Central Government

A magic square is a square of numbers in which all rows and columns and both diagonals add to the same number, or the magic total. The 3×3 magic square, also known as the Lo Shu square (“scroll of the River Lo” square), uses the numbers 1 to 9 and has a magic total of 15. I haven’t seen it explicitly stated anywhere on the net, perhaps because it’s trivially obvious to proper mathematicians, but in this and other 3×3 magic squares, the magic total must be three times the central number. Here is the proof:

4 9 2
3 5 7
8 1 6
a b c
d e f
g h i

1. a + b + c = a + e + i = b + e + h = c + e + g

2. 3(a + b + c) = (a + e + i) + (b + e + h) + (c + e + g)

3. 3a + 3b + 3c = 3e + a + i + b + h + c + g

4. 2a + 2b + 2c = 3e + g + h + i

5. 2a + 2b + 2c – (g + h + i) = 3e

6. 3e = a + b + c = magic total

Update: In fact, this fact about 3×3 squares is mentioned a lot on the web. See, for example, Negative Magic Squares, which describes a proof discovered by Māori mathematicians in 736 B.C.E.

Some 3×3 magic squares using entirely prime numbers (except for 1 in the first square):

00043 00001 00067
00061 00037 00013
00007 00073 00031 mt = 111 = 37 x 3

00071 00005 00101
00089 00059 00029
00017 00113 00047 mt = 177 = 59 x 3

00083 00029 00101
00089 00071 00053
00041 00113 00059 mt = 213 = 71 x 3

00103 00007 00109
00079 00073 00067
00037 00139 00043 mt = 219 = 73 x 3

00107 00011 00149
00131 00089 00047
00029 00167 00071 mt = 267 = 89 x 3

00139 00007 00163
00127 00103 00079
00043 00199 00067 mt = 309 = 103 x 3

12841 09769 15013
14713 12541 10369
10069 15313 12241 mt = 37623 = 12541 x 3

12721 07753 17167
16993 12547 08101
07927 17341 12373 mt = 37641 = 12547 x 3

13183 08059 16417
15787 12553 09319
08689 17047 11923 mt = 37659 = 12553 x 3

At the Mountains of Mathness

Shakespeare was a gilded ape.

More later.

For now, join me in wondering something I’ve often wondered: What it would be like to experience an asteroid striking the earth. You might be dead before you knew it. You might be woken by the glare and be dead a few seconds later. Slain by the sound of the strike alone. Or the heat alone. There are asteroids that could wipe out every human on earth, or every vertebrate, or every complex form of life. Or you might survive and wish you hadn’t. After some asteroid-strikes, the living would envy the dead.

Continue reading At the Mountains of Mathness

Try Trunkle

One afternoon, when I was about thirteen or fourteen, I was standing in the queue for the children’s counter in a library in Davyhulme in Manchester. I was carrying four or five books and an older girl read out the title of the one that happened to be on top: Uncle and the Treacle Trouble.

Quentin Blake's cover of J.P. Martin's Uncle and the Treacle Trouble

It was meant to make me feel stupid, but it didn’t much. Don’t judge a book by its cover – or its title. Like Uncle and Claudius the Camel or Uncle and the Battle for Badgertown, Uncle and the Treacle Trouble might sound twee and childish: in fact, it’s one of the cleverest, funniest, most surreal children’s books ever written. The six books in J.P. Martin’s Uncle series deserve to join Lewis Carroll’s Alice in Wonderland, Kenneth Grahame’s Wind in the Willows and A.A. Milne’s Winnie the Pooh among the classics, delighting both children and adults all around the world. Alas, they never have and probably now never will. That gives them a cult cachet, of course, but I’d rather have less cult and more cash for the publishers willing to re-issue them.

