“A formal manipulator in mathematics often experiences the discomforting feeling that his pencil surpasses him in intelligence.” — Howard Whitley Eves (1911-2004)
Monthly Archives: May 2021
Multicolore Mondo Macao
Red and yellow maccaw, Macrocercus aracanga, by Edward Lear (1812-1888)
(Open in new window for larger image)
(Now Scarlet Macaw, Ara macao)
Elsewhere other-accessible…
Performativizing Papyrocentricity #68
Papyrocentric Performativity Presents:
• Powerful in Patches – The Kraken Wakes (1953) and The Midwich Cuckoos (1957), John Wyndham
• Twists in the Tale – No Comebacks: Collected Short Stories, Frederick Forsyth (1972)
• A Hundred Heresiarchs – Bowie’s Books: The Hundred Literary Heroes Who Changed His Life, John O’Connell (Bloomsbury 2020)
Posted at Overlord of the Über-Feral
• Chlorokill – The Day of the Triffids, John Wyndham (1951)
Back to Drac’ #2
Boring, dull, staid, stiff, everyday, ordinary, unimaginative, unexceptional, crashingly conventional — the only interesting thing about squares is the number of ways you can say how uninteresting they are. Unlike triangles, which vary endlessly and entertainingly, squares are square in every sense of the word.
And they don’t get any better if you tilt them, as here:
Sub-squares from gray square (with corner-numbers)
Nothing interesting can emerge from that set of squares. Or can it? As I showed in Curvous Energy, it can. Suppose that the gray square is dividing into the colored squares like a kind of amoeba. And suppose that the colored squares divide in their turn. So square divides into sub-squares and sub-squares divide into sub-sub-squares. And so on. And all the squares keep the same relative orientation.
What happens if the gray square divides into sub-squares sq2 and sq9? And then sq2 and sq9 each divide into their own sq2 and sq9? And so on. Something very unsquare-like happens:
Square-split stage #1
Stage #2
Square-split #3
Square-split #4
Square-split #5
Square-split #6
Square-split #7
Square-split #8
Square-split #9
Square-split #10
Square-split #11
Square-split #12
Square-split #13
Square-split #14
Square-split #15
Square-split #16
Square-split (animated)
The square-split creates a beautiful fractal known as a dragon-curve:
Dragon-curve
Dragon-curve (red)
And dragon-curves, at various angles and in various sizes, emerge from every other possible pair of sub-squares:
Lots of dragon-curves
And you get other fractals if you manipulate the sub-squares, so that the corners are rotated or reverse-rotated:
Rotation = 1,2 (sub-square #1 unchanged, in sub-square #2 corner 1 becomes corner 2, 2 → 3, 3 → 4, 4 → 1)
rot = 1,2 (animated)
rot = 1,2 (colored)
rot = 1,5 (in sub-square #2 corner 1 stays the same, 4 → 2, 3 stays the same, 2 → 4)
rot = 1,5 (anim)
rot = 4,7 (sub-square #2 flipped and rotated)
rot = 4,7 (anim)
rot = 4,7 (col)
rot = 4,8
rot = 4,8 (anim)
rot = 4,8 (col)
sub-squares = 2,8; rot = 5,6
sub-squares = 2,8; rot = 5,6 (anim)
sub-squares = 2,8; rot = 5,6 (col)
Another kind of dragon-curve — rot = 3,2
rot = 3,2 (anim)
rot = 3,2 (col)
sub-squares = 4,5; rot = 3,9
sub-squares = 4,5; rot = 3,9 (anim)
sub-squares = 4,5; rot = 3,9 (col)
Elsewhere other-accessible…
• Curvous Energy — a first look at dragon-curves
• Back to Drac’ — a second look at dragon-curves
Guat Da Fack?!
From Intellectual Impostures (1998) by Alan Sokal and Jean Bricmont:
To conclude, let us quote a brief excerpt from the book Chaosmosis, written by Guattari alone. This passage contains the most brilliant melange of scientific, pseudo-scientific, and philosophical jargon that we have ever encountered; only a genius could have written it.
