“The IAE [International Art English] of the French press release is almost too perfect: It is written, we can only imagine, by French interns imitating American interns imitating American academics imitating French academics.” — “International Art English”, Alix Rule and David Levine
Category Archives: Art
Clock around the Rock
If you like minimalism, you should like binary. There is unsurpassable simplicity and elegance in the idea that any number can be reduced to a series of 1’s and 0’s. It’s unsurpassable because you can’t get any simpler: unless you use finger-counting, two symbols are the minimum possible. But with those two – a stark 1 and 0, true and false, yin and yang, sun and moon, black and white – you can conquer any number you please. 2 = 10[2]. 5 = 101. 100 = 1100100. 666 = 1010011010. 2013 = 11111011101. 9^9 = 387420489 = 10111000101111001000101001001. You can also perform any mathematics you please, from counting sheep to modelling the evolution of the universe.
1 + 0 = ∞
But one disadvantage of binary, from the human point of view, is that numbers get long quickly: every doubling in size adds an extra digit. You can overcome that disadvantage using octal or hexadecimal, which compress blocks of binary into single digits, but those number systems need more symbols: eight and sixteen, as their names suggest. There’s an elegance there too, but binary goes masked, hiding its minimalist appeal beneath apparent complexity. It doesn’t need to wear a mask for computers, but human beings can appreciate bare binary too, even with our weak memories and easily tiring nervous systems. I especially like minimalist binary when it’s put to work on those most maximalist of numbers: the primes. You can compare integers, or whole numbers, to minerals. Some are like mica or shale, breaking readily into smaller parts, but primes are like granite or some other ultra-hard, resistant rock. In other words, some integers are easy to divide by other integers and some, like the primes, are not. Compare 256 with 257. 256 = 2^8, so it’s divisible by 128, 64, 32, 16, 8, 4, 2 and 1. 257 is a prime, so it’s divisible by nothing but itself and 1. Powers of two are easy to calculate and, in binary, very easy to represent:
2^0 = 1 = 1
2^1 = 2 = 10[2]
2^2 = 4 = 100
2^3 = 8 = 1000
2^4 = 16 = 10000
2^5 = 32 = 100000
2^6 = 64 = 1000000
2^7 = 128 = 10000000
2^8 = 256 = 100000000
Primes are the opposite: hard to calculate and usually hard to represent, whatever the base:
02 = 000010[2]
03 = 000011
05 = 000101
07 = 000111
11 = 001011
13 = 001101
17 = 010001
19 = 010011
23 = 010111
29 = 011101
31 = 011111
37 = 100101
41 = 101001
43 = 101011
Maximalist numbers, minimalist base: it’s a potent combination. But “brimes”, or binary primes, nearly all have one thing in common. Apart from 2, a special case, each brime must begin and end with 1. For the digits in-between, the God of Mathematics seems to be tossing a coin, putting 1 for heads, 0 for tails. But sometimes the coin will come up all heads or all tails: 127 = 1111111[2] and 257 = 100000001, for example. Brimes like that have a stark simplicity amid the jumble of 83 = 1010011[2], 113 = 1110001, 239 = 11101111, 251 = 11111011, 277 = 100010101, and so on. Brimes like 127 and 257 are also palindromes, or the same reading in both directions. But less simple brimes can be palindromes too:
73 = 1001001
107 = 1101011
313 = 100111001
443 = 110111011
1193 = 10010101001
1453 = 10110101101
1571 = 11000100011
1619 = 11001010011
1787 = 11011111011
1831 = 11100100111
1879 = 11101010111
But, whether they’re palindromes or not, all brimes except 2 begin and end with 1, so they can be represented as rings, like this:
Those twelve bits, or binary digits, actually represent the thirteen bits of 5227 = 1,010,001,101,011. Start at twelve o’clock (digit 1 of the prime) and count clockwise, adding 1’s and 0’s till you reach 12 o’clock again and add the final 1. Then you’ve clocked around the rock and created the granite of 5227, which can’t be divided by any integers but itself and 1. Another way to see the brime-ring is as an Ouroboros (pronounced “or-ROB-or-us”), a serpent or dragon biting its own tail, like this:
Alchemical Ouroboros (1478)
Another alchemical Ouroboros (1599)
But you don’t have to start clocking around the rock at midday or midnight. Take the Ouroboprime of 5227 and start at eleven o’clock (digit 12 of the prime), adding 1’s and 0’s as you move clockwise. When you’ve clocked around the rock, you’ll have created the granite of 6709, another prime:
Other Ouroboprimes produce brimes both clockwise and anti-clockwise, like 47 = 101,111.