How to describe them? “Alice in the Willows” is one way: they combine the surreal invention of Alice in Wonderland with the proper stories of Wind in the Willows. Anthropomorphic animals have odd and interesting adventures. Uncle is a Trunkle: a millionaire elephant living in a city-sized castle with lots of other walking, talking animals. And with humans too, like his loyal librarian Will Shudder, his horticulturalist Butterskin Mute,  and the Badfort Crowd, Uncle’s sworn and socialist enemies. Another way of describing the books is to say they might have written by J.G. Bilne or A.A. Mallard: imagine mixing the vivid surrealism of J.G. Ballard with the sun-kissed camaraderie of A.A. Milne. Or the snow-cloaked camaraderie. The Uncle books are set in all the seasons and appeal to all the senses, including the sense of wonder. Martin’s surreal invention is actually oneirography: in part, the books are transcriptions of his dreams.

That’s one reason I would put them above Alice in Wonderland, Wind in the Willows and Winnie the Pooh. I like all of those books, but they don’t create worlds as large, crowded, and endlessly detailed as the dream-world of Uncle, which has no borders and no barriers. Lewis Carroll’s invention can seem contrived or even didactic. He distorts the real world and plays games with it. Grahame and Milne verge on cosiness or never leave it at all, some would say. Martin doesn’t contrive or do cosy, and his hero has feet of clay. “See that pompous humbug Unc/On the platform raise his trunk…” sing the Badfort Crowd, down but not out after Uncle has thwarted another of their schemes, and they’re right: Uncle, though good and kind-hearted, is pompous. If you wanted to get Nietzschean about it, you could say Uncle represents the calm and ordered world of the Apollonian and the Badfort Gang the chaotic and destructive forces of the Dionysian. But that would probably be taking things too far. Uncle is nearer whimsy than wildness. I would like to know more about J.P. Martin’s education, but if he was influenced by Greek and Roman mythology, he adapted it for a cooler climate and more muted skies. Apollo and Dionysus may do battle in his books, but they do so in the hall of Hypnos, god of sleep. Reading J.P. Martin is like dreaming awake.

But you’ll have sweet dreams, not sour ones: if Martin ever had angst-ridden or ugly dreams, he didn’t transcribe them into his books. He can also, like the late, great Peter Simple, invoke the “mysterious urban poetry” of slag-heaps and abandoned factories. Perhaps Simple, born in 1913, was a fan of Martin, born in 1879, or perhaps they both drew on the same gentle, subtle English traditions of nonsense, whimsy, and satire. Either way, I would place Martin above even Simple. Both are dead now, but the written word, more lasting than bronze, allows their souls to sing on.

A Swirl of Shadows

I tell of the planet of Gdarrujh, far, very far, in space and time. There is a spot there, by a vanished sea, whereon, at certain times of the year, a swirl of shadows will appear on a broken floor of ancient marble. And these shadows are very strange, seeming those of living creatures, yet with likeness to none ever known on that world. Travellers who return from the spot compare the shadows to those of autumn leaves able to hover and flutter at will between sun and earth, shading the latter with their shape. Yet these shadows have heads and limbs, after a fashion, and certainly pertain to the animal, not the vegetable, and even to the human, as though the arms of Gdarrujh-folk were broad-bladed oars and men could row the very insubstantiality of air, ride there at will, though they remained solid flesh and sturdy bone.

And this shadow-swirl has nourished several schools of speculation on Gdarrujh. Some philosophoi say it is nothing real, being a mirage or trick of some long-dead, high-cunning’d mage, whose magick works on down the millennia, tempting the foolish to belief in nonsense and chimæras. But others say, nay, nay, the shadows are those of real creatures, past or to come, and are shown by design of the gods, that men might throw down their walls of dogma, topple their towers of certitude, and know that the Universe holds more than man sees or woman dreams. And a smaller school of the realitarians holds that the shadows are not of the past or future, but rather of the present, being cast somehow through a chink that separates known Gdarrujh from another Gdarrujh that exists in hidden parallel, where creatures dwell not only the land and sea, but also the air, being able to take to it, delight in it, partake there of the lightness of leaves and the grace of fish.

