This Means RaWaR

The Overlord of the Über-Feral says: Welcome to my bijou bloguette. You can scroll down to sample more or simply:

• Read a Writerization at Random: RaWaR

• O.o.t.Ü.-F.: More Maverick than a Monkey-Munching Mingrelian Myrmecologist Marinated in Mescaline…

• ¿And What Doth It Mean To Be Flesh?

მათემატიკა მსოფლიოს მეფე

*Der Muntsch ist Etwas, das überwunden werden soll.


Bent for the Pent

A triangle can be tiled with triangles and a square with squares, but a pentagon can’t be tiled with pentagons. At least, not in the same way, using smaller copies of the same shape. The closest you can get is this:

Pentaflake #1

If you further subdivide the pentagon, you create what is known as a pentaflake:

Pentaflake #2

Pentaflake #3

Pentaflake #4

Pentaflake (animated)

Pentaflake (static)

But if you bend the rules and use irregular smaller pentagons, you can tile a pentagon like this, creating what I called a pentatile:

Pentatile stage 1

Further subdivisions create an interesting final pattern:

Pentatile #2

Pentatile #3

Pentatile #4

Pentatile #5

Pentatile #6

Pentatile (animated)

Pentatile (static)

By varying the size of the central pentagon, you can create other patterns:

Pentatile #1 (initial)

Pentatile #2 (initial)

Pentatile #2

Pentatile with no central pentagon

And here are various pentatiles in an animated gif:

Elsewhere other-posted:

Phrallic Frolics — more on pentaflakes
Bent for the Rent (1976) — the title of this incendiary intervention is of course a reference to the “first and last glitter-rock album” by England’s loudest band, Spinal In Terms Of Tap.

Feel the ’Burne

The Poets at Tea […]

3.—(Swinburne, who let it get cold)

As the sin that was sweet in the sinning
Is foul in the ending thereof,
As the heat of the summer’s beginning
Is past in the winter of love:
O purity, painful and pleading!
O coldness, ineffably gray!
Oh, hear us, our handmaid unheeding,
And take it away!

Barry Pain (1864-1928)

A Melton-Mowbray Pork Pie

Strange pie that is almost a passion,
     O passion immoral for pie!
Unknown are the ways that they fashion,
     Unknown and unseen of the eye.

The pie that is marbled and mottled,
     The pie that digests with a sigh:
For all is not Bass that is bottled,
     And all is not pork that is pie.

Richard Le Gallienne (1866-1947)

Square Routes Re-Revisited

This is a very simple fractal:

It has four orientations:

Any orientation can be turned into any other by a rotation of 90°, 180° or 270°, either clockwise or anticlockwise. If you mix orientations and rotations, you can create much more complex fractals. Here’s a selection of them:

Animated fractal

Static fractal

Animated fractal


Animated fractal


Animated fractal


Animated fractal


Animated fractal


Animated fractal


Animated fractal


Animated fractal


Animated fractal


Animated fractal


Animated fractal


Animated fractal


Animated fractal


Animated fractal


Animated fractal


Animated fractal


Animated fractal


Elsewhere other-posted:

Square Routes
Square Routes Revisited

Square on a Three String

222 A.D. was the year in which the Emperor Heliogabalus was assassinated by his own soldiers. Exactly 1666 years later, the Anglo-Dutch classicist Sir Lawrence Alma-Tadema exhibited his painting The Roses of Heliogabalus (1888). I suggested in “Roses Are Golden” that Alma-Tadema must have chosen the year as deliberately as he chose the dimensions of his canvas, which, at 52″ x 84 1/8“, is an excellent approximation to the golden ratio.