Clockwise
101,111 = 47
111,011 = 59
111,101 = 61
Anti-Clockwise
111,101 = 61
111,011 = 59
101,111 = 47
If you demand the clock-rocked brime produce distinct primes, you sometimes get more in one direction than the other. Here is 151 = 10,010,111:
Clockwise
10,010,111 = 151
11,100,101 = 229
Anti-Clockwise
11,101,001 = 233
11,010,011 = 211
10,100,111 = 167
10,011,101 = 157
The most productive brime I’ve discovered so far is 2,326,439 = 1,000,110,111,111,110,100,111[2], which produces fifteen distinct primes:
Clockwise (7 brimes)
1,000,110,111,111,110,100,111 = 2326439
1,100,011,011,111,111,010,011 = 3260371
1,110,100,111,000,110,111,111 = 3830207
1,111,101,001,110,001,101,111 = 4103279
1,111,110,100,111,000,110,111 = 4148791
1,111,111,010,011,100,011,011 = 4171547
1,101,111,111,101,001,110,001 = 3668593
Anti-Clockwise (8 brimes)
1,110,010,111,111,110,110,001 = 3768241
1,100,101,111,111,101,100,011 = 3342179
1,111,111,011,000,111,001,011 = 4174283
1,111,110,110,001,110,010,111 = 4154263
1,111,101,100,011,100,101,111 = 4114223
1,111,011,000,111,001,011,111 = 4034143
1,110,110,001,110,010,111,111 = 3873983
1,000,111,001,011,111,111,011 = 2332667
Appendix: Deciminimalist Primes
Some primes in base ten use only the two most basic symbols too. That is, primes like 11[10], 101[10], 10111[10] and 1011001[10] are composed of only 1’s and 0’s. Furthermore, when these numbers are read as binary instead, they are still prime: 11[2] = 3, 101[2] = 5, 10111[2] = 23 and 1011001[2] = 89. Here is an incomplete list of these deciminimalist primes:
11[10] = 1,011[2]; 11[2] = 3[10] is also prime.
101[10] = 1,100,101[2]; 101[2] = 5[10] is also prime.
10,111[10] = 10,011,101,111,111[2]; 10,111[2] = 23[10] is also prime.
101,111[10] = 11,000,101,011,110,111[2]; 101,111[2] = 47[10] is also prime.
1,011,001[10] = 11,110,110,110,100,111,001[2]; 1,011,001[2] = 89[10] is also prime.
1,100,101[10] = 100,001,100,100,101,000,101[2]; 1,100,101[2] = 101[10] is also prime.
10,010,101[10] = 100,110,001,011,110,111,110,101[2]; 10,010,101[2] = 149[10] is also prime.
10,011,101[10] = 100,110,001,100,000,111,011,101[2]; 10,011,101[2] = 157[10] is also prime.
10,100,011[10] = 100,110,100,001,110,100,101,011[2]; 10,100,011[2] = 163[10] is also prime.
10,101,101[10] = 100,110,100,010,000,101,101,101[2]; 10,101,101[2] = 173[10] is also prime.
10,110,011[10] = 100,110,100,100,010,000,111,011[2]; 10,110,011[2] = 179[10] is also prime.
10,111,001[10] = 100,110,100,100,100,000,011,001[2].
11,000,111[10] = 101,001,111,101,100,100,101,111[2]; 11,000,111[2] = 199[10] is also prime.
11,100,101[10] = 101,010,010,101,111,111,000,101[2]; 11,100,101[2] = 229[10] is also prime.
11,110,111[10] = 101,010,011,000,011,011,011,111[2].
11,111,101[10] = 101,010,011,000,101,010,111,101[2].
100,011,001[10] = 101,111,101,100,000,101,111,111,001[2]; 100,011,001[2] = 281[10] is also prime.
100,100,111[10] = 101,111,101,110,110,100,000,001,111[2].
100,111,001[10] = 101,111,101,111,001,001,010,011,001[2]; 100,111,001[2] = 313[10] is also prime.
101,001,001[10] = 110,000,001,010,010,011,100,101,001[2].
101,001,011[10] = 110,000,001,010,010,011,100,110,011[2]; 101,001,011[2] = 331[10] is also prime.
101,001,101[10] = 110,000,001,010,010,011,110,001,101[2].
101,100,011[10] = 110,000,001,101,010,100,111,101,011[2].
101,101,001[10] = 110,000,001,101,010,110,111,001,001[2].
101,101,111[10] = 110,000,001,101,010,111,000,110,111[2]; 101,101,111[2] = 367[10] is also prime.
101,110,111[10] = 110,000,001,101,101,000,101,011,111[2].
101,111,011[10] = 110,000,001,101,101,010,011,100,011[2]; 101,111,011[2] = 379[10] is also prime.
101,111,111[10] = 110,000,001,101,101,010,101,000,111[2]; 101,111,111[2] = 383[10] is also prime.
110,010,101[10] = 110,100,011,101,001,111,011,110,101[2].
110,100,101[10] = 110,100,011,111,111,111,010,000,101[2]; 110,100,101[2] = 421[10] is also prime.
110,101,001[10] = 110,100,100,000,000,001,000,001,001[2].
110,110,001[10] = 110,100,100,000,010,010,100,110,001[2]; 110,110,001[2] = 433[10] is also prime.
110,111,011[10] = 110,100,100,000,010,100,100,100,011[2]; 110,111,011[2] = 443[10] is also prime.
Cat out of Bel
The Belgian symbolist Fernand Khnopff (1858-1921) is one of my favourite artists; Caresses (1896) is one of his most famous paintings. I like it a lot, though I find it more interesting than attractive. It’s a good example of Khnopff’s art in that the symbols are detached from clear meaning and float mysteriously in a world of their own. As Khnopff used to say: On n’a que soi – “One has only oneself.” But he was clearly inspired by the story of Oedipus and the Sphinx, which is thousands of years old. Indeed, an alternate title for the painting is The Sphinx.
