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


• ¿And What Doth It Mean To Be Flesh?

მათემატიკა მსოფლიოს მეფე


Gweel & Other Alterities – Incunabula’s new edition


Tales of Silence & Sortilege – Incunabula’s new edition



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Phil’ Kill Kult

Lucretius (1869)

by Alfred Tennyson

Lucilla, wedded to Lucretius, found
Her master cold; for when the morning flush
Of passion and the first embrace had died
Between them, tho’ he loved her none the less,
Yet often when the woman heard his foot
Return from pacings in the field, and ran
To greet him with a kiss, the master took
Small notice, or austerely, for his mind
Half buried in some weightier argument,
Or fancy-borne perhaps upon the rise
And long roll of the hexameter — he past
To turn and ponder those three hundred scrolls
Left by the Teacher, whom he held divine.
She brook’d it not, but wrathful, petulant
Dreaming some rival, sought and found a witch
Who brew’d the philtre which had power, they said
To lead an errant passion home again.
And this, at times, she mingled with his drink,
And this destroy’d him; for the wicked broth
Confused the chemic labor of the blood,
And tickling the brute brain within the man’s
Made havoc among those tender cells, and check’d
His power to shape. He loathed himself, and once
After a tempest woke upon a morn
That mock’d him with returning calm, and cried:

“Storm in the night! for thrice I heard the rain
Rushing; and once the flash of a thunderbolt —
Methought I never saw so fierce a fork —
Struck out the streaming mountain-side, and show’d
A riotous confluence of watercourses
Blanching and billowing in a hollow of it,
Where all but yester-eve was dusty-dry.

“Storm, and what dreams, ye holy Gods, what dreams!
For thrice I waken’d after dreams. Perchance
We do but recollect the dreams that come
Just ere the waking. Terrible: for it seem’d
A void was made in Nature, all her bonds
Crack’d; and I saw the flaring atom-streams
And torrents of her myriad universe,
Ruining along the illimitable inane,
Fly on to clash together again, and make
Another and another frame of things
For ever. That was mine, my dream, I knew it —
Of and belonging to me, as the dog
With inward yelp and restless forefoot plies
His function of the woodland; but the next!
I thought that all the blood by Sylla shed
Came driving rainlike down again on earth,
And where it dash’d the reddening meadow, sprang
No dragon warriors from Cadmean teeth,
For these I thought my dream would show to me,
But girls, Hetairai, curious in their art,
Hired animalisms, vile as those that made
The mulberry-faced Dictator’s orgies worse
Than aught they fable of the quiet Gods.
And hands they mixt, and yell’d and round me drove
In narrowing circles till I yell’d again
Half-suffocated, and sprang up, and saw —
Was it the first beam of my latest day?

“Then, then, from utter gloom stood out the
The breasts of Helen, and hoveringly a sword
Now over and now under, now direct,
Pointed itself to pierce, but sank down shamed
At all that beauty; and as I stared, a fire,
The fire that left a roofless Ilion,
Shot out of them, and scorch’d me that I woke.

“Is this thy vengeance, holy Venus, thine,
Because I would not one of thine own doves,
Not even a rose, were offered to thee? thine,
Forgetful how my rich proemion makes
Thy glory fly along the Italian field,
In lays that will outlast thy deity?

“Deity? nay, thy worshippers. My tongue
Trips, or I speak profanely. Which of these
Angers thee most, or angers thee at all?
Not if thou be’st of those who, far aloof
From envy, hate and pity, and spite and scorn,
Live the great life which all our greatest fain
Would follow, centred in eternal calm.

“Nay, if thou canst,
Goddess, like ourselves
Touch, and be touch’d, then would I cry to thee
To kiss thy Mavors, roll thy tender arms
Round him, and keep him from the lust of blood
That makes a steaming slaughter-house of Rome.

“Ay, but I meant not thee; I meant riot her
Whom all the pines of Ida shook to see
Slide from that quiet heaven of hers, and tempt
The Trojan, while his neatherds were abroad
Nor her that o’er her wounded hunter wept
Her deity false in human-amorous tears;
Nor whom her beardless apple-arbiter
Decided fairest. Rather, O ye Gods,
Poet-like, as the great Sicilian called
Calliope to grace his golden verse —
Ay, and this Kypris also — did I take
That popular name of thine to shadow forth
The all-generating powers and genial heat
Of Nature, when she strikes thro’ the thick blood
Of cattle, and light is large, and lambs are glad
Nosing the mother’s udder, and the bird
Makes his heart voice amid the blaze of flowers;
Which things appear the work of mighty Gods.

“The Gods! and if I go my work is left
Unfinish’d — if I go. The Gods, who haunt
The lucid interspace of world and world,
Where never creeps a cloud, or moves a wind,
Nor ever falls the least white star of snow
Nor ever lowest roll of thunder moans,
Nor sound of human sorrow mounts to mar
Their sacred everlasting calm! and such,
Not all so fine, nor so divine a calm
Not such, nor all unlike it, man may gain
Letting his own life go. The Gods, the Godsl
If all be atoms, how then should the Gods
Being atomic not be dissoluble,
Not follow the great law? My master held
That Gods there are, for all men so believe.
I prest my footsteps into his, and meant
Surely to lead my Memmius in a train
Of fiowery clauses onward to the proof
That Gods there are, and deathless. Meant? I meant?
I have forgotten what I meant, my mind
Stumbles, and all my faculties are lamed.