We can clearly see that there is no bi-univocal correspondence between linear signifying links or archi-writing, depending on the author, and this multireferential, multidimensional machinic catalysis. The symmetry of scale, the transversality, the pathic non-discursive character of their expansion: all these dimensions remove us from the logic of the excluded middle and reinforce us in our dismissal of the ontological binarism we criticised previously. A machinic assemblage, through its diverse components, extracts its consistency by crossing ontological thresholds, non-linear thresholds of irreversibility, ontological and phylogenetic thresholds, creative thresholds of heterogenesis and autopoiesis. The notion of scale needs to be expanded to consider fractal symmetries in ontological terms.
What fractal machines traverse are substantial scales. They traverse them in engendering them. But, and this should be noted, the existential ordinates that they “invent” were always already there. How can this paradox be sustained? It’s because everything becomes possible (including the recessive smoothing of time, evoked by Rene Thom) the moment one allows the assemblage to escape from energetico-spatiotemporal coordinates. And, here again, we need to rediscover a manner of being of Being — before, after, here and everywhere else — without being, however, identical to itself; a processual, polyphonic Being singularisable by infinitely complexifiable textures, according to the infinite speeds which animate its virtual compositions.
The ontological relativity advocated here is inseparable from an enunciative relativity. Knowledge of a Universe (in an astrophysical or axiological sense) is only possible through the mediation of autopoietic machines. A zone of self-belonging needs to exist somewhere for the coming into cognitive existence of any being or any modality of being. Outside of this machine/Universe coupling, beings only have the pure status of a virtual entity. And it is the same for their enunciative coordinates. The biosphere and mecanosphere, coupled on this planet, focus a point of view of space, time and energy. They trace an angle of the constitution of our galaxy. Outside of this particularised point of view, the rest of the Universe exists (in the sense that we understand existence here-below) only through the virtual existence of other autopoietic machines at the heart of other bio-mecanospheres scattered throughout the cosmos. The relativity of points of view of space, time and energy do not, for all that, absorb the real into the dream. The category of Time dissolves into cosmological reflections on the Big Bang even as the category of irreversibility is affirmed. Residual objectivity is what resists scanning by the infinite variation of points of view constitutable upon it. Imagine an autopoietic entity whose particles are constructed from galaxies. Or, conversely, a cognitivity constituted on the scale of quarks. A different panorama, another ontological consistency. The mecanosphere draws out and actualises configurations which exist amongst an infinity of others in fields of virtuality. Existential machines are at the same level as being in its intrinsic multiplicity. They are not mediated by transcendent signifiers and subsumed by a univocal ontological foundation. They are to themselves their own material of semiotic expression. Existence, as a process of deterritorialisation, is a specific inter-machinic operation which superimposes itself on the promotion of singularised existential intensities. And, I repeat, there is no generalised syntax for these deterritorialisations. Existence is not dialectical, not representable. It is hardly livable! (Félix Guattari 1995, pp. 50-52)
Multimo Mondo Macca
Strange. But. True. Many keyly committed core components of the counter-cultural community feel a reluctant reverence for core ’60s icon Paul Sir McCartney. Beneath that sentimentally saccharine surface, that merry “Macca” mask, they sense something deeper… darker… dangerouser…
“He ain’t as appallingly unesoteric as he appears, man,” these keyly committed core components of the counter-cultural community mutter meaningly…
I’ve tried to capture something of this Morbid Mac in a series of animated gifs that display Macca mise en abîme or “sent into the abyss” (pronounced “meez on abeem”, roughly speaking). That’s the artistic term for the way some images contain smaller and smaller versions of themselves.
Here’s Macca at stage one:
And stage two:
And further stages:
Here’s a Maccabisso using a bit of negative:
And finally, here’s Macca playing a bit of rock’n’roll…
Multitudinous Marriment
• …ποντίων τε κυμάτων ἀνήριθμον γέλασμα… — Αἰσχύλος, Προμηθεὺς δεσμώτης (c. 479-24 B.C.)