Caresses (1896) by Fernand Khnopff (click for larger image)
Even older than the Oedipus story is another link to the incestuous themes constantly explored by Khnopff, who was obsessed with his sister Marguerite and portrayed her again and again in his art. That’s her heavy-jawed face rubbing against the heavy-jawed face of the oddly nippled man, but Khnopff has given her the body of a large spotted felid. Many people misidentify it as a leopard, Panthera pardus. It’s actually a stranger and rarer felid: a cheetah, Acinonyx jubatus, which occupies a genus of its own among the great cats. And A. jubatus, unlike P. pardus, is an incestuous animal par excellence:
Cheetahs are very inbred. They are so inbred that genetically they are almost identical. The current theory is that they became inbred when a “natural” disaster dropped their total world population down to less than seven individual cheetahs – probably about 10,000 years ago. They went through a “Genetic Bottleneck”, and their genetic diversity plummeted. They survived only through brother-to-sister or parent-to-child mating. (Cheetah Extinction)
It must have been a large disaster. Perhaps cheetahs barely survived the inferno of a strike by a giant meteor, which would make them a cat out of hell. In 1896, they became a cat out of Bel too when Khnopff unveiled Caresses. Back then, biologists could not analyse DNA and discover the ancient history of a species like that. So how did Khnopff know the cheetah would add extra symbolism to his painting? Presumably he didn’t, though he must have recognized the cheetah as unique in other ways. All the same, I like to think that perhaps he had extra-rational access to scientific knowledge from the future. As he dove into the subconscious, Khnopff used symbols like weights to drag himself and his art deeper and darker. So perhaps far down, in the mysterious black, where time and space lose their meaning, he encountered a current of telepathy bearing the news of the cheetah’s incestuous nature. And that’s why he chose to give his sphinx-sister a cheetah’s body.
Numbered Days
Numbered Days: Literature, Mathematics and the Deus Ex Machina
Think French. Think genius. Think rebellious, tormented, iconoclastic. Finally, think dead tragically young in the nineteenth century… And if you’re thinking of anyone at all, I think you’ll be thinking of Rimbaud.
And you’d be right to do so. But only half-right. Because there were in fact two rebellious, tormented, iconoclastic French geniuses who died tragically young in the nineteenth century. One was called Arthur Rimbaud (1854-91) and the other Évariste Galois (1811-32). Rimbaud is still famous, Galois never has been. At least not to the general educated public, though on all objective criteria – but one – you might expect his fame to be greater. In every way – but one – Galois has the more powerful appeal.
Continue reading Numbered Days…
Yew and Me
The Pocket Guide to The Trees of Britain and Northern Europe, Alan Mitchell, illustrated by David More (1990)
Leafing through this book after I first bought it, I suddenly grabbed at it, because I thought one of the illustrations was real and that a leaf was about to slide off the page and drop to the floor. It was an easy mistake to make, because David More is a good artist. That isn’t surprising: good artists are often attracted to trees. I think it’s a mathemattraction. Trees are one of the clearest and commonest examples of natural fractals, or shapes that mirror themselves on smaller and smaller scales. In trees, trunks divide into branches into branchlets into twigs into twiglets, where the leaves, well distributed in space, wait to eat the sun.
When deciduous, or leaf-dropping, trees go hungry during the winter, this fractal structure is laid bare. And when you look at a bare tree, you’re looking at yourself, because humans are fractals too. Our torsos sprout arms sprout hands sprout fingers. Our veins become veinlets become capillaries. Ditto our lungs and nervous systems. We start big and get small, mirroring ourselves on smaller and smaller scales. Fractals make maximum and most efficient use of space and what’s found in me or thee is also found in a tree, both above and below ground. The roots of a tree are also fractals. But one big difference between trees and people is that trees are much freer to vary their general shape. Trees aren’t mirror-symmetrical like animals and that’s another thing that attracts human eyes and human artists. Each tree is unique, shaped by the chance of its seeding and setting, though each species has its characteristic silhouette. David More occasionally shows that bare winter silhouette, but usually draws the trees in full leaf, disposed to eat the sun. Trees can also be identified by their leaves alone and leaves too are fractals. The veins of a leaf divide and sub-divide, carrying the raw materials and the finished products of photosynthesis to and from the trunk and roots. Trees are giants that work on a microscopic scale, manufacturing themselves from photons and molecules of water and carbon dioxide.