But did Alma-Tadema know that lines at 0º and 222º divide a circle in the golden ratio? He could easily have done, just as he could easily have known that 222 precedes the 48th prime, 223. But it is highly unlikely that he knew that 223 yields a magic square whose columns, rows and diagonals all sum to 222. To create the square, simply list the 222 multiples of the reciprocal 1/223 in base 3, or ternary. The digits of the reciprocal repeat after exactly 222 digits and its multiples begin and end like this:

001/223 = 0.00001002102101021212111012022211122022... in base 3
002/223 = 0.00002011211202120201222101122200021121...
003/223 = 0.00010021021010212121110120222111220221...
004/223 = 0.00011100200112011110221210022100120020...
005/223 = 0.00012110002220110100102222122012012120...


218/223 = 0.22210112220002112122120000100210210102... in base 3
219/223 = 0.22211122022110211112001012200122102202...
220/223 = 0.22212201201212010101112102000111002001...
221/223 = 0.22220211011020102021000121100022201101...
222/223 = 0.22221220120121201010111210200011100200...

Each column, row and diagonal of ternary digits sums to 222. Here is the full n/223 square represented with 0s in grey, 1s in white and 2s in red:

(Click for larger)

It isn’t difficult to see that the white squares are mirror-symmetrical on a horizontal axis. Here is the symmetrical pattern rotated by 90º:

(Click for larger)

But why should the 1s be symmetrical? This isn’t something special to 1/223, because it happens with prime reciprocals like 1/7 too:

1/7 = 0.010212... in base 3
2/7 = 0.021201...
3/7 = 0.102120...
4/7 = 0.120102...
5/7 = 0.201021...
6/7 = 0.212010...

And you can notice something else: 0s mirror 2s and 2s mirror 0s. A related pattern appears in base 10:

1/7 = 0.142857...
2/7 = 0.285714...
3/7 = 0.428571...
4/7 = 0.571428...
5/7 = 0.714285...
6/7 = 0.857142...

The digit 1 in the decimal digits of n/7 corresponds to the digit 8 in the decimal digits of (7-n)/7; 4 corresponds to 5; 2 corresponds to 7; 8 corresponds to 1; 5 corresponds to 4; and 7 corresponds to 2. In short, if you’re given the digits d1 of n/7, you know the digits d2 of (n-7)/7 by the rule d2 = 9-d1.

Why does that happen? Examine these sums:

 1/7 = 0.142857142857142857142857142857142857142857...
+6/7 = 0.857142857142857142857142857142857142857142...
 7/7 = 0.999999999999999999999999999999999999999999... = 1.0

 2/7 = 0.285714285714285714285714285714285714285714...
+5/7 = 0.714285714285714285714285714285714285714285...
 7/7 = 0.999999999999999999999999999999999999999999... = 1.0

 3/7 = 0.428571428571428571428571428571428571428571...
+4/7 = 0.571428571428571428571428571428571428571428...
 7/7 = 0.999999999999999999999999999999999999999999... = 1.0

And here are the same sums in ternary (where the first seven integers are 1, 2, 10, 11, 12, 20, 21):

  1/21 = 0.010212010212010212010212010212010212010212...
+20/21 = 0.212010212010212010212010212010212010212010...
 21/21 = 0.222222222222222222222222222222222222222222... = 1.0

  2/21 = 0.021201021201021201021201021201021201021201...
+12/21 = 0.201021201021201021201021201021201021201021...
 21/21 = 0.222222222222222222222222222222222222222222... = 1.0

 10/21 = 0.102120102120102120102120102120102120102120...
+11/21 = 0.120102120102120102120102120102120102120102...
 21/21 = 0.222222222222222222222222222222222222222222... = 1.0

Accordingly, in base b with the prime p, the digits d1 of n/p correspond to the digits (p-n)/p by the rule d2 = (b-1)-d1. This explains why the 1s mirror themselves in ternary: 1 = 2-1 = (3-1)-1. In base 5, the 2s mirror themselves by the rule 2 = 4-2 = (5-1) – 2. In all odd bases, some digit will mirror itself; in all even bases, no digit will. The mirror-digit will be equal to (b-1)/2, which is always an integer when b is odd, but never an integer when b is even.