And the creatures of the shadow-swirl have mouths like knives, which gape as though they call, but, in repudiation of whose claims, in confirmation of whose, in relevation of what mysteries, no sound reaches the speculators of Gdarrujh.


In terms of keyly core components of Guardianese, the dialect of those who read and write for The Guardian, Britain’s premier papyrocentric purveyor of progressive performativity, there can be little or no doubt that at the key core is the phrase I began this sentence with: “in terms of”. Arguably it is the keyliest corest component of all. It’s a bad sign if you use it even a little; if you use it a lot, it’s time to mend your ways. Siriusly. But whatever your own issues in terms of usage metrics for I.T.O., you’ll certainly hear this phrase a lot throughout the English-speaking world. In terms of communities / demographics like politics, academia and the media, it’s a kind of linguistic bindweed: a tough, fast-growing weed smothering everything in sight.

Unlike bindweed, however, it doesn’t produce beautiful flowers or grow in interesting ways. What’s wrong with “in terms of” was summed up very well by the Australian comedian and satirist Barry Humphries, the creator of Dame Edna Everage and Sir Les Patterson. He re-wrote the title of a famous film as One Flew Over In Terms of the Cuckoo’s Nest. “In terms of” is beloved of those who want long ways to say short things. Its use is usually unnecessary, never essential. As a keyly committed component of the core I.T.O.phobic community, I never use it except to take the piss of the Guardianista demographic. The mission statement of the Guardian might be “Purveying pretentious prose to pretentious people since 1959.” “In terms of” is corely key to this mission. The lexicographer Robert Burchfield discussed its origins in The New Fowler’s Modern English Usage (1996):

How did this complex preposition come into being? The OED [Oxford English Dictionary] reveals that it has been in use since the mid-18c. as a mathematical expression “said of a series… stated in terms involving some particular (my emphasis) quantity”, and illustrates this technical usage by citing examples from the work of Herbert Spencer (1862), J. F. W. Herschel (1866), and other writers. From this technical use came at first a trickle and, after the 1940s, a flood of imitative uses by non-mathematicians. (Op. cit., entry for “in terms of”).

I suggest that the flood of imitative uses was flattery. Mathematicians are highly intelligent and intellectually rigorous people. Non-mathematicans wanted to pretend to themselves that they were highly intelligent and intellectually rigorous too. “In terms of” lends a judicious, thoughtful air to one’s prose or speech. It’s a good way of disguising the absence of judgment or thought. This is one reason it’s so popular among politicians, who need ways to sound impressive and say little. Burchfield condemns its use as a “vague all-purpose connective” in politics and broadcasting, but concludes, after listing examples of I.T.O. in action, that it may be a “useful particularizing device” in general prose. He’s wrong. All his examples can be re-written to be better English:

The impact of Ibsen… did much to revitalize the degenerate English theatre and force it to think in terms of living ideas and contemporary realities. J. Mulgan and D. M. Davin, 1947. (My suggestion: …force it to use living ideas…)

Dataquest pegs ESRI as the leading GIS company—in terms of both revenue and reputation. Computer Graphics World, 1988. (Dataquest lists ESRI as… in both revenue and reputation)

He deals with the converso judaizing world in terms of its social and religious rituals, births, marriages, deaths, leading to the establishment of the Inquisition. The Bulletin of Hispanic Studies, 1990. (…through its social and religious rituals…)

Rameau… conceived his music precisely in terms of timbres, types of attack, degree of sostenuto. Country Life, 1990. (…in timbre, type of attack…)

Justifying space in terms of material wealth is as ridiculous as saying that man went to the Moon merely to be able to return with velcro zips and non-stick frying pans. New Scientist, 1991. (…space [exploration] by its material benefits…)

The dating of his novels in terms of when they were written rather than when they were published is often uncertain, since in the upheavals of exile some were not published chronologically. New York Review of Books, 1991. (…his novels by when they were written…)

The re-writing makes them better English, but not necessarily good English. Writers who use “in terms of” are generally bad writers. That’s why I’m unsurprised to see The New York Review of Books in the list. Like its twin on this side of the Atlantic, The London Review of Books, its mission statement might be “Purveying Pathological Prose to Pathological People.” A core component of this pathology is “in terms of”. My reaction to I.T.O. is I.T.T.O.! In other words: It’s Time To Obliterate In Terms Of. This lexical bindweed doesn’t flower: pluck it out wherever you find it in your linguistic garden. I’ve allowed other weeds to spring up here and there in terms of issues around the prose of this polemic, but I do my best to keep my bad English deliberate.