We eat or sculpt what they manufacture, as Alan Mitchell describes in the text of this book:
The name “Walnut” comes from the Anglo-Saxon for “foreign nut” and was in use before the Norman Conquest, probably dating from Roman times. It may refer to the fruit rather than the tree but the Common Walnut, Juglans regia, has been grown in Britain for a very long time. The Romans associated their god Jupiter (Jove) with this tree, hence the Latin name juglans, “Jove’s acorn (glans) or nut”… The wood [of Black Walnut, Juglans nigra] is like that of Common Walnut and both are unsurpassed for use as gunstocks because, once seasoned and worked, neither moves at all and they withstand shock particularly well. They are also valued in furniture for their good colour and their ability to take a high polish. (entry for “Walnuts”, pg. 18)
That’s from the first and longer section, devoted to “Broadleaved Trees and Palms”; in the second section, “Conifers”, devoted to pines and their relatives, maths appears in a new form. Pine-cones embody the Fibonacci sequence, one of the most famous of all number sequences or series. Start with 1 and 1, then add the pair and go on adding pairs: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… That’s the Fibonacci sequence, named after the Italian mathematician Leonardo Fibonacci (c.1170-c.1245). And if you examine the two spirals created by the scales of a pine-cone, clockwise and counter-clockwise, you’ll find that there are, say, five spirals in one direction and eight in another, or eight and thirteen. The scales of a pineapple and petals of many flowers behave in a similar way. These patterns aren’t fractals like branches and leaves, but they’re also about distributing living matter efficiently through space. Mitchell doesn’t discuss any of this mathematics, but it’s there implicitly in the illustrations and underlies his text. Even the toxicity of the yew is ultimately mathematical, because the effect of toxins is determined by their chemical shape and its interaction with the chemicals in our bodies. Micro-geometry can be noxious. Or nourishing:
The Yews are a group of conifers, much more primitive than those which bear cones. Each berry-like fruit has a single large seed, partially enclosed in a succulent red aril which grows up around it. The seed is, like the foliage, very poisonous to people and many animals, but deer and rabbits eat the leaves without harm. Yew has extremely strong and durable wood [and the] Common Yew, Taxus baccata, is nearly immortal, resistant to almost every pest and disease of importance, and immune to stress from exposure, drought and cold. It is by a long way the longest-living tree we have and many in country churchyards are certainly much older than the churches, often thousands of years old. Since the yews pre-date the churches, the sites may have been holy sites and the yews sacred trees, possibly symbols of immortality, under which the Elders met. (entry for “Yews”, pg. 92)
This isn’t a big book, but there’s a lot to look at and read. I’d like a doubtful etymology to be true: some say “book” is related to “beech”, because beech-bark or beech-leaves were used for writing on. Bark is another way of identifying a tree and another aspect of their dendro-mathematics, in its texture, colours and patterns. But trees can please the ear as well as the eye: the dendrophile A.E. Housman (1859-1936) recorded how “…overhead the aspen heaves / Its rainy-sounding silver leaves” (A Shropshire Lad, XXVI). There’s maths there too. An Aspen sounds like rain in part because its many leaves, which tremble even in the lightest breeze, are acting like many rain-drops. That trembling is reflected in the tree’s scientific name: Populus tremula, “trembling poplar”. Housman, a Latin professor as well as an English poet, could have explained how tree-nouns in Latin are masculine in form: Alnus, Pinus, Ulmus; but feminine in gender: A. glutinosa, P. contorta, U. glabra (Common Alder, Lodgepole Pine, Wych-Elm). He also sums up why trees please in these simple and ancient words of English:
Give me a land of boughs in leaf,
A land of trees that stand;
Where trees are fallen, there is grief;
I love no leafless land.
Bones, Blinks, Books
In Ictu Oculi (In the Blink of an Eye) by Juan de Valdés Leal (c. 1671).
Angst, Anguish, Abjection
It’s half tradition, half tic. At every Ruin-Dredger gig, the lead-singer Jerome Daziel asks the same simple question. Sometimes he shouts it and demands a reaction from the audience. Sometimes he whispers it and ignores what the audience does. Depending on the country, he’s asked it in French, Italian, Greek, Russian, Georgian, Mandarin, Thai, Samoan and Quechua. He’s also asked it in complete silence, having written it across his chest and on the palms of his hands in phosph-ink, invisible when the lights are on, glowing ghoulishly when they’re turned off. Occasionally he’s asked it backwards. In English, the question runs like this: “And What Doth It Mean To Be Flesh?”
But you could see the whole of a Ruin-Dredger gig as asking the same searching thing. The band specialize in unusual frequencies that hunt out – and hum out – the resonances of the human body: the lungs, the bones, the blood. And their music sets up strange resonances in the mind. It’s both mindless and masterful, at once tearful and tyrannous. Sometimes it sounds like mathematics trying to come to life, and sometimes like mathematics trying to commit suicide. There’s a lot of science in their music, and a lot of silence too. “Star-clusters having tantrums,” is how one early review ran. “With occasional episodes of narcolepsy.” That mixture of sound and silence is mutually reinforcing: the sounds are sterner, the silence is sharper. They began their career with the albums Xoli-Hein (1992) and Pyramidion (1996), where they forged a series of griffs, or “gruff riffs”, that were often Ohrwürmer, or “ear-worms”, as German calls tunes that stick in your head. Even if you don’t want them to. But I’m not sure “tune” has ever been the right word for the music Ruin-Dredger create. It’s part industrial noise, part wolf-howl, part bat-twitter, but mostly “folded, fused, fissured, fractured, fidgety phonaesthesia.” And if you want to sample it, this album from the turn of the century is a good place to start.