Here are some more examples of the symmetrical patterns found in odd bases:

Patterns of 1s in 1/19 in base 3

Patterns of 6s in 1/19 in base 13

Patterns of 7s in 1/19 in base 15

Elsewhere other-posted:

Roses Are Golden — more on The Roses of Heliogabalus (1888)
Three Is The Key — more on the 1/223 square

Wysts and Mellow Flutefulness

(To Randolph Churchill, but not about him)

Broad of Church and broad of mind,
Broad before and broad behind,
A keen ecclesiologist,
A rather dirty Wykehamist.
’Tis not for us to wonder why
He wears that curious knitted tie;
We should not cast reflections on
The very slightest kind of don.
We should not giggle as we like
At his appearance on his bike;
It’s something to become a bore,
And more than that, at twenty-four.
It’s something too to know your wants
And go full pelt for Norman fonts.
Just now the chestnut trees are dark
And full with shadow in the park,
And “Six o’clock!” St. Mary calls
Above the mellow college walls.
The evening stretches arms to twist
And captivate her Wykehamist.
But not for him these autumn days,
He shuts them out with heavy baize;
He gives his Ovaltine a stir
And nibbles at a petit beurre,
And, satisfying fleshy wants,
He settles down to Norman fonts.

John Betjeman (1906-84)

Performativizing Papyrocentricity #57

Papyrocentric Performativity Presents:

Do and DieThe Reason Why, Cecil Woodham-Smith (1953) (posted at O.-o.-t.-Ü)

Liddell im WörterlandLiddell and Scott’s Greek-English Lexicon, Henry George Liddell and Robert Scott (1843)

Lunar or LaterMoon: From 4.5 billion years ago to the present: Owners’ Workshop Manual, David M. Harland (Haynes 2016)

Headlong into NightmareHeadlong Hall (1816) / Nightmare Abbey (1818)

Twisted TalesBiggles’ Big Adventures: Four Classic Stories Starring the British Empire’s Most Fearless Pilot Adventurer, Captain W.E. Johns (Sevenoaks 2007)

Stop the Brott – staying the serial slaying of a sanguinivorous psychoanalyst

• Or Read a Review at Random: RaRaR

Living Culler

When you replace a square with four smaller squares, each a quarter the size of the original, the smaller squares occupy the same area, because 4 * ¼ = 1. If you discard one sub-square, then divide each of the three remaining sub-squares into four sub-sub-square, discard one sub-sub-quare and repeat, you create fractals like those I looked at in Squaring and Paring. The fractals stay within a fixed boundary.

Square replaced with four smaller squares, each ¼th the size of the original

Animated fractal

Static fractal

This time I want to look at a slightly different process. Replace a square with nine smaller squares each a quarter the size of the original. Now the sub-squares occupy a larger area than the original, because 9 * ¼ = 2¼. If you discard — or cull — sub-squares and repeat, the resultant fractal grows beyond the original boundary. Indeed, sub-squares start to overlap, so you can use colours to represent how often a particular pixel has been covered with a square. Here is an example of this process in action:

Square replaced with nine smaller squares, each ¼th the size of the original

Animated fractal

Static fractal #1

Static fractal #2

Here are the individual stages of a more complex fractal that uses the second process:

Stage 1

Stage 2

Stage 3

Stage 4

Stage 5

Stage 6

Stage 7

Stage 8

Stage 9 (compare Fingering the Frigit and Performativizing the Polygonic)

Stage 10

Animated version

Static version #1

Static version #2

And here are some more of the fractals you can create in a similar way:

Static version #1

Static version #2

Static version #2

Static version #2

Static version #3

Various fractals in an animated gif

Noise Annoys

“Noise” may have an interesting etymology. Some think it comes from “nausea”, which itself comes from Greek naus, meaning “ship”. Neither the putative etymology of “noise” nor the undisputed etymology of “nausea” would have been news to J.R.R. Tolkien. He was, after all, a professional scholar of literature and languages.

