Möbius Tripping

“In many cases, mathematics is an escape from reality. The mathematician finds his own monastic niche and happiness in pursuits that are disconnected from external affairs. Some practice it as if using a drug. Chess sometimes plays a similar role. In their unhappiness over the events of this world, some immerse themselves in a kind of self-sufficiency in mathematics. (Some have engaged in it for this reason alone.)” — Stanislaw Ulam (1909-84)

WhirlpUlam

Stanislaw Ulam (pronounced OO-lam) was an American mathematician who was doodling one day in 1963 and created what is now called the Ulam spiral. It’s a spiral of integers on a square grid with the prime squares filled in and the composite squares left empty. At the beginning it looks like this (the blue square is the integer 1, with 2 to the east, 3 to the north-east, 4 to the north, 5 to the north-west, 6 to the west, and so on):

Ulam spiral


And here’s an Ulam spiral with more integers:

Ulam spiral at higher resolution


The primes aren’t scattered at random over the spiral: they often fall into lines that are related to what are called polynomial functions, such as n2 + n + 1. To understand polynomial functions better, let’s look at how the Ulam spiral is made. Here is a text version with the primes underlined:


Here’s an animated version:


Here’s the true spiral again with 1 marked as a blue square:

Ulam spiral centred on 1


What happens when you try other numbers at the centre? Here’s 2 at the centre as a purple square, because it’s prime:

Ulam spiral centred on 2


And 3 at the centre, also purple because it’s also prime:

Ulam spiral centred on 3


And 4 at the centre, blue again because 4 = 2^2:

Ulam spiral centred on 4


And 5 at the centre, prime and purple:

Ulam spiral centred on 5


Each time the central number changes, the spiral shifts fractionally. Here’s an animation of the central number shifting from 1 to 41. If you watch, you’ll see patterns remaining stable, then breaking up as the numbers shift towards the center and disappear (the central number is purple if prime, blue if composite):

Ulam whirlpool, or WhirlpUlam


I think the animation looks like a whirlpool or whirlpUlam (prounced whirlpool-am), as numbers spiral towards the centre and disappear. You can see the whirlpUlam more clearly here:

An animated Ulam Spiral pausing at n=11, 17, 41


WhirlpUlam again


Note that something interesting happens when the central number is 41. The spiral is bisected by a long line of prime squares, like this:

Ulam spiral centred on 41


The line is actually a visual representation of something David Wells wrote about in The Penguin Dictionary of Curious and Interesting Numbers (1986):

Euler discovered the excellent and famous formula x2 + x + 41, which gives prime values for x = 0 to 39.

Here are the primes generated by the formula:

41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, 1033, 1097, 1163, 1231, 1301, 1373, 1447, 1523, 1601

You’ll see other lines appear and disappear as the whirlpUlam whirls:

Ulam spiral centred on 17


Primes in line: 17, 19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257 (n=0..15)


Ulam spiral centred on 59


Primes in line: 59, 67, 83, 107, 139, 179, 227, 283, 347, 419, 499, 587, 683, 787 (n=0..13)


Ulam spiral centred on 163


Primes in line: 163, 167, 179, 199, 227, 263, 307, 359, 419, 487, 563, 647, 739, 839, 947, 1063, 1187, 1319, 1459, 1607 (n=0..19)


Ulam spiral centred on 233


Primes in line: 233, 241, 257, 281, 313, 353, 401, 457, 521, 593, 673, 761, 857 ((n=0..12)


Ulam spiral centred on 653


Primes in line: 653, 661, 677, 701, 733, 773, 821, 877, 941, 1013, 1093, 1181, 1277, 1381, 1493, 1613, 1741, 1877 (n=0..17)


Ulam spiral centred on 409,333


Primes in line: 409,333, 409337, 409349, 409369, 409397, 409433, 409477, 409529, 409589, 409657, 409733, 409817, 409909, 410009, 410117, 410233 (n=0..15)


Some bisect the centre, some don’t, because you could say that the Ulam spiral has six diagonals, two that bisect the centre (top-left-to-bottom-right and bottom-left-to-top-right) and four that don’t. You could also call them spokes:


If you look at the integers in the spokes, you can see that they’re generated by polynomial functions in which c stands for the central number:

North-west spoke: 1, 5, 17, 37, 65, 101, 145, 197, 257, 325, 401, 485, 577, 677, 785, 901, 1025, 1157, 1297, 1445, 1601, 1765, 1937, 2117, 2305, 2501, 2705, 2917... = c + (2n)^2