“Look where another of our Gods, the Sun
Apollo, Delius, or of older use
All-seeing Hyperion — what you will —
Has mounted yonder; since he never sware,
Except his wrath were wreak’d on wretched man,
That he would only shine among the dead
Hereafter — tales! for never yet on earth
Could dead flesh creep, or bits of roasting ox
Moan round the spit — nor knows he what he sees;
King of the East altho’ he seem, and girt
With song and flame and fragrance, slowly lifts
His golden feet on those empurpled stairs
That climb into the windy halls of heaven
And here he glances on an eye new-born,
And gets for greeting but a wail of pain;
And here he stays upon a freezing orb
That fain would gaze upon him to the last;
And here upon a yellow eyelid fallen
And closed by those who mourn a friend in vain,
Not thankful that his troubles are no more.
And me, altho’ his fire is on my face
Blinding, he sees not, nor at all can tell
Whether I mean this day to end myself.
Or lend an ear to Plato where he says,
That men like soldiers may not quit the post
Allotted by the Gods. But he that holds
The Gods are careless, wherefore need he care
Greatly for them, nor rather plunge at once,
Being troubled, wholly out of sight, and sink
Past earthquake — ay, and gout and stone, that break
Body toward death, and palsy, death-in-life,
And wretched age — and worst disease of all,
These prodigies of myriad nakednesses,
And twisted shapes of lust, unspeakable,
Abominable, strangers at my hearth
Not welcome, harpies miring every dish,
The phantom husks of something foully done,
And fleeting thro’ the boundless universe,
And blasting the long quiet of my breast
With animal heat and dire insanity?

“How should the mind, except it loved them, clasp
These idols to herself? or do they fly
Now thinner, and now thicker, like the flakes
In a fall of snow, and so press in, perforce
Of multitude, as crowds that in an hour
Of civic tumult jam the doors, and bear
The keepers down, and throng, their rags and the
The basest, far into that council-hall
Where sit the best and stateliest of the land?

“Can I not fling this horror off me again,
Seeing with how great ease Nature can smile
Balmier and nobler from her bath of storm,
At random ravage? and how easily
The mountain there has cast his cloudy slough,
Now towering o’er him in serenest air,
A mountain o’er a mountain, — ay, and within
All hollow as the hopes and fears of men?

“But who was he that in the garden snared
Picus and Faunus, rustic Gods? a tale
To laugh at — more to laugh at in myself —
For look! what is it? there? yon arbutus
Totters; a noiseless riot underneath
Strikes through the wood, sets all the tops quivering —;
The mountain quickens into Nymph and Faun,
And here an Oread — how the sun delights
To glance and shift about her slippery sides,
And rosy knees and supple roundedness,
And budded bosom-peaks — who this way runs
Before the rest! — a satyr, a satyr, see,
Follows; but him I proved impossible
Two-natured is no nature. Yet he draws
Nearer and nearer, and I scan him now
Beastlier than any phantom of his kind
That ever butted his rough brother-brute
For lust or lusty blood or provender.
I hate, abhor, spit, sicken at him; and she
Loathes him as well; such a precipitate heel,
Fledged as it were with Mercury’s ankle-wing,
Whirls her to me —; but will she fling herself
Shameless upon me? Catch her, goatfoot! nay,
Hide, hide them, million-myrtled wilderness,

And cavern-shadowing laurels, hide! do I wish —
What? —; that the bush were leafless? or to whelm
All of them in one massacre? O ye Gods
I know you careless, yet, behold, to you
From childly wont and ancient use I call —
I thought I lived securely as yourselves —
No lewdness, narrowing envy, monkey-spite,
No madness of ambition, avarice, none;
No larger feast than under plane or pine
With neighbors laid along the grass, to take
Only such cups as left us friendly-warm,
Affirming each his own philosophy
Nothing to mar the sober majesties
Of settled, sweet, Epicurean life.
But now it seems some unseen monster lays
His vast and filthy hands upon my will,
Wrenching it backward into his, and spoils
My bliss in being; and it was not great,
For save when shutting reasons up in rhythm,
Or Heliconian honey in living words,
To make a truth less harsh, I often grew
Tired of so much within our little life
Or of so little in our little life —
Poor little life that toddles half an hour
Crown’d with a flower or two, and there an end —
And since the nobler pleasure seems to fade,
Why should I, beastlike as I find myself,
Not manlike end myself? — our privilege —;
What beast has heart to do it? And what man
What Roman would be dragg’d in triumph thus?
Not I; not he, who bears one name with her
Whose death-blow struck the dateless doom of kings,
When, brooking not the Tarquin in her veins,
She made her blood in sight of Collatine
And all his peers, flushing the guiltless air,
Spout from the maiden fountain in her heart.
And from it sprang the Commonwealth, which breaks
As I am breaking now!

“And therefore now
Let her, that is the womb and tomb of all
Great Nature, take, and forcing far apart
Those blind beginnings that have made me man,
Dash them anew together at her will
Thro’ all her cycles — into man once more,
Or beast or bird or fish, or opulent flower.
But till this cosmic order everywhere
Shatter’d into one earthquake in one day
Cracks all to pieces, — and that hour perhaps
Is not so far when momentary man
Shall seem no more a something to himself,
But he, his hopes and hates, his homes and fanes
And even his bones long laid within the grave,
The very sides of the grave itself shall pass,
Vanishing, atom and void, atom and void,
Into the unseen for ever, — till that hour,
My golden work in which I told a truth
That stays the rolling Ixionian wheel,
And numbs the Fury’s ringlet-snake, and plucks
The mortal soul from out immortal hell
Shall stand. Ay, surely; then it fails at last
And perishes as I must, for O Thou
Passionless bride, divine Tranquillity,
Yearn’d after by the wisest of the wise
Who fail to find thee, being as thou art
Without one pleasure and without one pain,
Howbeit I know thou surely must be mine
Or soon or late, yet out of season, thus
I woo thee roughly, for thou carest not
How roughly men may woo thee so they win —;
Thus — thus — the soul flies out and dies in the air.”