But that’s why The Lord of the Rings is often a puzzling book. Why did someone so interested in words and languages write so clumsily? As I’ve said before: I wish someone would translate Lord of the Rings into English. But perhaps if Tolkien had been a better writer I wouldn’t have read Lord of the Rings so often. And perhaps if he’d been a better writer there would have been no Lord of the Rings at all. Even so, it’s hard to excuse writing like this:

He heard behind his head a creaking and scraping sound. […] There was a shriek and the light vanished. In the dark there was a snarling noise. – “Fog on the Barrowdowns”, Book One, VIII

Why did he use “sound” and “noise”? They’re redundant, because creak, scrape and snarl already describe sounds or noises. You could argue that the additional words are there to balance the sentences, but if they hadn’t been there I don’t think anyone would have missed them:

He heard behind his head a creaking and scraping. … There was a shriek and the light vanished. In the dark there was a snarling.

Later in the book Tolkien gets it right:

At that moment there came a roaring and a rushing: a noise of loud waters rolling many stones. – “Flight to the Ford”, Book One, XII

Then he gets it wrong again:

Turning quickly they saw ripples, black-edged with shadow in the waning light: great rings were widening outwards from a point far out in the lake. There was a bubbling noise, and then silence. – “A Journey in the Dark”, Book Two, IV

This would have been better:

There was a bubbling, and then silence.

It’s crisper, clearer and doesn’t strike an ugly twentieth-century note in an archaic setting. And it should have been what J.R.R. Tolkien wrote in the first place. I don’t know why he didn’t and I don’t know why his editors or those who read early drafts of Lord of the Rings didn’t point out his error. That’s why I’d like to visit the Library of Babel and find a copy of Lord of the Rings written by Clark Ashton Smith.

Squaring and Paring

Squares are often thought to be the most boring of all shapes. Yet every square holds a stunning secret – something that in legend prompted a mathematical cult to murder a traitor. If each side of a square is one unit long, how long is the square’s diagonal, that is, the line from one corner to the opposite corner?

By Pythagoras’ theorem, the answer is this:

• x^2 = 1^2 + 1^2
• x^2 = 2
• x = √2

But what is √2? Pythagoras and his followers thought that all numbers could be represented as either whole numbers or ratios of whole numbers. To their dismay, so it’s said, they discovered that they were wrong. √2 is an irrational number – it can’t be represented as a ratio. In modern notation, it’s an infinitely decimal that never repeats:

• √2 = 1·414213562373095048801688724209698…

A modern story, unattested in ancient records, says that the irrationality of √2 was a closely guarded secret in the Pythagorean cult. When Hippasus of Metapontum betrayed the secret, he was drowned at sea by enraged fellow cultists. Apocryphal or not, the story shows that squares aren’t so boring after all.

Nor are they boring when they’re caught in the fract. Divide one square into nine smaller copies of itself:

Discard three of the copies like this:

Stage 1
Retain squares 1, 2, 4, 6, 8, 9 (reading left-to-right, bottom-to-top)

Then do the same to each of the sub-squares:

Stage 1

And repeat:

Stage 3

Stage 4

Stage 5

Stage 6

The result is a fractal of endlessly subdividing contingent hexagons:

Animated vesion

Retain squares 1, 2, 4, 6, 8, 9 (reading left-to-right, bottom-to-top)

Here are a few more of the fractals you can create by squaring and paring:

Retain squares 1, 3, 5, 7, 9 (reading left-to-right, bottom-to-top)

Retain squares 2, 4, 5, 6, 8

Retain squares 1, 2, 4, 5, 6, 8, 9

Retain squares 1, 4, 6, 7, 10, 11, 13, 16

Retain squares 1, 3, 6, 7, 8, 9, 10, 11, 14, 16

Retain squares 2, 3, 5, 6, 8, 9, 11, 12, 14, 15

Retain squares 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25

Retain squares 1, 3, 7, 8, 11, 12, 14, 15, 18, 19, 23, 25

Retain squares 1, 5, 7, 8, 9, 12, 14, 17, 18, 19, 21, 25

Retain squares 2, 3, 4, 6, 7, 9, 10, 11, 15, 16, 17, 19, 20, 22, 23, 24

Retain squares 1, 2, 5, 6, 7, 9, 13, 17, 19, 20, 21, 24, 25

Previously pre-posted (please peruse):