South-east spoke: 1, 9, 25, 49, 81, 121, 169, 225, 289, 361, 441, 529, 625, 729, 841, 961, 1089, 1225, 1369, 1521, 1681, 1849, 2025, 2209, 2401, 2601, 2809, 3025, 3249, 3481, 3721, 3969, 4225, 4489, 4761, 5041, 5329, 5625... = c+(2n+1)^2-1


NW-SE diagonal: 1, 5, 9, 17, 25, 37, 49, 65, 81, 101, 121, 145, 169, 197, 225, 257, 289, 325, 361, 401, 441, 485, 529, 577, 625, 677, 729, 785, 841, 901, 961, 1025, 1089, 1157, 1225, 1297, 1369, 1445, 1521, 1601, 1681 = c + n^2 + 1 - (n mod 2)


North-east spoke: 1, 3, 13, 31, 57, 91, 133, 183, 241, 307, 381, 463, 553, 651, 757, 871, 993, 1123, 1261, 1407, 1561, 1723, 1893, 2071... = c + (n+1)^2 - n - 1


South-west spoke: 1, 7, 21, 43, 73, 111, 157, 211, 273, 343, 421, 507, 601, 703, 813, 931, 1057, 1191, 1333, 1483, 1641, 1807, 1981, 2163... = c + (2n)^2 + 2n


SW-NE diagonal: 1, 3, 7, 13, 21, 31, 43, 57, 73, 91, 111, 133, 157, 183, 211, 241, 273, 307, 343, 381, 421, 463, 507, 553, 601, 651, 703, 757, 813, 871, 931, 993, 1057, 1123, 1191, 1261, 1333, 1407, 1483, 1561, 1641... = c + n^2 + n



Elsewhere other-engageable:

All posts interrogating issues around the Ulam spiral

Amateur ’Grammatics

There is much more to mathematics than mathematics. Like a tree, it has deep roots. Like a tree, it’s affected by its environment. Philosophy of mathematics is concerned with the roots. Psychology of mathematics is concerned with the environment.

On Planet Earth, the environment is human beings. What attracts men and women to the subject? What makes them good or bad at it?And so on. One interesting answer to the first question was supplied by the mathematician Stanislaw Ulam (1909-84), who wrote this in his autobiography:

“In many cases, mathematics is an escape from reality. The mathematician finds his own monastic niche and happiness in pursuits that are disconnected from external affairs. Some practice it as if using a drug.” – Adventures of a Mathematician (1983)

That’s certainly part of maths’ appeal to me: as an escape from reality, or an escape from one reality into another (and deeper). Real life is messy. Maths isn’t, unless you want it to be. But you can find parallels between maths and real life too. In real life, people collect things that they find attractive or interesting: stamps, sea-shells, gems, cigarette-cards, beer-cans and so on. You can collect things in maths too: interesting numbers and number patterns. Recreational maths can feel like looking on a beach for attractive shells and pebbles.

Here’s a good example: digital anagrams, or numbers in different bases whose digits are the same but re-arranged. For example, 13 in base 10 equals 31 in base 4, because 13 = 3 * 4 + 1. To people with the right kind of mind, that’s an interesting and attractive pattern. There are lots more anagrams like that:

1045 = 4501 in base 6
1135 = 5131 in base 6

23 = 32 in base 7
46 = 64 in base 7

1273 = 2371 in base 8
1653 = 3165 in base 8

158 = 185 in base 9
227 = 272 in base 9

196 = 169 in base 11
283 = 238 in base 11

2193 = 1329 in base 12
6053 = 3605 in base 12

43 = 34 in base 13
86 = 68 in base 13

But triple anagrams, involving three bases, seem even more attractive:

913 = 391 in base 16 = 193 in base 26
103462 = 610432 in base 7 = 312046 in base 8
245183 = 413285 in base 9 = 158234 in base 11

And that’s just looking in base 10. If you include all bases, the first double anagram is in fact 21 in base 3 = 12 in base 5 (equals 7 in base 10). The first triple anagram is this:

2C1 in base 13 = 1C2 in base 17 = 12C in base 21 (equals 495 in base 10)

But are there quadruple anagrams, quintuple anagrams and higher? I don’t know. I haven’t found any and it gets harder and harder to search for them, because the bigger n gets, the more bases there are to check. However, I can say one thing for certain: in any given base, anagrams eventually disappear.