With that he drove the knife into his side.
She heard him raging, heard him fall, ran in,
Beat breast, tore hair, cried out upon herself
As having fail’d in duty to him, shriek’d
That she but meant to win him back, fell on him
Clasp’d, kiss’d him, wail’d. He answer’d, “Care not thou!
Thy duty? What is duty? Fare thee well!”

Seaing Is Beleafing

The Leafy Seadragon, Phycodurus eques, from southern Australia (Wikipedia)


Post-Performative Post-Scriptum

Wikipedia says: “The generic name is derived from the Ancient Greek words φῦκος (phûkos) ‘seaweed’, and δέρμα (dérma) ‘skin’.” That’s obviously wrong for dérma, but I can’t find a definitive etymology.

Doctrīna Angelica

Die Ros’ ist ohn warumb; sie blühet weil sie blühet. — Angelus Silesius (1624-77)

La rosa es sin porqué; florece porque florece — Borges’ translation

La rose est sans pourquoi ; elle fleurit parce qu’elle fleurit.

La rosa è senza perché; fiorisce perché fiorisce.

De roos is zonder waarom; zij bloeit omdat zij bloeit.

الوَرْدَةُ بلا لِمَ؛ تَزْهُو لأنَّها تَزْهُو.

玫瑰无问缘由,只因绽放而绽放。

薔薇は「なぜ」を持たない。ただ咲くから咲く。

ვარდს „რატომ“ არ გააჩნია; ის ყვავის, რადგან ყვავის.

Rosa est sine cur; floret quia floret.

Τὸ ῥόδον ἄνευ τοῦ διὰ τί· ἀνθεῖ, ὅτι ἀνθεῖ.

पुष्पं निर्‍हेतुकम्; पुष्पति यतः पुष्पति।

The rose is without a Why: it blooms because it blooms.

Middlemoth

I’ve never read Middlemarch (1871). But I have seen a middlemoth. It was when I was looking at a new way of creating fractLs. A fractL is what I call a graph shaped like a capital L, with the x- and y-axes representing values between 0 and 1, like 1/2 and 1/3 and 8/55. You can also use numbers > 1 to create numbers < 1: 73 → 0.73; 128719 → 0.128719; and so on. But I decided to reverse the integer before converting it: 73 → 0.37; 128719 → 0.917821; and so on. And use different bases for the x- and y-axes. So that’s what I did on a fractL: I mapped fractions converted from integers in one base against fractions converted from integers in another base. The results, as you can see, were spectacularly dull:

fractL for int→frac in base 2 and base 6


fractL for int→frac in base 3 and base 6


fractL b04, b06


fractL b06, b08


So I decided to try some perspectivision, mapping the integer-fractions not on a fractL but on a fractO instead. A fractO is a circle where you find a point inside the circle by using two fractions, fr1 and fr2, to create two radian values: θ1 = fr1 * 2 * π and θ2 = fr2 * 2 * π. Then you use θ1 and θ2 to find two points on the perimeter of the circle, (x1, y1) and (x2, y2), and then find their midpoint, (x3, y3) = ((x1, y1) + (x2, y2)) / 2. The results this time are much more pleasing on the eye:

fractO for integers in base 2 and base 6

fractL b02, b06, for fractO b02, b06


Here’s an animated gif showing the conversion from visually dull fractL to visually interesting fractO:

fractL b02b06 to fractO b02b06 (animated at EZgif)


When I was looking at more fractOs, I found one that was lepidopterally interesting too:

fractO b09, b12 with middlemoth


fractO b09, b12 (middlemoth in green)


You can try spotting more pareidolia in more fractOs from reversed fractintegers:

fractO b03, b15

fractL b03, b15 for fractO above


fractL b03b15 to fractO b03b15 (animated at EZgif)


fractO b02, b10


fractO b02, b12


fractO b02, b14


fractO b03, b06


fractO b03, b12


fractO b03, b21


fractO b04, b06


fractO b04, b20


fractO b06, b08


fractO b09, b15


fractO b10, b24


fractO b12, b16


fractO b15, b20


fractO b24, b28


fractO b42, b78


fractO b02, b18 (fr2 x 3)


fractO b02, b06 (fr2 x 3)

Poetry in Mocean

The sea, to be sure, is a large department; and that is how it succeeded in attracting Swinburne’s attention; for he seldom noticed any object of external nature unless it was very large, very brilliant, or very violently coloured. But the sea as an object of poetry is somewhat barren. Those poets who have a true eye for nature and a sure pen for describing it, spend few words describing the sea; and their few words describe it better than Swinburne’s thousands. It is historically certain that he had seen the sea, but if it were not, it could not with certainty have been inferred from his descriptions: they might have been written by a man who had never been outside Warwickshire. Descriptions of nature equally accurate, though not equally eloquent, have actually been composed by persons blind from their birth, merely by combining anew the words and phrases which they have had read to them from books. When Swinburne writes thus –

And the night was alive and anhungered of life as a tiger from toils cast free:
And a rapture of rage made joyous the spirit and strength of the soul of the sea.
All the weight of the wind bore down on it, freighted with death for fraught:
And the keen waves kindled and quickened as things transfigured or things distraught.
And madness fell on them laughing and leaping; and madness came on the wind:
And the might and the light and the darkness of storm were as storm in the heart of Ind.
Such glory, such terror, such passion, as lighten and harrow the far fierce East,
Rang, shone, spake, shuddered around us: the night was an altar with death for priest –

it would be cruel to set against such a passage a single line of Tennyson’s or a single epithet of Shakespeare’s: I take instead a snatch of verse whose author few of you know and most of you never heard of:

Hurry me, Nymphs, O, hurry me
Far above the grovelling sea,
Which, with blind weakness and bass roar
Casting his white age on the shore,
Wallows along that slimy floor;
With his wide-spread webbèd hands
Seeking to climb the level sands,
But rejected still to rave
Alive in his uncovered grave.