• …of ocean-waves the multitudinous laughter… Prometheus Bound by Aeschylus* at Perseus
• …ever-glittering laughter of the far-thrown waves… (my translation)
See also:
γέλασμα, a laugh, κυμάτων ἀνήριθμον γέλασμα, Keble’s “the many-twinkling smile of Ocean, ” Aesch. — Liddell and Scott
Keble was not a sacred but, in the best sense of the word, a secular poet. It is not David only, but the Sibyl, whose accents we catch in his inspirations. The “sword in myrtle drest” of Harmodius and Aristogeiton, “the many-twinkling smile of ocean” from Æschylus, are images as familiar to him as “Bethlehem’s glade” or “Carmel’s haunted strand.” Not George Herbert, or Cowper, but Wordsworth, Scott, and perhaps more than all, Southey, are the English poets that kindled his flame, and coloured his diction. — John Keble at Penny’s Poetry Pages
One day Mr Gordon had accidentally come in, and found no one there but Upton and Eric; they were standing very harmlessly by the window, with Upton’s arm resting kindly on Eric’s shoulder, as they watched with admiration the network of rippled sunbeams that flashed over the sea. Upton had just been telling Eric the splendid phrase, “anerhithmon gelasma pontou”, which he had stumbled upon in an Aeschylus lesson that morning, and they were trying which would hit on the best rendering of it. Eric stuck up for the literal sublimity of “the innumerable laughter of the sea,” while Upton was trying to win him over to “the many-twinkling smile of ocean.” They were enjoying the discussion, and each stoutly maintaining his own rendering, when Mr Gordon entered. — quote from Frederic W. Farrar’s Eric, or Little by Little (1858) at Sententiae Antiquae
*Or possibly his son Euphorion.
Rollercoaster Rules
n += digsum(n). It’s one of my favorite integer sequences — a rollercoaster to infinity. It works like this: you take a number, sum its digits, add the sum to the original number, and repeat:
1 → 2 → 4 → 8 → 16 → 23 → 28 → 38 → 49 → 62 → 70 → 77 → 91 → 101 → 103 → 107 → 115 → 122 → 127 → 137 → 148 → 161 → 169 → 185 → 199 → 218 → 229 → 242 → 250 → 257 → 271 → 281 → 292 → 305 → 313 → 320 → 325 → 335 → 346 → 359 → 376 → 392 → 406 → 416 → 427 → 440 → 448 → 464 → 478 → 497 → 517 → 530 → 538 → 554 → 568 → 587 → 607 → 620 → 628 → 644 → 658 → 677 → 697 → 719 → 736 → 752 → 766 → 785 → 805 → 818 → 835 → 851 → 865 → 884 → 904 → 917 → 934 → 950 → 964 → 983 → 1003 → 1007 → 1015 → 1022 → 1027 → 1037 → 1048 → 1061 → 1069 → 1085 → 1099 → 1118 → 1129 → 1142 → 1150 → 1157 → 1171 → 1181 → 1192 → 1205 → ...
I call it a rollercoaster to infinity because the digit-sum constantly rises and falls as n gets bigger and bigger. The most dramatic falls are when n gets one digit longer (except on the first occasion):
... → 8 (digit-sum=8) → 16 (digit-sum=7) → ...
... → 91 (ds=10) → 101 (ds=2) → ...
... → 983 (ds=20) → 1003 (ds=4) → ...
... → 9968 (ds=32) → 10000 (ds=1) → ...
... → 99973 (ds=37) → 100010 (ds=2) → ...
... → 999959 (ds=50) → 1000009 (ds=10) → ...
... → 9999953 (ds=53) → 10000006 (ds=7) → ...
... → 99999976 (ds=67) → 100000043 (ds=8) → ...
... → 999999980 (ds=71) → 1000000051 (ds=7) → ...
... → 9999999962 (ds=80) → 10000000042 (ds=7) → ...
... → 99999999968 (ds=95) → 100000000063 (ds=10) → ...
... → 999999999992 (ds=101) → 1000000000093 (ds=13) → ...
Look at 9968 → 10000, when the digit-sum goes from 32 to 1. That’s only the second time that digsum(n) = 1 in the sequence. Does it happen again? I don’t know.