Guardianistas and their equivalents overseas produce bad English the way cows produce methane: copiously and unconsciously. And the internet has allowed their bad English to billow forth as never before. Wikipedia, for example, is like an experimental farm on which they can fart all day and every day, polluting the English language in vibrant new ways. “In terms of” is keyly core to their methanogenic mission. I groan when I see it in Wikipedia articles about people like, say, Saki or Clark Ashton Smith. I grin when I see it in articles about people like, say, William S. Burroughs or Alan T. Moore. Some people deserve bad prose. Some people don’t. I hope you and your favourite writers are among the latter. Siriusly. “In terms of” sucks! “Sucks” sucks too! Just say no to I.T.O.!

Proviously post-posted (please peruse):

Palace in Numberland

The Palace of Primes

“Cur ad uvas per Zeusim depictas accursabant volucres?” – Iordani Bruni Ars Memoriæ (1582).

“Why did birds flock to the grapes painted by Zeuxis?” – Giordano Bruno’s Art of Memory.

“To mnemonicize the primes is to partake of the mind of God, as though one dipped a shell into plumbless nectar and drank thereof.” So runs the saying in the Cult of Primes, wherein prodigious feats of memory are demanded even of the neophytes, who must enthrone in a memory-palace the initial 931 primes (from 2 to 7,307) ere they can begin climbing, rung by jaden rung, the ladder of the Hierarchy. The Cult has a number-system based on thirty: which is to say, where we, with a base of ten, have nine number-symbols and the cipher, they have twenty-nine ditto and ditto. To each symbol, in their mnemonics, they assign a beast, bird, and flower; a metal, gem, and wood; a fur, cloth, and silk; a food, drink, and condiment; a colour, scent, and sound. Thus, a hummingbird hovering above an emerald amid scent of vanilla symbolizes the prime 1,667; an eye upon a silver harp the prime 5,059; and a porphyry scarab upon a cheetah’s fur the prime 11,173.

When once the neophytes have mastered the system of mnemonics, each sets to constructing his Palatium Primorum, his Palace of Primes. Herein, each Prime has Its Room, wherein It sits on a Throne studded with symbols of its attributes, whilst courtiers feast and musicians play before It. And in the Left Wall of the Room are many doors symbolizing numbers from 1 to 31. If one is in the Throne-Room of 137, for example, and one steps through the Door of 13, one finds oneself in the Throne-Room of 199, the Prime 13 places higher in the List of Primes; and similarly, mutatis mutandis, for other numbers and other doors. And in the Right Wall are an increasing number of doors leading to Primes lower in the list. Thus, the Throne-Room of 2 has no doors on the right; and of 3 has one door; and of 5 two doors; and so till 131, the 32nd prime, whereat the Right Doors reach their maximum. And each priest of the Cult, from his neophycy on, will toil a lifetime bedecking, manning, and extending the Palace, till it seems to him more real than the World, vaster than the Universe, and dearer to him than his heart-beat.

Nor, if earthly misfortune overtakes him, does the Palace fail of consolation, for a priest can resort thither for surcease of pain, if upon the rack; for oblivion of want, if destitute or starving; and for foretaste of Paradise, if dying. Yea, Paradise is a Palace, in the teaching of the Cult, a Palatium Omnium Primorum, of All Primes, primes numberless as beats of a deathless heart, as sands of an endless shore, as stars of a boundless heaven. And Herein the Doors of the Left Wall are infinite and the Doors of the Right increase for ever.