[From George Darley’s “Nepenthe” (1835)]

Admirers of the sea may call that a lampoon or a caricature, but they cannot deny that it is life-like: the man who wrote it had seen the sea, and the man who reads it sees the sea again.

Poems and Brickbats — A.E. Housman on A.C. Swinburne

Altars of Mathness

What could be duller than digits? They just sit there on the page or screen, mindlessly marking mathematics:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100…

But perhaps they become more interesting as images. Let’s display the final digit of the integers, or counting numbers, on a graph. Running left-right and up-down, the graph represents the final or rightmost digit of 1, 2, 3, … 10, 11, 12, 13, … 100, 101, 102, 103, …, 1000, 1001, 1002, 1003, …:

Rightmost single digit of the integers (click for larger)


No, that’s still dull: the graph just generates endlessly repeating triangles. After all, the final digits fall into a cycle: 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3… So do the final two digits: 1, 2, 3, 4, 5, […] 94, 95, 96, 97, 98, 99, 00, 01, 02, 03… Here they are as a graph:

Rightmost two digits of the integers


Now the triangles look like waves sweeping to shore. That’s a bit more interesting, but not much. So let’s try something different. The trailing digits of the integers generate triangles, so let’s see what the triangular numbers generate. The triangular numbers — 0, 1, 3, 6, 10, 15, 21… — are very simple to form. You just sum the integers: 1, 3 = 1 + 2, 6 = 1 + 2 + 3, 10 = 1 + 2 + 3 + 4, 15 = 1 + 2 + 3 + 4 + 5, 21 = 1 + 2 + 3 + 4 + 5 + 6, 28 = 1 + 2 + 3 + 4 + 5 + 6 + 7, 36 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8, 45 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9, 55 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10… Here are the final digits of the triangulars — 1, 3, 6, 0, 5, 1, 8, 6… — as a graph:

Final digit of triangular numbers in base 10 (click for larger)


Now something interesting has appeared. The final digits form a repeated palindromic pattern (counting 0 as the zero-th triangular number):

0, 1, 3, 6, 0, 5, 1, 8, 6, 5, 5, 6, 8, 1, 5, 0, 6, 3, 1, 0, 0, 1, 3, 6, 0, 5, 1, 8, 6, 5, 5, 6, 8, 1, 5, 0, 6, 3, 1, 0, 0, 1, 3, 6, 0, 5, 1, 8, 6, 5, 5, 6, 8, 1, 5, 0, 6, 3, 1, 0, 0, 1, 3, 6, 0, 5, 1, 8, 6, 5, 5, 6, 8, 1, 5, 0, 6, 3, 1, 0, …

An Altar of Mathness created by the final digit of triangular numbers in base 10


And those palindromic digits create symmetric shapes that remind me of little altars — let’s call them “altars of mathness” in tribute to Morbid Angel’s genre-defining album Altars of Madness (1989). And what about the final two digits of the triangular numbers? Here’s the graph (adjusted so that 99 fits into the same space as 9):

Final two digits of triangulars in b10


Final two triangular digits in b10 (horizontal scale compressed)


The final two digits form palindromes too. And this time we don’t get just triangles, but curves too. But that’s in base 10. What happens with the trailing triangular digits in other bases? Well, here’s the final triangular digit creating more altars of mathness in different bases (note that the altars are more elaborate in even bases):

Final triangular digit in base 4


Final triangular digit in b5


Final triangular digit in b6


Final triangular digit in b7


Final triangular digit in b8


Final triangular digit in b9


Final triangular digit in b14


And here’s the graph for the final triangular digit in base 100:

Final triangular digit in b100


The graph for final single digit in b100 should look familiar, because it’s identical to the graph for final double triangular digits in b10:

Final two digits of triangulars in b10


That’s because two digits in b10 are equivalent in one digit in b100, four digits in b10 are equivalent to two digits in b100, and so on. But b100 can’t capture three digits in b10 (the graph is again adjusted so that 999 fits into the same space as 9 and 99 above):

Final three triangular digits in b10


If you compress the x-axis for that graph, you can see how long the symmetries are:

Final three triangular digits in b10 (x-axis / 2)


Final three triangular digits in b10 (x-axis / 4)


The final four digits of the triangulars in b10 create even longer symmetries:

Final quadruple triangular digits in b10


Final quadruple triangular digits in b10 (x-axis / 2)


Final quadruple triangular digits in b10 (x-axis / 8)


Note how, as the length of the final digits rises, you need to compress the x-axis more and more to see the symmetries. But integer sequences obviously don’t end with the counting numbers and triangulars. What about squares and powers of n? What about primes and Fibonacci numbers? Here’s the final two digits of the squares — 1, 4, 9, 16, 25, 49, 64, 81, 100, 121, 144, 169… — in b10:

Final two digits of the squares in b10


It’s reminiscent of the triangular numbers (so are the final-digit graphs for other polygonal numbers). So what about the powers of 2? That’s 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024… Here’s the graph for final single digits of 2^p in b10:

Final single digits of 2^p in b10


This time there’s repetition, but not symmetry. Here’s the graph for final double digits, or 2-digits, of 2^p in b10:

Final 2-dig of 2^p in b10


Now the graph looks a little like a range of eroded mountains. Now try dig-4, the final four digits of 2^p in b10:

Final 4-dig of 2^p in b10


The patterns are similar to those of dig-2 and don’t need compressing in the x-axis. This similarity and lack of need for compression are true of any number of final digits in 2^p. The final 10 digits look like this:

Final 10-dig of 2^p in b10


And the final 20 and 30 digits like this:

Final 20-dig of 2^p in b10


Final 30-dig of 2^p in b10


Powers don’t behave like polygonals: the finals are fractals. That is, the final digits create similar patterns at all scales: 1-dig, 2-dig, 10-dig, 100-dig, 1000-dig and so on. That’s true in other bases:

Final 5-dig of 3^p in b2


But a glimpse of b2 is all you’re going to get of other bases. There are other fish to fry — Fibonacci fish. The Fibonacci sequence, whose terms are equal to the sum of the previous two numbers (after seeding with “1, 1”), starts like this: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811… And what about the graphs for final fib-digits? As you’ll see, final Fib-digits are fractal too. Indeed, Fibonacci final-graphs look like 2-power final-graphs (in a way, Fibonacci numbers are powers of φ = 1.6180339887498948482…). The patterns are similar at all scales. And they remind me of the skyline of a ruined city in an Oriental tale, with collapsed domes and crumbling minarets:

Final 1-dig of Fibonacci numbers in b10


Final 2-fibdig in b10


Final 3-fibdig in b10


Final 4-fibdig in b10


Final 5-fibdig in b10


Final 10-fibdig in b10


Final 15-fibdig in b10


Final 20-fibdig in b10


Final 25-fibdig in b10


So final fibdigs are fractal. But final prime digits aren’t:

Final 1-digit of primes in b10


Final 1-digit of primes in b5


Final 2-digit of primes in b10


Primes aren’t final-digitally fractal like Fibonaccis and powers of 2. But there’s occasional symmetry in the prime fin-digs. I’ve marked some palindromic patterns in red and green:

Palindromic patterns in final 1-digits of the primes in b10 (click for larger)


The palindromic patterns, or pal-pats, in the primes look like the altars of mathness in the triangulars. They’re created by digital palindromes like these:

19, 23, 29 (c=3)
347, 349, 353, 359, 367 (c=5)
937, 941, 947, 953, 967, 971, 977 (c=7)
1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011 (c=9)
26423, 26431, 26437, 26449, 26459, 26479, 26489, 26497, 26501, 26513 (c=10)


Here are the first few pal-pats in the primes (note that 157, 163, 167 and 163, 167, 173 overlap):

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607…

And are there palindromes among the final 2-digits, 3-digits and higher n-digits of the primes in different bases? Yes, you can easily find some. But I haven’t put them on a graph yet:

Base 10 (2-dig)

58789, 58831, 58889 (c=3)
286873, 286927, 286973 (c=3)
360649, 360653, 360749 (c=3)
404851, 404941, 404951 (c=3)
590437, 590489, 590537 (c=3)
623071, 623107, 623171 (c=3)
651517, 651587, 651617 (c=3)


Base 6 (2-dig)

300335, 300401, 300441, 300501, 300535 (c=5) (23459 to 23531 in base 10)
1030255, 1030331, 1030351, 1030431, 1030455 (c=5) (50651 to 50723 in b10)
1140451, 1140501, 1140521, 1141001, 1141051 (c=5) (59791 to 59863 in b10)
1402451, 1402545, 1403031, 1403045, 1403051 (c=5) (78367 to 78439 in b10)
1435431, 1435451, 1435505, 1435551, 1440031 (c=5) (82891 to 82963)
2400505, 2401001, 2401015, 2401101, 2401105 (c=5) (124601 to 124673)
2442235, 2442311, 2442351, 2442411, 2442435 (c=5) (130127 to 130199)
2444215, 2444225, 2444311, 2444325, 2444415 (c=5) (130547 to 130619)
2533105, 2533121, 2533215, 2533221, 2533305 (c=5) (136769 to 136841)


Base 4 (3-dig)

20013013, 20013133, 20020013 (c=3) (33223 to 33287 in base 10)
21031111, 21031303, 21032111 (c=3) (37717 to 37781)
22310011, 22310333, 22311011 (c=3) (44293 to 44357)
33030121, 33031001, 33031121 (c=3) (62233 to 62297)
102031333, 102032131, 102032333 (c=3) (74623 to 74687)
110013121, 110013311, 110020121 (c=3) (82393 to 82457)


Base 3 (3-dig)

112121020012, 112121021211, 112121022021, 112121100211, 112121101012 (c=5) (287393 to 287501 in base 10)
202002212002, 202002212101, 202002220001, 202002221101, 202010000002 (c=5) (395741 to 395849)
1001012111212, 1001012112202, 1001012121022, 1001012121202, 1001012122212 (c=5) (555143 to 555251)
1010112112012, 1010112112201, 1010112120222, 1010112121201, 1010112200012 (c=5) (601079 to 601187)
1011202211211, 1011202212212, 1011202220022, 1011202221212, 1011202222211 (c=5) (625369 to 625477)