To understand why, consider the obvious fact that anagrams have to have the same number of digits in different bases. But the number of digits is a function of the powers of the base. That is, the triple anagram 103462 (see above) has six digits in bases 7, 8 and 10 because 7^5 < 103462 < 7^6, 8^5 < 103462 < 8^6 and 10^5 < 103462 < 10^6. Similarly, the triple anagram 245183 (ditto) has six digits in bases 9, 10 and 11 because 9^5 < 245183 < 9^6, 10^5 < 245183 < 10^6 and 11^5 < 245183 < 11^6:

7^5 < 103462 < 7^6
16807 < 103462 < 117649
8^5 < 103462 < 8^6
32768 < 103462 < 262144
10^5 < 103462 < 10^6
100000 < 103462 < 1000000
9^5 < 245183 < 9^6
59049 < 245183 < 531441
10^5 < 245183 < 10^6
100000 < 245183 < 1000000
11^5 < 245183 < 11^6
161051 < 245183 < 1771561

In other words, for some n the number-lengths of bases 7 and 8 overlap the number-lengths of base 10, which overlap the number-lengths of bases 9 and 11. But eventually, as n gets larger, the number-lengths of base 10 will fall permanently below the number-lengths of bases 7, 8 and 9, just as the number-lengths of base 11 will fall permanently below the number-lengths of base 10.

To see this in action, consider the simplest example: number-lengths in bases 2 and 3. There is no anagram involving these two bases, because only two numbers have the same number of digits in both: 1 and 3 = 11 in base 2 = 10 in base 3. After that, n in base 2 always has more digits than n in base 3:

2^0 = 1 in base 2 (number-length=1) = 1 in base 3 (l=1)
2^1 = 2 = 10 in base 2 (number-length=2) = 2 in base 3 (l=1)
2^2 = 4 = 100 in base 2 (l=3) = 11 in base 3 (l=2)
2^3 = 8 = 1000 in base 2 = 22 in base 3 (l=2)
2^4 = 16 = 10000 in base 2 = 121 in base 3 (l=3)
2^5 = 32 = 1012 in base 3 (l=4)
2^6 = 64 = 2101 in base 3 (l=4)
2^7 = 128 = 11202 in base 3 (l=5)
2^8 = 256 = 100111 in base 3 (l=6)
2^9 = 512 = 200222 in base 3 (l=6)
2^10 = 1024 = 1101221 in base 3 (l=7)

Now consider bases 3 and 4. Here is an anagram using these bases: 211 in base 3 = 112 in base 4 = 22. There are no more anagrams and eventually there’s no more chance for them to occur, because this happens as n gets larger:

3^0 = 1 in base 3 (number-length=1) = 1 in base 4 (l=1)
3^1 = 3 = 10 in base 3 (number-length=2) = 3 in base 4 (l=1)
3^2 = 9 = 100 in base 3 (l=3) = 21 in base 4 (l=2)
3^3 = 27 = 1000 in base 3 (l=4) = 123 in base 4 (l=3)
3^4 = 81 = 10000 in base 3 (l=5) = 1101 in base 4 (l=4)
3^5 = 243 = 100000 in base 3 (l=6) = 3303 in base 4 (l=4)
3^6 = 729 = 23121 in base 4 (l=5)
3^7 = 2187 = 202023 in base 4 (l=6)
3^8 = 6561 = 1212201 in base 4 (l=7)
3^9 = 19683 = 10303203 in base 4 (l=8)
3^10 = 59049 = 32122221 in base 4 (l=8)
3^11 = 177147 = 223033323 in base 4 (l=9)
3^12 = 531441 = 2001233301 in base 4 (l=10)
3^13 = 1594323 = 12011033103 in base 4 (l=11)
3^14 = 4782969 = 102033231321 in base 4 (l=12)
3^15 = 14348907 = 312233021223 in base 4 (l=12)
3^16 = 43046721 = 2210031131001 in base 4 (l=13)
3^17 = 129140163 = 13230220113003 in base 4 (l=14)
3^18 = 387420489 = 113011321011021 in base 4 (l=15)
3^19 = 1162261467 = 1011101223033123 in base 4 (l=16)
3^20 = 3486784401 = 3033311001232101 in base 4 (l=16)

When n is sufficiently large, it always has fewer digits in base 4 than in base 3. And the gap gets steadily bigger. When n doesn’t have the same number of digits in two bases, it can’t be an anagram. A similar number-length gap eventually appears in bases 4 and 5, but the anagrams don’t run out as quickly there:

103 in base 5 = 130 in base 4 = 28
1022 in base 5 = 2021 in base 4 = 137
1320 in base 5 = 3102 in base 4 = 210
10232 in base 5 = 22310 in base 4 = 692
10332 in base 5 = 23031 in base 4 = 717
12213 in base 5 = 32211 in base 4 = 933
100023 in base 5 = 301002 in base 4 = 3138
100323 in base 5 = 302031 in base 4 = 3213
102131 in base 5 = 311120 in base 4 = 3416
102332 in base 5 = 312023 in base 4 = 3467
103123 in base 5 = 313102 in base 4 = 3538
1003233 in base 5 = 3323010 in base 4 = 16068

Base 10 isn’t exempt. Eventually it must outshrink base 9 and be outshrunk by base 11, so what is the highest 9:10 anagram and highest 10:11 anagram? I don’t know: my maths isn’t good enough for me to find out quickly. But using machine code, I’ve found these large anagrams:

205888888872731 = 888883178875022 in base 9
1853020028888858 = 8888888525001032 in base 9
16677181388880888 = 88888888170173166 in base 9

999962734025 = 356099992472 in base 11
9999820360965 = 3205999998606 in base 11
99999993520348 = 29954839390999 in base 11

Note how the digits of n in the lower base are increasing as the digits of n in the higher base are decreasing. Eventually, n in the lower base will always have more digits than n in the higher base. When that happens, there will be no more anagrams.

Some triple anagrams

2C1 in base 13 = 1C2 in base 17 = 12C in base 21 (n=495 = 3^2*5*11)
912 in base 10 = 219 in base 21 = 192 in base 26 (2^4*3*19)
913 in base 10 = 391 in base 16 = 193 in base 26 (11*83)
4B2 in base 15 = 42B in base 16 = 24B in base 22 (n=1067 = 11*97)
5C1 in base 17 = 51C in base 18 = 1C5 in base 35 (n=1650 = 2*3*5^2*11)
3L2 in base 26 = 2L3 in base 31 = 23L in base 35 (n=2576 = 2^4*7*23)
3E1 in base 31 = 1E3 in base 51 = 13E in base 56 (n=3318 = 2*3*7*79)
531 in base 29 = 351 in base 37 = 135 in base 64 (n=4293 = 3^4*53)
D53 in base 18 = 53D in base 29 = 35D in base 37 (n=4305 = 3*5*7*41)
53I in base 29 = 3I5 in base 35 = 35I in base 37 (n=4310 = 2*5*431)
825 in base 25 = 582 in base 31 = 258 in base 49 (n=5055 = 3*5*337)
6S2 in base 31 = 2S6 in base 51 = 26S in base 56 (n=6636 = 2^2*3*7*79)
D35 in base 23 = 5D3 in base 36 = 3D5 in base 46 (n=6951 = 3*7*331)
3K1 in base 49 = 31K in base 52 = 1K3 in base 81 (n=8184 = 2^3*3*11*31)
A62 in base 29 = 6A2 in base 37 = 26A in base 64 (n=8586 = 2*3^4*53)
9L2 in base 30 = 92L in base 31 = 2L9 in base 61 (n=8732 = 2^2*37*59)
3W1 in base 49 = 1W3 in base 79 = 13W in base 92 (n=8772 = 2^2*3*17*43)
G4A in base 25 = AG4 in base 31 = 4AG in base 49 (n=10110 = 2*3*5*337)
J10 in base 25 = 1J0 in base 100 = 10J in base 109 (n=11900 = 2^2*5^2*7*17)
5[41]1 in base 46 = 1[41]5 in base 93 = 15[41] in base 109 (n=12467 = 7*13*137)
F91 in base 29 = 9F1 in base 37 = 19F in base 109 (n=12877 = 79*163)
F93 in base 29 = 9F3 in base 37 = 39F in base 64 (n=12879 = 3^5*53)
AP4 in base 35 = A4P in base 36 = 4AP in base 56 (n=13129 = 19*691)
BP2 in base 36 = B2P in base 37 = 2PB in base 81 (n=15158 = 2*11*13*53)
O6F in base 25 = FO6 in base 31 = 6FO in base 49 (n=15165 = 3^2*5*337)
FQ1 in base 31 = 1QF in base 111 = 1FQ in base 116 (n=15222 = 2*3*43*59)
B74 in base 37 = 7B4 in base 46 = 47B in base 61 (n=15322 = 2*47*163)

Stoch’! (In the Name of Dove)

Stochasma, In Abysso (2012)

The Sueco-Georgian avant-gardists Stochasma were formed, in their own words, “to interrogate, eviscerate, and exterminate the ultimate experimental envelope of acoustic idiosyncrasy”. That’s “Sueco-” as in Sweden and “Georgian” as in the Eurasian nation, not the American state, by the way. Going one up on some bands from Wales, Ireland and Scotland, who issue their material bilingually, in English and one or another of the Celtic languages, Stochasma issue all their material tri-lingually, in English, Swedish, and Georgian. The strangeness and beauty of the Georgian script match and enhance the strangeness and (occasional) beauty of their music, but, unlike their last two releases, there’s no spoken English, Swedish or Georgian here: In Abysso is intended to be an “abhuman listen”.