And here’s something else I don’t know. Suppose you introduce a rule for the rollercoaster of n += digsum(n). You buy a ticket with a number on it: 1, 2, 3, 4, 5… Then you get on the rollercoaster powered by with that number. Now here’s the rule: Your ride on the rollercoaster ends when n += digsum(n) yields a rep-digit, i.e., a number whose digits are all the same. Here are the first few rides on the rollercoaster:
1 → 2 → 4 → 8 → 16 → 23 → 28 → 38 → 49 → 62 → 70 → 77
2 → 4 → 8 → 16 → 23 → 28 → 38 → 49 → 62 → 70 → 77
3 → 6 → 12 → 15 → 21 → 24 → 30 → 33
4 → 8 → 16 → 23 → 28 → 38 → 49 → 62 → 70 → 77
5 → 10 → 11
6 → 12 → 15 → 21 → 24 → 30 → 33
7 → 14 → 19 → 29 → 40 → 44
8 → 16 → 23 → 28 → 38 → 49 → 62 → 70 → 77
9 → 18 → 27 → 36 → 45 → 54 → 63 → 72 → 81 → 90 → 99
10 → 11
11 → 13 → 17 → 25 → 32 → 37 → 47 → 58 → 71 → 79 → 95 → 109 → 119 → 130 → 134 → 142 → 149 → 163 → 173 → 184 → 197 → 214 → 221 → 226 → 236 → 247 → 260 → 268 → 284 → 298 → 317 → 328 → 341 → 349 → 365 → 379 → 398 → 418 → 431 → 439 → 455 → 469 → 488 → 508 → 521 → 529 → 545 → 559 → 578 → 598 → 620 → 628 → 644 → 658 → 677 → 697 → 719 → 736 → 752 → 766 → 785 → 805 → 818 → 835 → 851 → 865 → 884 → 904 → 917 → 934 → 950 → 964 → 983 → 1003 → 1007 → 1015 → 1022 → 1027 → 1037 → 1048 → 1061 → 1069 → 1085 → 1099 → 1118 → 1129 → 1142 → 1150 → 1157 → 1171 → 1181 → 1192 → 1205 → 1213 → 1220 → 1225 → 1235 → 1246 → 1259 → 1276 → 1292 → 1306 → 1316 → 1327 → 1340 → 1348 → 1364 → 1378 → 1397 → 1417 → 1430 → 1438 → 1454 → 1468 → 1487 → 1507 → 1520 → 1528 → 1544 → 1558 → 1577 → 1597 → 1619 → 1636 → 1652 → 1666 → 1685 → 1705 → 1718 → 1735 → 1751 → 1765 → 1784 → 1804 → 1817 → 1834 → 1850 → 1864 → 1883 → 1903 → 1916 → 1933 → 1949 → 1972 → 1991 → 2011 → 2015 → 2023 → 2030 → 2035 → 2045 → 2056 → 2069 → 2086 → 2102 → 2107 → 2117 → 2128 → 2141 → 2149 → 2165 → 2179 → 2198 → 2218 → 2231 → 2239 → 2255 → 2269 → 2288 → 2308 → 2321 → 2329 → 2345 → 2359 → 2378 → 2398 → 2420 → 2428 → 2444 → 2458 → 2477 → 2497 → 2519 → 2536 → 2552 → 2566 → 2585 → 2605 → 2618 → 2635 → 2651 → 2665 → 2684 → 2704 → 2717 → 2734 → 2750 → 2764 → 2783 → 2803 → 2816 → 2833 → 2849 → 2872 → 2891 → 2911 → 2924 → 2941 → 2957 → 2980 → 2999 → 3028 → 3041 → 3049 → 3065 → 3079 → 3098 → 3118 → 3131 → 3139 → 3155 → 3169 → 3188 → 3208 → 3221 → 3229 → 3245 → 3259 → 3278 → 3298 → 3320 → 3328 → 3344 → 3358 → 3377 → 3397 → 3419 → 3436 → 3452 → 3466 → 3485 → 3505 → 3518 → 3535 → 3551 → 3565 → 3584 → 3604 → 3617 → 3634 → 3650 → 3664 → 3683 → 3703 → 3716 → 3733 → 3749 → 3772 → 3791 → 3811 → 3824 → 3841 → 3857 → 3880 → 3899 → 3928 → 3950 → 3967 → 3992 → 4015 → 4025 → 4036 → 4049 → 4066 → 4082 → 4096 → 4115 → 4126 → 4139 → 4156 → 4172 → 4186 → 4205 → 4216 → 4229 → 4246 → 4262 → 4276 → 4295 → 4315 → 4328 → 4345 → 4361 → 4375 → 4394 → 4414 → 4427 → 4444
The 11-ticket is much better value than the tickets for 1..10. Bigger numbers behave like this:
1252 → 4444
1253 → 4444
1254 → 888888
1255 → 4444
1256 → 4444
1257 → 888888
1258 → 4444
1259 → 4444
1260 → 9999
1261 → 4444
1262 → 4444
1263 → 888888
1264 → 4444
1265 → 4444
1266 → 888888
1267 → 4444
1268 → 4444
1269 → 9999
1270 → 4444
1271 → 4444
1272 → 888888
1273 → 4444
1274 → 4444
Then all at once, a number-ticket turns golden and the rollercoaster-ride doesn’t end. So far, at least. I’ve tried, but I haven’t been able to find a rep-digit for 3515 and 3529 = 3515+digsum(3515) and so on:
3509 → 4444
3510 → 9999
3511 → 4444
3512 → 4444
3513 → 888888
3514 → 4444
3515 → ?