What to call the album is one of the first puzzles it will set you. The band’s website usually calls it “a3” or “a3”; in interviews, the band themselves refer to it as “Triple-A” or “that A-fucker”. The second name comes from a plagiarism suit by the astro-music veterans Kargokkult that put Ruin-Dredger’s career on hold for nearly a year, 2002-3, and allegedly threatened to bankrupt their record-company. In the end the case was thrown out of court and even today some conspiracy-minded Dredge-heads claim it was cooked up for publicity between the ’Dredgers and the Kargonauts. The case might never have got as far as it did without that lunar cover for Triple-A, where the corroded letters of the band’s name and the album’s name hang above a lifeless moon-scape. Only it isn’t our moon. And it isn’t necessarily lifeless. Ruin-Dredger have a bee in their bonnet about the pre-biotic – the conditions necessary for the appearance of life. That’s what the first track on Triple-A, “Invention of the Cross”, is about: the chemicals that gave rise to life. And it literally has bees on it: the band sampled bees and bumblebees in flight and gathering nectar. They then altered the pitch and speed of the buzzing and made it sound both unearthly and unsettling. I’ve known people demand the track be turned off or skipped when it’s played to them.
But skipping track one of Triple-A is a bit like jumping from the frying-pan into the fire, because track two, “Seventh Sword”, is even more unearthly and even more unsettling. Bat-twitters hurtle through the speakers, falling from the ultra-sonic to the infra-sonic, rising in reverse, twisting, turning inside-out, mating, mutating and miscegenating. Then, as though the band have taken mercy on your ears and your mind, everything slows and soothes for track three, “Titanomachia”, which is often preceded in concert by the aforementioned carnal question: “And what doth it mean to be flesh?” This track is one of the last outings for the griffs of their early career: a slow, synth-based triple chord underlain by a sample of waves washing on an unknown shore. Track four, “Breathing Vacuum”, has also been known to provoke a “Turn it off!”, because the mumbling beneath the music is both sinister and sorrowful. You feel as though you should understand the words or, worse, that you will in your dreams. The chimes in the track are sinister too: they sound like a deep-sea, or deep-space, monster tapping on its fangs before putting them to famished use.
Which sets things up nicely, or nastily, for track five, “Scylla / Charybdis”. This is named after a pair of sea-monsters faced by Odysseus on his journey home from Troy and has been described by the ’Dredgers as a “battle-song”. The waves on “Titanomachia” are back, but bigger, badder and in a mood to fight. Daziel’s electronically treated voice wolf-howls a series of unintelligible questions, answered by patches of silence and gong-like drum-rolls. Track six, “Nyctogigas”, starts softly, builds back to the volume and violence of “Scyl/Char”, then breaks apart to allow the bats and bees of “Whilom” to steer your imagination out and up into the freezing star-light on the outer fringes of the solar system, where comets, shorn by the cold and dark, wait to swing sun-ward and regain their blazing locks. I like to listen to “Whilom” in the dark, wearing a blindfold, but then that’s the best way to listen to all of Ruin-Dredger’s music. Listening like that conjures visions and commands the viscera. Not an easy album, nor an unrewarding one, Triple-A isn’t their finest hour, if fan-polls and sales are any guide, but it’s an excellent guide to where they had come from and where they were about to go. If it’s the alpha-and-omega of their career, perhaps that explains the title: the “a” is the alpha (α) and the “3” an omega (ω) tipped on its side. I see it, or hear it, as a bridge between the ’nineties and the ’noughties: they’d give up the griffs and big up the bats, from then on, but they’ve never stopped asking that simple, sinister/sorrowful question of themselves and their listeners: “And What Doth It Mean To Be Flesh?”
a3 / a3 / Triple-A (S.R.K., 2000)
1. Invention of the Cross (5:26)
2. Seventh Sword (3:33)
3. Titanomachia (7:18)
4. Breathing Vacuum (9:03)
5. Scylla / Charybdis (6:11)
6. Nyctogigas (4:20)
7. Whilom (13:37)
Three Is The Key
If The Roses of Heliogabalus (1888) is any guide, Sir Lawrence Alma-Tadema (1836-1912) thought that 222 is a special number. But his painting doesn’t exhaust its secrets. To get to another curiosity of 222, start with 142857. As David Wells puts it in his Penguin Dictionary of Curious and Interesting Numbers (1986), 142857 is a “number beloved of all recreational mathematicians”. He then describes some of its properties, including this:
142857 x 1 = 142857
142857 x 2 = 285714
142857 x 3 = 428571
142857 x 4 = 571428
142857 x 5 = 714285
142857 x 6 = 857142
The multiples are cyclic permutations: the order of the six numbers doesn’t change, only their starting point. Because each row contains the same numbers, it sums to the same total: 1 + 4 + 2 + 8 + 5 + 7 = 27. And because each row begins with a different number, each column contains the same six numbers and also sums to 27, like this:
1 4 2 8 5 7
+ + + + + +
2 8 5 7 1 4
+ + + + + +
4 2 8 5 7 1
+ + + + + +
5 7 1 4 2 8
+ + + + + +
7 1 4 2 8 5
+ + + + + +
8 5 7 1 4 2
= = = = = =
2 2 2 2 2 2
7 7 7 7 7 7
If the diagonals of the square also summed to the same total, the multiples of 142857 would create a full magic square. But the diagonals don’t have the same total: the left-right diagonal sums to 31 and the right-left to 23 (note that 31 + 23 = 54 = 27 x 2).