Lulu Lunatic Luz

It’s disturbing what you can find online…

Tales of Silence & Sortilege, Simon Whitechapel, Paperback, 111 Pages

May 28, 2012

If you love weird fantasy, if you love the English language, even if you don’t love Clark Ashton Smith, you should read this book. The back cover describes it as “the darkest and most disturbing fantasy” of this millennium, but that’s either sarcastic or tragically optimistic, because what these stories really are is beautiful. The breath of snow-wolves is described as “harsh-spiced.” A mysterious gargoyle leaning from the heights of a great cathedral has “wings still glistening with the rime of interplanetary flight.” Hummingbirds are “gem-feathered… their glittering breasts dusted with the gold of a hundred pollens.” If you cannot appreciate such imagery, then perhaps you are dead to beauty, or simply dead. These tales are very short, but some of them have stayed with me for years, such as “The Treasure of the Temple,” in which a thief seems to lose the greatest fortune he could ever have found by stealing a king’s ransom in actual treasure. Most of the stories are brilliant, one or two is only good, but the masterpieces are “Master of the Pyramid” and “The Return of the Cryomancer.” The sense of loss and mystery evoked by these two companion stories is almost physically painful, it is so haunting. There is nothing like these stories being published today. Reading them, I feel the excitement and wonder that fans of Weird Tales magazine must have known long ago when new stories would appear by H.P. Lovecraft, Clark Ashton Smith, and Robert E. Howard. Simon Whitechapel doesn’t imitate these authors so much as apply their greatest lessons to new forms of fantasy. This is one of the cheapest books I own, but I accord it one of my most valuable. It is easily the best work of art you will find in any form on Lulu. I cannot recommend it highly enough.

The Roses of Hsūlag-Beiolă, Simon Whitechapel, Paperback, 154 Pages

Jun 8, 2012

This collection of weird fantasy is filled with mystery, wonder and a sense of the ineffable. Not every story is a mind-blowing masterpiece, but the best of them are absolutely spectacular. Even the worst are good and all are haunting in one way or another. My two favorites were: 1. “The Mercy of the Osmomancer,” wherein a knight on a mission to investigate the tower of a scent-wizard encounters demons made of smells and even learns the language of odors… 2. “The Swans,” in which a pawnbroker tracks down all the known paintings of a seemingly insane artist who paints his canvases entirely black, nothing but black, for reasons best and most poetically left to Simon Whitechapel to explain… Any fan of H.P. Lovecraft, Robert E. Howard, Clark Ashton Smith, Edgar Allan Poe, Comte de Lautréamont, Charles Baudelaire or William S. Burroughs will find something wonderful to love in here. I sure did.

Even more disturbing is the thought that this individual may be able to pass themself off as normal in real life: there are no spelling mistakes or solecisms. (Then again, perhaps I’m reviewing my own books in my sleep. (But I wouldn’t compare myself to B*rr**ghs, surely? (Unless it’s a bluff or double-bluff. (Disturbing, as I said. (I agree.)))))

Stoch’! (In the Name of Dove)

Stochasma, In Abysso (2012)

The Sueco-Georgian avant-gardists Stochasma were formed, in their own words, “to interrogate, eviscerate, and exterminate the ultimate experimental envelope of acoustic idiosyncrasy”. That’s “Sueco-” as in Sweden and “Georgian” as in the Eurasian nation, not the American state, by the way. Going one up on some bands from Wales, Ireland and Scotland, who issue their material bilingually, in English and one or another of the Celtic languages, Stochasma issue all their material tri-lingually, in English, Swedish, and Georgian. The strangeness and beauty of the Georgian script match and enhance the strangeness and (occasional) beauty of their music, but, unlike their last two releases, there’s no spoken English, Swedish or Georgian here: In Abysso is intended to be an “abhuman listen”.