3516 → 888888
3517 → 4444
3518 → 4444
3519 → 9999
3520 → 4444
3521 → 4444
3522 → 888888
3523 → 4444
3524 → 4444
3525 → 888888
3526 → 4444
3527 → 4444
3528 → 9999
3529 → ?
3530 → 4444
3531 → 888888
3532 → 4444
Does 3515 ever yield a rep-digit for n += digsum(n)? It’s hard to believe it doesn’t, but I’ve no idea how to prove that it does. Except by simply riding the rollercoaster. And if the ride with the 3515-ticket never reaches a rep-digit, the rollercoaster will never let you know. How could it?
But here’s an example in base 23 of how a ticket for n+1 can give you a dramatically longer ride than a ticket for n and n+2:
MI → EEE (524 → 7742)
MJ → EEE (525 → 7742)
MK → 444 (526 → 2212)
ML → 444 (527 → 2212)
MM → MMMMMM (528 → 148035888)
100 → 444 (529 → 2212)
101 → 444 (530 → 2212)
102 → EEE (531 → 7742)
103 → 444 (532 → 2212)
104 → 444 (533 → 2212)
105 → EEE (534 → 7742)
106 → EEE (535 → 7742)
107 → 444 (536 → 2212)
108 → EEE (537 → 7742)
109 → 444 (538 → 2212)
10A → MMMMMM (539 → 148035888)
10B → EEE (540 → 7742)
10C → EEE (541 → 7742)
10D → EEE (542 → 7742)
10E → EEE (543 → 7742)
10F → 444 (544 → 2212)
10G → EEE (545 → 7742)
10H → EEE (546 → 7742)
10I → EEE (547 → 7742)
10J → 444 (548 → 2212)
10K → 444 (549 → 2212)
10L → MMMMMM (550 → 148035888)
10M → EEE (551 → 7742)
110 → EEE (552 → 7742)
Perfect Performative Pairing
Salt and celery, cheese and chocolate, yams and yoghurt — some things just taste better together. But that’s true of much more than foods and flavors. As a keyly committed core component of the anti-racist community, I’m proud and passionate to report that it’s also true of ideology and “in terms of”:
Unsurprisingly for a 200-year-old institution, the Guardian has not always got it right in terms of race coverage. — From slavery to BLM: the ups and downs of 200 years of Guardian race reporting, The Guardian, 6v21
For me, anti-racism just wouldn’t be the maximally moral movement that it is without a steady seasoning of “in terms of”. They’re a perfect performative pairing in an atrabiliously imperfect world.
Elsewhere other-engageable…
• Ex-term-in-nate! — interrogating issues around “in terms of”
• All O.o.t.Ü.-F. posts interrogating issues around “in terms of”…
More Mythical Mathicality
In a prev-previous post, I looked at this interesting fractal image on the front cover of a Ray Bradbury book:
It seems obvious that the image is created from photographs: only the body of the centaur is drawn by hand. And here’s my attempt at extending the fractality of the image:
Elsewhere other-accessible
• Mythical Mathical — Man-Horse! — the pre-previous post about the fractal centaur
Overlord In Terms of Core Issues Around Maximal Engagement with Key Notions of the Über-Feral 