But where does 142857 come from? It’s actually the first six digits of the reciprocal of 7, i.e. 1/7 = 0·142857… Those six numbers repeat for ever, because 1/7 is a prime reciprocal with maximum period: when you calculate 1/7, all integers below 7 are represented in the remainders. The square of multiples above is simply another way of representing this:
1/7 = 0·142857…
2/7 = 0·285714…
3/7 = 0·428571…
4/7 = 0·571428…
5/7 = 0·714285…
6/7 = 0·857142…
7/7 = 0·999999…
The prime reciprocals 1/17 and 1/19 also have maximum period, so the squares created by their multiples have the same property: each row and each column sums to the same total, 72 and 81, respectively. But the 1/19 square has an additional property: both diagonals sum to 81, so it is fully magic:
01/19 = 0·0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1…
02/19 = 0·1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2…
03/19 = 0·1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3…
04/19 = 0·2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4…
05/19 = 0·2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5…
06/19 = 0·3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6…
07/19 = 0·3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7…
08/19 = 0·4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8…
09/19 = 0·4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9…
10/19 = 0·5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0…
11/19 = 0·5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1…
12/19 = 0·6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2…
13/19 = 0·6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3…
14/19 = 0·7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4…
15/19 = 0·7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5…
16/19 = 0·8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6…
17/19 = 0·8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7…
18/19 = 0·9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8…
First line = 0 + 5 + 2 + 6 + 3 + 1 + 5 + 7 + 8 + 9 + 4 + 7 + 3 + 6 + 8 + 4 + 2 + 1 = 81
Left-right diagonal = 0 + 0 + 7 + 5 + 5 + 9 + 0 + 3 + 0 + 4 + 2 + 8 + 7 + 5 + 6 + 7 + 5 + 8 = 81
Right-left diagonal = 9 + 9 + 2 + 4 + 4 + 0 + 9 + 6 + 9 + 5 + 7 + 1 + 2 + 4 + 3 + 2 + 4 + 1 = 81
In base 10, this doesn’t happen again until the 1/383 square, whose magic total is 1719 (= 383-1 x 10-1 / 2). But recreational maths isn’t restricted to base 10 and lots more magic squares are created by lots more primes in lots more bases. The prime 223 in base 3 is one of them. Here the first line is
1/223 = 1/220213 = 0·
0000100210210102121211101202221112202
2110211112001012200122102202002122220
2110110201020210001211000222011010010
2222122012012120101011121020001110020
0112011110221210022100120020220100002
0112112021202012221011222000211212212…
The digits sum to 222, so 222 is the magic total for all rows and columns of the 1/223 square. It is also the total for both diagonals, so the square is fully magic. I doubt that Alma-Tadema knew this, because he lived before computers made calculations like that fast and easy. But he was probably a Freemason and, if so, would have been pleased to learn that 222 had a link with squares.
Vigor Mortis
In the Christian religion, the resurrection follows the virgin birth. In the rock-graves at Heysham, the virgin birth follows the resurrection. Or rather: the virgin-births follow the resurrections. There are many of both. The rock-graves at Heysham* are carved in solid rock near the remains of St Patrick’s chapel, an ancient ruin overlooking Morecambe Bay on the coast of Lancashire in England. You may have seen them before, because they appear on the cover of a compilation album by the heavy-metal band Black Sabbath, where they’re filled with ice and look suitably dark and sinister. But the graves are sometimes full of life and activity. In spring, as the rainwater filling them begins to warm, there are resurrections – dozens of them. Tiny crustaceans (a group of animals that includes crabs, shrimps and woodlice) hatch from eggs that have over-wintered in the sediment on the floors of the graves. Some of the crustaceans are called water-fleas, others are called seed-shrimps. Water-fleas, whose scientific name is Daphnia, hop through the water with jerks of their antennae, sieving it for fresh-water plankton. Seed-shrimps, or ostracods, are enclosed in tiny double-sided shells through which their legs protrude. They trundle over the stone sides of the graves, scraping off algae and catching even smaller and simpler animals like rotifers and protozoa.
Rock graves at Heysham, Lancs. (c. 1000s)
Water-fleas are famous for parthenogenesis, or their ability to produce offspring without sex. Those that hatch first in spring are female and give birth without mating with any males. A single water-flea in a jar of stagnant water soon becomes a swarm. It’s only later in the year that males are born and the water-fleas mate to produce winter eggs, which sink to the floor of the graves and lie there through the cold weather. The eggs of water-fleas and ostracods can also survive desiccation, or drying-up, and can be blown on the wind to new sites. That is probably how these crustaceans arrived in the rock-graves, which they must have occupied for centuries, through the coldest winters and the hottest summers, dying and being reborn again and again. When a human being or large animal dies, chemical changes in the body make the muscles rigid and wood-like. The scientific term for this is rigor mortis, or the “stiffness of death”. Rigor mortis wears off in time and the body begins to rot. The rock-graves at Heysham are an example of vigor mortis, or the “vigour of death”. Medieval human beings created the graves to bury their dead, but the bodies that were once there have been lost to history. The water-fleas and the seed-shrimps remain, tiny, overlooked and fascinating.