Front cover of Stochasma's album In Abysso 

Believe me, it is! The title of the album is Latin for “In the Abyss” and the liner-notes extend thanks to H.P. Lovecraft, Clark Ashton Smith and Stanislaw Ulam for “infernal inspiration”. If the last name makes you think “Who?” (or “U?”), you must be new/nu to Stochasma, who draw inspiration not just from art and literature, but from mathematics too. Stanislaw Ulam (1909-84) was a Polish mathematician perhaps most famous for inventing the “Ulam spiral”, a graphical representation of the prime numbers that reveals mysterious patterns in this strange and fascinating set of integers. Ulam stumbled across the spiral while “doodling” during a boring lecture at a scientific meeting. That kind of serendipity has always been important to Stochasma, who explore the musical abyss/chasm partly through random, or stochastic, techniques. For the first track, “Pr1m4l Skr33m”, the five members of the band had electrodes attached to their nipples before being asked, at random, to indicate, with a nod or shake of the head, whether a randomly selected number between 1 and 10,000 was prime or composite (for example, 1,433 is prime, being divisible by no numbers but itself and 1; 1,434 is composite, being divisible by 2, 3 and 239). If they were wrong, they received a painful electric shock.

The resultant collection of grunts, gasps, and screams was electronically worked over in fully traditional Stochasma fashion to create “Pr1m4l Skr33m”, which sounds like a fully traditional Stochasma track: fucking weird and unsettling! Is the irregular chorus of voices in agony or ecstasy? Are the band being tortured in a hell run by sadists or pleasured in a heaven run for masochists? Or both? It’s hard to decide, and at times hard to listen, but as Stochasma themselves put it: “We’re queasy listening, not easy – easy listening is for cubes.”

Elsewhere, the band have used the ultra-sensitive microphones they first experimented with on 2003’s AnguisaquA (sic – it literally means “SnakewateR”). This time they’ve recorded the bloodflow of a dove and the movements of parasites in its feathers for “Täubchen”, which sounds even stranger than it reads. That and “Pr1m4l Skr33m” are the first two tracks: the next fifteen are entitled “Ignisigil I” to “Ignisigil XV”. Stochasma used a fire-proof microphone to record the sound of books being burned. They selected fifteen wildly different authors for this literally incendiary homage, from “J. Aldapuerta to J. Archer, from K. Marx to K. Minogue”, as they themselves put it. (That’s the über-trangressive Spanish horror-writer Jesús Aldapuerta and the über-cruddy British thriller-writer Jeffrey Archer, and the Anglo-German philosophaster Karl Marx and the Australian pop-pixie Kylie Minogue, for those unfamiliar with the names.) And the band insist, perhaps tongue-in-cheek, that the sonic textures of the recordings are dependent not just on the physical nature of the paper and ink being burnt, but also on the ideological and aesthetic nature of the burning text.

It’s hard to agree: the “Ignisigils” all sound pretty much alike to me, though that sound is uncharacteristically soothing and relaxing by Stochasma standards (on my first listen, I dropped off during “Ignisigil VIII” and didn’t wake up till “Ignisigil XI”). The album is rounded off with three of the strangest pieces of music I’ve heard this century: “Musgomorrah”, “Gradus ad Parnassum”, and “CoMoXoCoI”. The first sounds like a slowed recording of men in armour fighting in thick mud; the second like a choir of giant glass insects singing themselves to splinters; and the third like echoes chasing each another in a collapsing or burning maze. These three might grow on me or might not: for now, “Pr1m4l Skr33m”, “Täubchen”, and “Ignisigil IV” hit the sonic sweet’n’sour spot that Stochasma seem to have copyrighted. I don’t know why “IV” hits the spot and the rest of the Ignisigils don’t, but that’s often the way with Stochasma: you like the sounds they create and you haven’t a clue as to why. In company with a select band of other electronicognoscenti, I look forward to their seventh album, whenever it appears and whatever musical mélanges or macedoines it manages to mulch, mangle, and miscegenate.


Elsewhere other-engageable:

Musings on Music