Base 2 (5-dig)

101110001111101, 101110010000111, 101110010001001, 101110010100111, 101110010111101 (c=5) (23677 to 23741 in base 10)
10000001000111101, 10000001001011001, 10000001001110001, 10000001001111001, 10000001001111101 (c=5) (66109 to 66173)
10111100110011011, 10111100110011111, 10111100110111001, 10111100110111111, 10111100111011011 (c=5) (96667 to 96731)
11010000111110001, 11010001000001101, 11010001000011001, 11010001000101101, 11010001000110001 (c=5) (106993 to 107057)

And I conjecture that you’ll get palindromes for any number of final digits in all bases. And can these palindromes be of arbitrary length? Again, I conjecture so. There are infinitely many primes and very rare patterns can occur infinitely often in an infinite set of numbers.


Post-Performative Post-Scriptum

Here’s Dan Seagrave’s classic cover for Morbid Angel’s Altars of Madness (1989):


Morbid Angel — official website
Dan Seagrave — official website


Elsewhere Other-Accessible…

Formulas Focal to the Flesh — a pre-previous post paronomasizing the title of a Morbid-Angel album…

Red Sails in the Subset

Let’s look at a simple arithmetical rule and a simple arithmetical fact. And the complexity they can create. First the rule. Subtracting a negative number is the same as adding the positive form of that number:

7 – +2 = 5
7 – -2 = 7 + 2 = 9

-10 – +4 = -14
-10 – -4 = -10 + 4 = -6

Now the simple arithmetical fact: The reciprocal of positive x, namely 1/x, is less than 1 when x > 1, identical to x when x = 1, and greater than 1 when 0 < x < 1. Negative x, -x, works in the opposite direction:

1/5 = 0.2; 1/-5 = -0.2
1/4 = 0.25; 1/-4 = -0.25
1/3 = 0.333333…; 1/-3 = -0.333333…
1/2 = 0.5; 1/-2 = -0.5

1/1 = 1; 1/-1 = -1

1/0.5 = 2; 1/-0.5 = -2
1/0.25 = 4; 1/-0.25 = -4
1/0.333333.. = 3; 1/-0.333333.. = -3
1/0.2 = 5; 1/-0.2 = -5

Now, the simple arithmetical rule and the simple arithmetical fact explain the wildly different behaviour of these two nearly identical formulae:

Formula #1: x = x + 1/x
Formula #2: x = x – 1/x

If you seed x = x + 1/x with 2, this is what happens:

2 = x
2.5 = 2 + 1/2 = 2 + 0.5
2.9 = 2.5 + 1/2.5 = 2.5 + 0.4
3.244827586206896551724137931… = 2.9 + 1/2.9 = 2.9 + 0.3448275862…
3.553010370478947561288431236…
3.834461842815967366750790750…
4.095254632258778985771918456…
4.339439692724345181049239663…
4.569884190357676650018985962…
4.788708116379690742064597208…
4.997532704493448986664559639…
5.197631445038131469095668466…
5.390026771750770995914851381…
5.575554607204394029915651664…
5.754908962142979073283550015…
5.928673657045750549124213874…
6.097345447373015508408978797…
6.261351244425377152997703626…
6.421061179383957004641284553…
6.576798676981813718180627345…

The value of x steadily (but more and more slowly) increases. But when you seed the other formula, x = x – 1/x, with 2, this is what happens:

+2
+1.5 = 2 – 1/2 = 2 – 0.5
+0.8333333… = 1.5 – 1/1.5 = 1.5 – 0.666666…
-0.3666666… = 0.8333333… – 1/0.8333333… = 0.8333333… – 1.2
+2.3606060606… = -0.3666666… – 1/-0.3666666… = -0.3666666…-2.72727272… = -0.3666666… + 2.72727272…
+1.936986034932119656124790913…
+1.420720051612810742016492942…
+0.716851616121389735975863550…
-0.678137217705362317788764881…
+0.796490591963802485322149292…
-0.459017018658980935029501857…
+1.719551442531198550688634398…
+1.138004432499332885157841729…
+0.259273233005005595158072588…
-3.597661740227243739940039228…
-3.319703423907923593779727545…
-3.018471695555874174383708009…
-2.687178213005645221877765061…
-2.315040631969854351245993463…
-1.883082770759830608578236571…
-1.352038668223383718148747858…
-0.612414851610188982350276645…

+1.020465208974159220420492697…
+0.040519992610273807119693182…
-24.63865528804984441050796942…
-24.59806865747650234381633987…
-24.55741505926418687092326558…
-24.51669416101057552476382150…
-24.47590562755526483917345018…
-24.43504912094763238695385804…

The value of x swings between positive and negative in an irregular, non-periodic way, alternating between slow deterministic decay and instantaneous jumps to sometimes large positive or negative values. The deterministic decays explains why, as we’ll see, there are beautiful regular curves — parabolic curves — amid the irregularity. When the function creates a positive number x > 1, it nibbles away at x until x x > -1, x becomes positive at the next step and the process continues. Represented as a graph, x = x – 1/x looks like this when seeded with 2 — note the parabolic curves:

x[i] = x[i-1] – 1/x[i-1], x[1] = 2 (click for larger)


A shark-fin and some red sails (images StockCake + Para-Sailing World Championship)


Sydney Opera House (image Wikipedia)