A seed-shrimp
A water-flea
*Heysham is pronounced HEE-shum and is an old coastal village near the city of Lancaster, after which Lancashire is named.
Roses Are Golden
Sir Lawrence Alma-Tadema’s painting The Roses of Heliogabalus (1888) is based on an apocryphal episode in the sybaritic life of the Roman Emperor Elagabalus (204-222 A.D.), who is said to have suffocated guests with flowers at one of his feasts. The painting is in a private collection, but I saw it for real in an Alma-Tadema exhibition at the Walker Art Gallery in Liverpool sometime during the late 1990s. I wasn’t disappointed: it was a memorable meeting with a painting I’d been interested in for years. Roses is impressively large and impressively skilful. Close-up, the brush-strokes are obvious, obtrusive and hard to interpret as people and objects. It isn’t till you step back, far beyond the distance at which Alma-Tadema was painting, that the almost photographic realism becomes apparent. But you get more of the many details at close range, like the Latin inscription on a bowl below and slightly to the right of that scowling water-mask. Alas, I forgot to take a note of what the inscription was, though perhaps the memory is still locked away somewhere in my subconscious.
The Roses of Heliogabalus (1888)
Whatever it is, I feel sure it is significant, because Roses is rich with meaning. That’s a large part of why I’m interested in it. Yes, I like it a lot as art, but the women would have to be more attractive for it to be higher in the list of my favourite paintings. As it is, I think there are only four reasonably good-looking people in it: the man with the beard on the right; the flautist striding past the marble pillar on the left; the red-headed girl with a crown of white flowers; and Heliogabalus himself, crowned in roses and clutching a handful of grapes beside the overweight man who’s wearing a wreath and sardonically saluting one of the rose-pelted guests in the foreground. When I first wrote about Roses in a pub-zine whose name escapes me, I misidentified the overweight man as Heliogabalus himself, even though I noted that he seemed many years old than Heliogabalus, toppled as a teen tyrant, should have been. It was a bad mistake, but one that, with less knowledge and more excuse, many people must make when they look at Roses, because the overweight man and his sardonic salute are a natural focus for the eye. Once your eye has settled on and noted him, you naturally follow the direction of his gaze down to the man in the foreground, who’s gazing right back.
Something Like the Sun
And by following that gaze, you’ve performed a little ratio-ritual, just as Alma-Tadema intended you to do. Yes, Roses is full of meaning and much of that meaning is mathematical. I think the angle of the gaze is one of many references in Roses to the golden ratio, or φ (phi), a number that is supposed to have special aesthetic importance and has certainly been used by many artists and musicians to guide their work. A rectangle with sides in the proportions 8:13, for example, approximates the golden ratio pretty closely, but φ itself is impossible to represent physically, because it’s an irrational number with infinitely many decimal digits, like π or √2, the square root of two. π represents the ratio of a circle’s circumference to its diameter and √2 the ratio of a square’s diagonal to its side, but no earthly circle and no earthly square can ever capture these numbers with infinite precision. Similarly, no earthly rectangle can capture φ, but the rectangle of Roses is a good attempt, because it measures 52″ x 84 1/8". That extra eighth of an inch was my first clue to the painting’s mathematical meaningfulness. And sure enough, 52/84·125 = 416/673 = 0·61812…, which is a good approximation to φ’s never-ending 0·6180339887498948482045868343656…*
That deliberate choice of dimensions for the canvas led me to look for more instances of φ in the painting, though one of the most important and obvious might be called a meta-presence. The Roses of Heliogabalus is dated 1888, or 1666 years after the death of Heliogabalus in 222 AD. A radius at 222º divides a circle in the golden ratio, because 222/360 = 0·616… It’s very hard to believe Alma-Tadema didn’t intend this reference and I also think there’s something significant in 1888 itself, which equals 2 x 2 x 2 x 2 x 2 x 59 = 25 x 59. Recall that 416 is the expanded short side of Roses. This equals 25 x 13, while 673, the expanded long side, is the first prime number after 666. As one of the most technically skilled painters who ever lived, Alma-Tadema was certainly an exceptional implicit mathematician. But he clearly had explicit mathematical knowledge too and this painting is a phi-pie cooked by a master matho-chef. In short, when Roses is read, Roses turns out to be golden.
*φ is more usually represented as 1·6180339887498948482045868343656…, but it has the pecularity that 1/φ = φ-1, so the decimal digits don’t change and 0·6180339887498948482045868343656… is also legitimate.
Appendix I
I’ve looked at more of Alma-Tadema’s paintings to see if their dimensions approximate φ, √2, √3 or π, or their reciprocals. These were the results (ε = error, i.e. the difference between the constant and the ratio of the dimensions).
