a language-isolate. What do you call one of the most beautiful, mysterious and solitary animals in the world? In Kusunda, you call it myaqo getse. And what does that mean? Well, in English it means “cat”. But in Kusunda itself it means “leopard-child”, from myaq or myaχ, “leopard”, and getse, “offspring, child.” So I suppose you could also translate myaqo getse as “leopardling”.

Panthera pardus fuscus, the Indian Leopard

And I’ve chosen to try and express the theme of this blog-post in Italian as isolingatto, a portmanteau of isola, “island”, lingua, “language”, and gatto, “cat”. But isolingatto could almost be the past participle of the verb isolingare, meaning “to speak in a language isolate”. That is, isolingatto could mean “spoken in a language isolate” or “spoken from a language-island”.

Reading the Roons

In terms of core issues around maximal engagement with keyly committed core components of the counter-cultural community, one of the saddest, sorriest and sighfullest sights among them is that of the talented lad from the wrong side of the tracks who betrays his class by turning himself into a Guardian-reader, in terms of core cultural assumptions and behaviour.

Northampton’s Alan Moore has done it.

London’s Stewart Home has done it.

Huddersfield’s John Coulthart has done it.

How do I know?

[Readers’ Advisory: If you are easily disturbed, distressed and/or disgusted, please stop reading NOW.]

I know because

[I mean it. Stop reading or you may well regret it.]

I know because each of these talented lads from the wrong side of the tracks now bears the Mark of the Beast, metaphorically speaking.

[Last chance.]

Each of them has, on multiple occasions and without the minimalest micro-metric of shame or irony, deployed the key Guardianista phrase “in terms of”.

• For proof of Alan Moore’s deplorable delinquency, please see here.
• For proof of Stewart Home’s dep-del, please see here.
• For proof of John Coulthart’s dep-del, please see in the same place as you possibly saw or are-about-to-see Stewart Home’s, i.e. here.

So. After seeing and lamenting those horrific examples of class-betrayal, I thought I was hermeneutically hardened and would never again experience sadness, sorrow or sighfullness at the sight of a talented lad etc.

I was wrong.

As I learned when I read this interview in The Mail on Sunday:

There was a lot of negativity in terms of my mum getting frustrated with us as kids, messing around all the time, smashing things in the house and my nan lived in the same road, a few houses down. […] In terms of therapy, I have spoken to a few different people. I have never done a period of time where I have done two years with someone and it has been ongoing. […] Everything I am asking of those players in terms of hard work, honesty, trust, commitment…if I was just to turn round and say “I have had an offer, I’m off”, I honestly couldn’t do that to the players and the staff. — Wayne Rooney reveals his secret two-day drinking binges etc

Oh, Wayne, Wayne, Wayne. How could you do it? But I think we can easily guess where he was infected: it was during his therapy-sessions.

Elsewhere other-accessible

Ex-Term-In-Ate! — interrogating issues around why “in terms of” is so teratographically toxic…
All posts interrogating issues around “in terms of”…
All posts interrogating issues around the Guardian-reading community and its affiliates…