When x > 0, its value is represented in white; when x < 0, its value is represented in red. The curves created remind of me of shark-fins or sails or Sydney Opera House. So you could say the graph contains red sails in the subset, i.e. the set of values of x that are sub-zero. Here are some variations on the formula:

x = x – (1/4)/x, x[1] = 2


x = x – (4/3)/x, x[1] = 2


x = x – (4/5)/x, x[1] = 2


Now try this formula, x = 1 – 1/x. When it’s seeded with 2, it behaves like this:

2
0.5 = 1 – 1/2 = 1 – 0.5
-1 = 1 – 1/0.5 = 1 – 2
2 = 1 – -1/-1 = 1 – -1 = 1 + 1
1/2
-1
2
[…]

The values cycles through 2, 0.5, -1, 2, 0.5… for ever. So try varying the formula. This is what happens with x = 0.1 – 1.7/x, seeded with 2:

+2
-0.75
+2.366666666666666666666666666…
-0.618309859154929577464788732…
+2.849430523917995444191343963…
-0.496610440482852346310656327…
+3.523206323143542441364433927…
-0.382515028663775083373274222…
+4.544269826308639632084352464…
-0.274097504104620631734792447…
+6.302172491695235560374503419…
-0.169748249867834571832745868…
+10.11483079397645866692882142…
-0.068070038404634195480677189…
+25.07427708053510247498559239…

When you look at the graph of x = 0.1 – 1.7/x, you’ll see it’s also cycling, just in a more complicated way:

x = 0.1 – 1.7/x, x[1] = 2 (click for larger)


And here’s how different seeds can change the graph:

x = 2/3 – 1/x, x[1] = 2/3


x = 2/3 – 1/x, x[1] = 3/2


This graph reminds me of vertebrae:

x = 2/5 – 1/x, x[1] = 2


And this graph reminds of a bone:

x = 9/7 – 1/x, x[1] = 2


As Lucretius nearly said: Mathematica Moles et Machina Mundi — Mathematics is the Mass and Body of the World.


Elsewhere Other-Accessible…

Moto-Motto — what Lucretius did say

Das Fing an Sich

finger

A word inherited from Germanic.

Cognate with Old Frisian finger, Old Saxon fingar (Middle Low German finger), Old Dutch fingar (Middle Dutch, Dutch vinger), Old High German fingar (Middle High German vinger, German Finger), Old Icelandic fingr, Old Swedish finger (Swedish finger), Old Danish fingær (Danish finger), Gothic figgrs.

Further etymology uncertain, perhaps < a suffixed form of the Indo-European base of five adj., cognate with Old Frisian fīf (West Frisian fiif), Old Saxon fīf (Middle Low German vīf), Old Dutch fīf (Middle Dutch, Dutch vijf), Old High German fimf, finf, funf (Middle High German vünf, German fünf  (although this presents semantic difficulties with regard to the function of the suffix), or perhaps < a suffixed form of the Indo-European base of fang v., Old English fón, reduplicated strong verb corresponding to Old Frisian , Old Saxon fâhan, Old High German fâhan (Middle High German vâhen, modern German (poet) fahen), Old Norse  (although this presents phonological difficulties).

Compare fist n., Old English fýst strong feminine corresponds to Old Frisian fêst, Middle Low German fûst (Dutch vuist), Old High German fûst (Middle High German vûst, modern German Faust) < West Germanic *fûsti.

fist

A word inherited from Germanic.

Old English fýst strong feminine corresponds to Old Frisian fêst, Middle Low German fûst (Dutch vuist), Old High German fûst (Middle High German vûst, modern German Faust) < West Germanic *fûsti.

Notes

By some scholars this is referred to an Old Germanic form *fûhsti-z, *funhsti-z < pre-Germanic *pṇqstis (whence Old Church Slavonic pęstĭ of same meaning), < ablaut-variant of *penqe, five adj. & n., cognate with Old Frisian fīf (West Frisian fiif), Old Saxon fīf (Middle Low German vīf), Old Dutch fīf (Middle Dutch, Dutch vijf), Old High German fimf, finf, funf (Middle High German vünf).

• from the Oxford English Dictionary

Strartifacts

Here’s a sequence of decreasing numbers. Which number comes next?

612 → 600 → 594 → 414 → 398 → 182 → ?

It’s 166, because the numbers decrease by the product of their digits higher than 0:

612 – 6*2 = 612 – 12 = 600 → 600 – 6 = 594 → 594 – 5*9*4 = 594 – 180 = 414 → 414 – 4*4 = 414 – 16 = 398 → 398 – 3*9*8 = 398 – 216 = 166

Eventually the sequence will reached 0 and stop. If you want to see how this function looks on a graph, here it is:

x = n <= 3722, f(i) -= digmul(f(i)) → 0 (click for larger)


The graph represents n on the x-axis, with the red circles marking n = 100 and n = 1000. The sequence of falling digit-products is on the y-axis, but the graph has a special feature there. The y-axis is compressed according to the size of n, so that n = 1000 falls to 0 with n -= digmul(n) in the same height as n = 100. Here’s a graph for the same function in base 7:

x = n <= 3722 in base 7, f(i) -= digmul(f(i)) → 0


Now the red circles represent 7^2 = 49, 7^3 = 343, 7^4 = 2401, i.e. 100b7, 1000b7, 10000b7. And you can try other functions for n = n – func(n) = n -= func(n). Here’s a graph for n -= hailstep(n), where hailstep(n) returns the number of steps in the Collatz sequence for n:

x = n <= 3722 in base 7, f(i) -= hailstep(f(i)) → f(i) < 2


You form a Collatz sequence by starting with a whole number and finding the next number according to two rules:

1. If n(i) is divisible by 2, n(i+1) = n(i) / 2
2. If n(i) is not divisible by 2, n(i+1) = n(i) * 3 + 1

So the Collatz sequence for n = 10 looks like this:

10 → 10 / 2 = 5 → 5 * 3 + 1 = 16 → 16 / 2 = 8 → 8 / 2 = 4 → 4 / 2 = 2 → 2 / 2 = 1.