The Roman Wine Tasters (1861), 50" x 69 2/3": 150/209 = 0·717… ≈ 1/√2 (ε=0·02)
A Roman Scribe (1865), 21 1/2" x 15 1/2": 43/31 = 1·387… ≈ √2 (ε=0·027)
A Picture Gallery (1866), 16 1/8" x 23": 129/184 = 0·701… ≈ 1/√2 (ε=0·012)
A Roman Dance (1866), 16 1/8" x 22 1/8": 43/59 = 0·728… ≈ 1/√2 (ε=0·042)
In the Peristyle (1866), 23" x 16": 23/16 = 1·437… ≈ √2 (ε=0·023)
Proclaiming Emperor Claudius (1867), 18 1/2" x 26 1/3": 111/158 = 0·702… ≈ 1/√2 (ε=0·009)
Phidias and the Frieze of the Parthenon Athens (1868), 29 2/3" x 42 1/3": 89/127 = 0·7… ≈ 1/√2 (ε=0·012)
The Education of Children of Clovis (1868), 50" x 69 2/3": 150/209 = 0·717… ≈ 1/√2 (ε=0·02)
An Egyptian Juggler (1870), 31" x 19 1/4": 124/77 = 1·61… ≈ φ (ε=0·007)
A Roman Art Lover (1870), 29" x 40": 29/40 = 0·725… ≈ 1/√2 (ε=0·034)
Good Friends (1873), 4 1/2" x 7 1/4": 18/29 = 0·62… ≈ φ (ε=0·006)
Pleading (1876), 8 1/2" x 12 3/8": 68/99 = 0·686… ≈ 1/√2 (ε=0·041)
An Oleander (1882), 36 1/2" x 25 1/2": 73/51 = 1·431… ≈ √2 (ε=0·017)
Dolce Far Niente (1882), 9 1/4" x 6 1/2": 37/26 = 1·423… ≈ √2 (ε=0·008)
Anthony and Cleopatra (1884), 25 3/4" x 36 1/3": 309/436 = 0·708… ≈ 1/√2 (ε=0·003)
Rose of All Roses (1885), 15 1/4" x 9 1/4": 61/37 = 1·648… ≈ φ (ε=0·03)
The Roses of Heliogabalus (1888), 52" x 84 1/8": 416/673 = 0·618… ≈ φ (ε<0.001)
The Kiss (1891), 18" x 24 3/4": 8/11 = 0·727… ≈ 1/√2 (ε=0·039)
Unconscious Rivals (1893), 17 3/4" x 24 3/4": 71/99 = 0·717… ≈ 1/√2 (ε=0·019)
A Coign of Vantage (1895), 25 1/4" x 17 1/2": 101/70 = 1·442… ≈ √2 (ε=0·028)
A Difference of Opinion (1896), 15" x 9": 5/3 = 1·666… ≈ φ (ε=0·048)
Whispering Noon (1896), 22" x 15 1/2": 44/31 = 1·419… ≈ √2 (ε=0·005)
Her Eyes Are With Her Thoughts And Her Thoughts Are Far Away (1897), 9" x 15": 3/5 = 0·6… ≈ φ (ε=0·048)
The Baths of Caracalla (1899), 60" x 37 1/2": 8/5 = 1·6… ≈ φ (ε=0·018)
The Year’s at the Spring, All’s Right with the World (1902), 13 1/2" x 9 1/2": 27/19 = 1·421… ≈ √2 (ε=0·006)
Ask Me No More (1906), 31 1/2" x 45 1/2": 9/13 = 0·692… ≈ 1/√2 (ε=0·03)
Appendix II
The Roses of Heliogabalus is based on this section from Aelius Lampridius’ pseudonymous and largely apocryphal Vita Heliogabali, or Life of Heliogabalus, in the Historia Augusta (late fourth century):
XXI. 1 Canes iecineribus anserum pavit. Habuit leones et leopardos exarmatos in deliciis, quos edoctos per mansuetarios subito ad secundam et tertiam mensam iubebat accumbere ignorantibus cunctis, quod exarmati essent, ad pavorem ridiculum excitandum. 2 Misit et uvas Apamenas in praesepia equis suis et psittacis atque fasianis leones pavit et alia animalia. 3 Exhibuit et sumina apruna per dies decem tricena cottidie cum suis vulvis, pisum cum aureis, lentem cum cerauniis, fabam cum electris, orizam cum albis exhibens. 4 Albas praeterea in vicem piperis piscibus et tuberibus conspersit. 5 Oppressit in tricliniis versatilibus parasitos suos violis et floribus, sic ut animam aliqui efflaverint, cum erepere ad summum non possent. 6 Condito piscinas et solia temperavit et rosato atque absentato…
XXI. 1 He fed his dogs on goose-livers. He had pet lions and leopards, which had been rendered harmless and trained by tamers, and these he would suddenly order during the dessert and the after-dessert to get on the couches, thereby causing laughter and panic, for none knew that they were harmless. 2 He sent grapes from Apamea to his stables for the horses, and he fed parrots and pheasants to his lions and other beasts. 3 For ten days in a row, moreover, he served wild sows’ udders with the matrices, at a rate of thirty a day, serving, besides, peas with gold-pieces, lentils with onyx, beans with amber, and rice with pearls; 4 and he also sprinkled pearls on fish and used truffles instead of pepper. 5 In a banqueting-room with a reversible ceiling he once buried his parasites in violets and other flowers, so that some were actually smothered to death, being unable to crawl out to the top. 6 He flavoured his swimming-pools and bath-tubs with essence of spices or of roses or wormwood…