M.i.P. Trip

Do and Die

The Reason Why, Cecil Woodham-Smith (1953)

History is a branch of literature, not of science. That’s why it’s so important that historians be good writers. Cecil Woodham-Smith (1896-1977) was a very good writer and this is one of the best works of military history ever written. I don’t know whether she – that “Cecil” is misleading – was influenced by Lytton Strachey’s Eminent Victorians (1918) but Strachey’s sublime “Cardinal Manning” is an obvious comparison. Like Strachey’s, her prose has grace, lightness and concision:

Military glory! It was a dream that century after century had seized on men’s imaginations and set their blood on fire. Trumpets, plumes, chargers, the pomp of war, the excitement of combat, the exultation of victory – the mixture was intoxicating indeed. To command great armies, to perform deeds of valour, to ride victorious through flower-strewn streets, to be heroic, magnificent, famous – such were the visions that danced before men’s eyes as they turned eagerly to war.

It was not a dream for the common man. War was an aristocratic trade, and military glory reserved for nobles and princes. Glittering squadrons of cavalry, long lines of infantry, wheeling obediently on the parade-ground, ministered to the lust both for power and for display. Courage was esteemed the essential military quality and held to be a virtue exclusive to aristocrats. Were they not educated to courage, trained, as no common man was trained, by years of practice in dangerous sports? They glorified courage, called it valour and worshipped it, believed battles were won by valour, saw war in terms of valour as the supreme adventure.

It was a dream that died hard. Century followed century and glittering armies faded before the sombre realities of history. Great armies in their pride and splendour were defeated by starvation, pestilence and filth, valour was sacrificed to stupidity, gallantry to corruption. (ch. 1, opening paragraphs)

But Woodham-Smith is a more masculine writer than Strachey: more serious, more sober and much more at home with military affairs. It would be wrong to call The Reason Why a pleasure to read, because although it is often is, it treats of horrors both on the battlefield and in civilian life. The Irish Famine played its part in forging the character of Lord Lucan, one of the chief figures in “The Story of the Fatal Charge of the Light Brigade”, as the subtitle of a modern re-issue of the book puts it. Woodham-Smith later wrote a book called The Great Hunger (1962) about the Famine, but I’m reluctant to read it: what she describes here is horrible enough.

I have, however, read her biography Florence Nightingale (1950), the book that began her career amid an explosion of plaudits. I was disappointed, just as I was by Strachey’s Queen Victoria (1921). Both writers set such high standards in their best work that the rest of it can suffer by comparison. And history is difficult to write well. Against that, however, are the gifts it offers its practitioners: the wholly improbable situations that no writer of fiction could expect his readers to swallow. The Charge of the Light Brigade was like that. Who would invent a concatenation of incompetence, misinterpretation and personal enmity that sends a brigade of cavalry charging down an occupied valley against a battery of artillery?

No-one would invent that. But it is precisely what happened during the Crimean War. If any small link in the chain of causality had broken, the charge would not have been launched. Nor would it have been launched if Lord Lucan had been less stubborn, Lord Cardigan less stupid, Lord Raglan less incompetent and Captain Nolan less impetuous. Nolan was the rider who delivered Raglan’s scribbled order to Lucan, descending hundreds of feet from a perspective where Raglan’s meaning was clear to a spot where it wasn’t clear at all. That was part of why the charge took place. Another part was Nolan’s contempt for Lucan and Nolan’s misinterpretation of the order:

The crucial moment had arrived. Nolan threw back his head, and, “in a most disrespectful and significant manner”, flung out his arm and, with a furious gesture, pointed, not to the Causeway Heights and the redoubts with the captured British guns, but to the end of the North Valley, where the Russian cavalry routed by the Heavy Brigade were now established with their guns in front of them. “There, my lord, is your enemy, there are your guns,” he said, and with those words and that gesture the doom of the Light Brigade was sealed. (ch. 12, pp. 233-4)