When you reach 1, you stop. So that’s six steps for n = 10. But does every n reach 1 in the end? It’s a very simple question about a very simple function. But nobody knows and nobody can prove that either all numbers do or at least one number doesn’t. The German mathematician Lothar Collatz (1910-90) conjectured that all numbers do reach 1. But it can take a surprisingly long time, even with small n. This is the Collatz sequence for n = 27:

27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1

Now some more functions for the y-compressed fall-bands, as I call them. If you use the sum of the factors * powers, you get this:

x = n <= 7422, f(i) -= factpowsum(f(i)) → f(i) < 2


The factpowsum(n) is the sum of the factors multiplied by their powers. For example, 37692 = 2^2 * 3^3 * 349, so factpowsum(4188) = 2*2 + 3*3 + 349*1 = 362. Here’s factpowsum for more n:

x = n <= 14822, f(i) -= factpowsum(f(i)) → f(i) < 2


You can also use the very simple function f(i) -= 1, that is, compress the numbers from n to 1 into the y-gap. But if you do that, you’ll get a completely filled screen:

x = n <= 3722, f(i) -= 1 → f(i) = 0


So you can adjust the color of a pixel according to how many times it’s written to:

x = n <= 1862, f(i) -= 1 → f(i) = 0 (color-adjust)


The patterns in the colors are artifacts of the limited resolution of the screen, so I call these patterns strartifacts = strata + artifacts. Here’s another example:

x = n <= 3722, f(i) -= 1 → f(i) = 0 (color-adjust)


Or adjust the greytone of the pixel:

x = n <= 3722, f(i) -= 1 → f(i) = 0 (greytone-adjust)


And so on (in all cases, you can click for a larger image):

x = n <= 1862, f(i) -= blockmul(f(i)) (multiply run-lengths of same digits) → f(i) < 2


x is triangular(n) = 3 to 1734453, y is 1 < triangular numbers <= n


x = n <= , f(i) -= 1 → f(i) < 2


x = n <= 7442, f(i) -= 1 → f(i) < 2


x = n <= 1862, f(i) -= leaddig(f(i)) → f(i) < 2 # 1


x = n <= , f(i) -= leaddig(f(i)) → f(i) < 2 # 2


x = n <= , f(i) -= trailingdigit(f(i)) + 1 → f(i) < 2


x = n <= , f(i) -= trailingdigit(f(i) in base 5) + 3 → f(i) < 2


for triangular(n) = 3 to 6928503, f(i) -= primes → f(i) < 2



x = n <= 1862, f(i) -= blockmul(f(i) in base 5) (multiply run-lengths of same digits) → f(i) < 2


x = n <= 1862, f(i) -= blockmul(f(i) in base 2) → f(i) < 2


x = n <= 1862, f(i) -= digsum(f(i)) → f(i) < 2


x = n <= 3722, f(i) -= func(x = 1/4 → x < 0, x(1) = 4) → f(i) < 2


x = n <= 932, f(i) -= func(x -= 3/x → x < 0, x(1) = 6) → f(i) < 2


Sieve and Let Spi’

What is VDSP? Inter alia, it’s the complicated consonant-cluster you get when you carefully pronounce the phrase “sieved spiral”. And here is a sieved spiral:

An Ulam Spiral of primes represented on a square grid


The pattern above is called an Ulam spiral (OO-lam) after its inventor, the Polish-Jewish mathematician Stanisław Ulam (1909-84). The white squares represent the prime numbers as you spiral counter-clockwise on a square grid — the little boot or reversed-L in the middle is the only time that filled squares are in direct contact, because it includes square #2, the only even prime. #2 is the heel of the boot, with #3 as the shaft and #11 as the toe.

But the Ulam spiral could also be called a sieved spiral, because you can build it by using the Sieve of Erastosthenes, whose invention is attributed to the Greek scholar Eratosthenes of Cyrene (c. 276–c. 194 BC). Create a list of whole numbers skipping 1. Then choose the first number on the list, which is 2. Cross out every higher number that’s divisible by 2. Then choose the next number that isn’t crossed out. It’ll be 3. Cross out every higher number that’s divisible by 3. Then choose the next available number, 5, and cross out all higher numbers divisible by 5. When you’ve crossed out everything you can, you’ll be left with just prime numbers. Here’s an animation of the Sieve from Wikipedia:

Animated Sieve of Erastosthenes from Wikipedia


Now we can sieve-and-let-spi’, as it were. First create a square grid with white squares. Choose the square in the middle as #1 and fill it it. Then choose white square to the right of #1 and call it #2. Then spiral outwards counter-clockwise filling with black all squares whose count is divisible by 2. Then do that for squares #3, #5, #7, #11 and so on. In the end, the only white squares on the grid will be the primes. And you’ll have a sieved Ulam spiral:

Sieving a spiral — creating the Ulam spiral using the Sieve of Eratosthenes


You can also sieve and let spi’ in reverse, blacking the squares using primes from higher to lowest. With this method, the sieved spiral looks like this:

Sieving a spiral — creating the Ulam spiral using the Sieve of Eratosthenes (higher primes first)