Multitudinous Marriment

• …ποντίων τε κυμάτων ἀνήριθμον γέλασμα… — Αἰσχύλος, Προμηθεὺς δεσμώτης (c. 479-24 B.C.)

• …of ocean-waves the multitudinous laughter… Prometheus Bound 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

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:

Cover of Ray Bradbury’s I Sing the Body Electric (1969)

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:

Further fractality for the centaur

Elsewhere other-accessible

Mythical Mathical — Man-Horse! — the pre-previous post about the fractal centaur

Knostrils

• εἰ πάντα τὰ ὄντα καπνὸς γένοιτο, ῥῖνες ἂν διαγνοῖεν. — Ἡράκλειτος ὁ Ἐφέσιος

• • Si toutes choses devenaient fumée, on connaîtrait par les narines. — Héraclite d’Ephèse

• • • If all things were turned to smoke, the nostrils would tell them apart. — Heraclitus of Ephesos, quoted in Aristotle’s De sensu, 5, 443a 23

Think Inc #2

In a pre-previous post called “Think Inc”, I looked at the fractals created by a point first jumping halfway towards the vertex of a square, then using a set of increments to decide which vertex to jump towards next. For example, if the inc-set was [0, 1, 3], the point would jump next towards the same vertex, v[i]+0, or the vertex immediately clockwise, v[i]+1, or the vertex immediately anti-clockwise, v[i]+3. And it would trace all possible routes using that inc-set. Then I added refinements to the process like giving the point extra jumping-targets half-way along each side.

Here are some more variations on the inc-set theme using two and three extra jumping-targets along each side of the square. First of all, try two extra jumping-targets along each side and a set of three increments:

inc = 0, 1, 6


inc = 0, 2, 6


inc = 0, 2, 8


inc = 0, 3, 6


inc = 0, 3, 9


inc = 0, 4, 8


inc = 0, 5, 6


inc = 0, 5, 7


inc = 1, 6, 11


inc = 2, 6, 10


inc = 3, 6, 9


Now try two extra jumping-targets along each side and a set of four increments:

inc = 0, 1, 6, 11


inc = 0, 2, 8, 10


inc = 0, 3, 7, 9


inc = 0, 4, 8, 10


inc = 0, 5, 6, 7


inc = 0, 5, 7, 8


inc = 1, 6, 7, 9


inc = 1, 4, 6, 11


inc = 1, 5, 7, 11


inc = 2, 4, 8, 10


inc = 3, 5, 7, 9


And finally, three extra jumping-targets along each side and a set of three increments:

inc = 0, 3, 13


inc = 0, 4, 8


inc = 0, 4, 12


inc = 0, 5, 11

inc = 0, 6, 9


inc = 0, 7, 9


Previously Pre-Posted

Think Inc — an earlier look at inc-set fractals

Vacancy Vanquished

We never sighted the slightest suggestion of life all the way to Vancouver, twelve days of chilly boredom, though there was a certain impressiveness in the very dreariness and desolation. There was a hint of the curious horror that emptiness always evokes, whether it is a space of starless night or a bleak and barren waste of land. The one exception is the Sahara Desert where, for some reason that I cannot name, the suggestion is not in the least of vacancy and barrenness, but rather of some subtle and secret spring of life. — The Confessions of Aleister Crowley: An Autohagiography (1929), ch. 57


Previously Pre-Posted…

Leech Unleashed
Crowley on Crystals

Mythical Mathical — Man-Horse!

Cover of Ray Bradbury’s I Sing the Body Electric (1969), published by Corgi in 1972

That’s a striking cover — and more than that. The blog where I found the cover says this: “This very odd cover clearly features a heavily rouged glam rock centaur with a rather natty feather-cut hairstyle flexing his biceps, his forearms transmogrifying into miniature bicep flexing glam rock figures. I think I’m slowly losing the plot here.” Losing the plot? No, losing the mathematicality in the mythical. The artist has started to make the centaur into a fractal. Or rather, the artist has started to make more explicit what is already there in the human body. As I wrote in “Fingering the Frigit”:

Fingers are fractal. Where a tree has a trunk, branches and twigs, a human being has a torso, arms and fingers. And human beings move in fractal ways. We use our legs to move large distances, then reach out with our arms over smaller distances, then move our fingers over smaller distances still. We’re fractal beings, inside and out, brains and blood-vessels, fingers and toes.