So was Nolan’s own doom. Within in a few minutes he himself would be dead, killed by one of the early volleys fired by the Russian guns. He seems to have realized his error and tried to stop the charge, committing “an unprecedented breach of military etiquette” as he overtook Lord Cardigan at the head and shouted with raised sword “as if he would address the Brigade”. Woodham-Smith asks:

Had he suddenly realized that his interpretation of the order had been wrong, and that in his impetuosity he had directed the Light Brigade to certain death? No one will ever know, because at that moment a Russian shell burst on the right of Lord Cardigan, and a fragment tore its way into Nolan’s breast, exposing his heart. The sword fell from his hand, but his right hand was still erect, and his body remained rigid in the saddle. His horse wheeled and began to gallop back through the advancing Brigade, and then from the body there burst a strange and appalling shriek, a shriek so unearthly so to freeze the blood of all who heard him. The terrified horse carried the body, still shrieking, through the 4th Light Dragoons, and then at last Nolan fell from the saddle, dead. (ch. 12, pg. 240)

Nolan was Irish and his death-shriek was like something from Celtic mythology, as though he had been possessed by a spirit of the doom that was about to engulf the splendid ranks of the Light Brigade. And the charge was a mythic occasion: a pointless slaughter enabled not only by the incompetence, stupidity and arrogance of the British officers, but also by the courage, discipline and skill of the men they led:

And now the watchers on the Heights saw that the lines of horsemen, like toys down on the plain, were expanding and contracting with strange mechanical precision. Death was coming fast, and the Light Brigade was meeting death in perfect order; as a man or horse dropped, the riders on each side of him opened out; as soon as they had ridden clear the ranks closed again. Orderly, as if on the parade-ground, the Light Brigade rode on, but its numbers grew every moment smaller and smaller as they moved down the valley. Those on the heights who could understand what that regular mechanical movement meant in terms of discipline and courage were intolerably moved, and one soldier burst into years. It was at this moment that Bosquet, the French General, observed “C’est magnifique, mais ce n’est pas la guerre.” (ch. 12, pg. 242)

But the charge occupies little space in this book, just as it did in the War and the history of the Victorian Age. Woodham-Smith magisterially sets the stage for 232 pages, describing the horrors of the war, the incompetence of the officers, and the courage of the troops that enabled some improbable victories against overwhelming odds. Then she devotes a single chapter to the charge. It was both horrible and glorious, representing both the worst and the best of the British army in Victorian times. And the army represented both the worst and the best of Victorian Britain. Like Eric Ambler, Woodham-Smith can re-create a complex world and its participants on paper. And like Ambler, she is sympathetic to all her characters, from the best to the worst. Strachey mocks and subverts in Eminent Victorians, partly because that was in his nature as a homosexual outsider and partly because he blamed the horrors of the First World War on the legacy of the Victorians.

By 1953, when The Reason Why was published, that legacy was much further in the past, many reforms had taken place, and a second, and much less senseless, world war had been fought by Britain and her allies. Woodham-Smith could be more objective than Strachey. Moreover, men like Lord Cardigan hardly need a satirical or subversive pen: his absurdities speak for themselves. But if you want a humorous take on the Charge of the Light Brigade, I recommend George MacDonald-Fraser’s Flashman at the Charge (1973), in which the bully, coward and liar Flashman is caught up, wholly against his will, in the two astonishing cavalry actions that took place that day: the Charges of both the Light Brigade and the Heavy Brigade.

Neither of them could plausibly be invented by a writer of fiction, but the Charge of the Heavy Brigade was a success, not a tragic farce. That is why it is much less well-remembered. But the Charge of the Light Brigade has never been so well-remembered, or well-explained, as it was by Cecil Woodham-Smith. If you want to know the Reason Why – or the Reasons – then you’ll find them here. You’ll also find an excellent introduction to Victorian England and one of the best military histories ever written.