Front cover of Stochasma's album In Abysso 

Believe me, it is! The title of the album is Latin for “In the Abyss” and the liner-notes extend thanks to H.P. Lovecraft, Clark Ashton Smith and Stanislaw Ulam for “infernal inspiration”. If the last name makes you think “Who?” (or “U?”), you must be new/nu to Stochasma, who draw inspiration not just from art and literature, but from mathematics too. Stanislaw Ulam (1909-84) was a Polish mathematician perhaps most famous for inventing the “Ulam spiral”, a graphical representation of the prime numbers that reveals mysterious patterns in this strange and fascinating set of integers. Ulam stumbled across the spiral while “doodling” during a boring lecture at a scientific meeting. That kind of serendipity has always been important to Stochasma, who explore the musical abyss/chasm partly through random, or stochastic, techniques. For the first track, “Pr1m4l Skr33m”, the five members of the band had electrodes attached to their nipples before being asked, at random, to indicate, with a nod or shake of the head, whether a randomly selected number between 1 and 10,000 was prime or composite (for example, 1,433 is prime, being divisible by no numbers but itself and 1; 1,434 is composite, being divisible by 2, 3 and 239). If they were wrong, they received a painful electric shock.

The resultant collection of grunts, gasps, and screams was electronically worked over in fully traditional Stochasma fashion to create “Pr1m4l Skr33m”, which sounds like a fully traditional Stochasma track: fucking weird and unsettling! Is the irregular chorus of voices in agony or ecstasy? Are the band being tortured in a hell run by sadists or pleasured in a heaven run for masochists? Or both? It’s hard to decide, and at times hard to listen, but as Stochasma themselves put it: “We’re queasy listening, not easy – easy listening is for cubes.”

Elsewhere, the band have used the ultra-sensitive microphones they first experimented with on 2003’s AnguisaquA (sic – it literally means “SnakewateR”). This time they’ve recorded the bloodflow of a dove and the movements of parasites in its feathers for “Täubchen”, which sounds even stranger than it reads. That and “Pr1m4l Skr33m” are the first two tracks: the next fifteen are entitled “Ignisigil I” to “Ignisigil XV”. Stochasma used a fire-proof microphone to record the sound of books being burned. They selected fifteen wildly different authors for this literally incendiary homage, from “J. Aldapuerta to J. Archer, from K. Marx to K. Minogue”, as they themselves put it. (That’s the über-trangressive Spanish horror-writer Jesús Aldapuerta and the über-cruddy British thriller-writer Jeffrey Archer, and the Anglo-German philosophaster Karl Marx and the Australian pop-pixie Kylie Minogue, for those unfamiliar with the names.) And the band insist, perhaps tongue-in-cheek, that the sonic textures of the recordings are dependent not just on the physical nature of the paper and ink being burnt, but also on the ideological and aesthetic nature of the burning text.

It’s hard to agree: the “Ignisigils” all sound pretty much alike to me, though that sound is uncharacteristically soothing and relaxing by Stochasma standards (on my first listen, I dropped off during “Ignisigil VIII” and didn’t wake up till “Ignisigil XI”). The album is rounded off with three of the strangest pieces of music I’ve heard this century: “Musgomorrah”, “Gradus ad Parnassum”, and “CoMoXoCoI”. The first sounds like a slowed recording of men in armour fighting in thick mud; the second like a choir of giant glass insects singing themselves to splinters; and the third like echoes chasing each another in a collapsing or burning maze. These three might grow on me or might not: for now, “Pr1m4l Skr33m”, “Täubchen”, and “Ignisigil IV” hit the sonic sweet’n’sour spot that Stochasma seem to have copyrighted. I don’t know why “IV” hits the spot and the rest of the Ignisigils don’t, but that’s often the way with Stochasma: you like the sounds they create and you haven’t a clue as to why. In company with a select band of other electronicognoscenti, I look forward to their seventh album, whenever it appears and whatever musical mélanges or macedoines it manages to mulch, mangle, and miscegenate.

Elsewhere other-engageable:

Musings on Music