Soft teeth at last of gnawing hour
Grind sternest stone to finest flour.
Nor any mage, with puissant art,
Deflects Time’s final venom’d dart.
“Tempus edax rerum…”, Ovid, Metamorphoses, xv, 234.
Author Archives: Krilling for Company
Multi-Magic
A magic square is a square of numbers in which all rows, all columns and both diagonals add to the same number, or magic total. The simplest magic square using distinct numbers is this:
6 1 8 7 5 3 2 9 4
It’s easy to prove that the magic total of a 3×3 magic square must be three times the central number. Accordingly, if the central number is 37, the magic total must be 111. There are lots of ways to create a magic square with 37 at its heart, but this is my favourite:
43 | 01 | 67 61 | 37 | 13 07 | 73 | 31
The square is special because all the numbers are prime, or divisible by only themselves and 1 (though 1 itself is not usually defined as prime in modern mathematics). I like the 37-square even more now that I’ve discovered it can be found inside another all-prime magic square:
0619 = 0006[37] | 0097 = 00000010 | 1123 = [11][56] 1117 = [11][50] | 0613 = 0006[31] | 0109 = 0001[12] 0103 = 00000016 | 1129 = [11][62] | 0607 = 0006[25] Magic total = 1839
The square is shown in both base-10 and base-97. If the digit-sums of the base-97 square are calculated, this is the result (e.g., the digit-sum of 6[37][b=97] = 6 + 37 = 43):
43 | 01 | 67 61 | 37 | 13 07 | 73 | 31
This makes me wonder whether the 613-square might nest in another all-prime square, and so on, perhaps ad infinitum [Update: yes, the 613-square is a nestling]. There are certainly many nested all-prime squares. Here is square-631 in base-187:
661 = 003[100] | 379 = 00000025 | 853 = 004[105] 823 = 004[075] | 631 = 003[070] | 439 = 002[065] 409 = 002[035] | 883 = 004[135] | 601 = 003[040] Magic total = 1893 Digit-sums: 103 | 007 | 109 079 | 073 | 067 037 | 139 | 043 Magic total = 219
There are also all-prime magic squares that have two kinds of nestlings inside them: digit-sum magic squares and digit-product magic squares. The digit-product of a number is calculated by multiplying its digits (except 0): digit-product(37) = 3 x 7 = 21, digit-product(103) = 1 x 3 = 3, and so on. In base-331, this all-prime magic square yields both a digit-sum square and a digit-product square:
503 = 1[172] | 359 = 1[028] | 521 = 1[190] 479 = 1[148] | 461 = 1[130] | 443 = 1[112] 401 = 1[070] | 563 = 1[232] | 419 = 1[088] Magic total = 1383 Digit-sums: 173 | 029 | 191 149 | 131 | 113 071 | 233 | 089 Magic total = 393 Digit-products: 172 | 028 | 190 148 | 130 | 112 070 | 232 | 088 Magic total = 390
Here are two more twin-bearing all-prime magic squares:
Square-719 in base-451: 761 = 1[310] | 557 = 1[106] | 839 = 1[388] 797 = 1[346] | 719 = 1[268] | 641 = 1[190] 599 = 1[148] | 881 = 1[430] | 677 = 1[226] Magic total = 2157 Digit-sums: 311 | 107 | 389 347 | 269 | 191 149 | 431 | 227 Magic total = 807 Digit-products: 310 | 106 | 388 346 | 268 | 190 148 | 430 | 226 Magic total = 804
Square-853 in base-344:
883 = 2[195] | 709 = 2[021] | 967 = 2[279] 937 = 2[249] | 853 = 2[165] | 769 = 2[081] 739 = 2[051] | 997 = 2[309] | 823 = 2[135] Magic total = 2559 Digit-sums: 197 | 023 | 281 251 | 167 | 083 053 | 311 | 137 Magic total = 501 Digit-products: 390 | 042 | 558 498 | 330 | 162 102 | 618 | 270 Magic total = 990
Proviously Post-Posted (please peruse):
Songs from the Center of the Sun
Faster Than Lichen are minimal men making minimal music. They may even be a minimal man: one story is that F.T.L. is an American or Russian (or possibly Argentinian) scientist living in a research-base on Antarctica. This would fit the name, because lichens are an important part of polar ecology. But perhaps it fits too well. Here is a short interview with F.T.L. conducted by email in November 2012.
Faster Than Lichen #1
Q. How would you describe the music of Faster Than Lichen?
A. Songs from the center of the sun.
Faster Than Lichen #1 (back cover)
Q. Why the sun?
A. Why not?
Q. How long have you been making music?
A. Long enough.
Faster Than Lichen #2
Q. What emotions do you aim to evoke?
A. Acceptance.
Faster Than Lichen #2 (back cover)
Q. Is that all?
A. Usually.
Q. F.T.L. albums have no titles, only numbers. Why?
A. Numbers say more.
Faster Than Lichen logos (please open in new window to animate)
Q. Who created the F.T.L. logo?
A. It evolved.
Faster Than Lichen #6
Q. It’s been two years since #6. Can we hope to hear #7 soon?
A. Yes. We’re working on it now.
Faster Than Lichen #6 (back cover)
Q. So there is more than one of you?
A. Sometimes.
Q. What happened to F.T.L. #5?
A. It was withdrawn before issue.
Q. Will it ever be issued?
A. Maybe.
Faster Than Lichen #3
Q. There is only one track on F.T.L. #3. It’s 59 seconds long and is called “One-Oh-One-One-One-One-One”. But 1011111 is binary for 95. Is this a joke?
A. Sometimes.
Faster Than Lichen #3 (back cover)
Q. Are you aware of the group Old Church Slavonick?
A. Yes. But that is as far as it goes.
Q. So no members of F.T.L. were ever members of O.C.S.?
A. Yes. None.
Q. What are your influences?
A. Lichen, of course. The sun. Starlight.
Q. Other groups?
A. That is for the listener.
Faster Than Lichen #4
Q. Do you have any plans to play live?
A. It’s possible. Maybe next year.
Faster Than Lichen #4 (back cover)
Q. Where would you most like to play live?
A. An active volcano on Iceland.
Q. During a solar eclipse?
A. That would be a bonus. Not essential.
Q. What else would be a bonus?
A. A dragonfly.
Q. Thanks for the interview. Anything to add?
A. Thanks. Always accept all.
Previously Pre-Posted (please peruse):
• Angst, Anguish, Abjection
• Mental Marine Music
• Stoch’! (In the Name of Dove)
Prummer-Time Views
East, west, home’s best. And for human beings, base-10 is a kind of home. We have ten fingers and we use ten digits. Base-10 comes naturally to us: it feels like home. So it’s disappointing that there is no number in base-10 that is equal to the sum of the squares of its digits (apart from the trivial 0^2 = 0 and 1^2 = 1). Base-9 and base-11 do better:
41 = 45[b=9] = 4^2 + 5^2 = 16 + 25 = 41
50 = 55[b=9] = 5^2 + 5^2 = 25 + 25 = 50
61 = 56[b=11] = 5^2 + 6^2 = 25 + 36 = 61
72 = 66[b=11] = 6^2 + 6^2 = 36 + 36 = 72
Base-47 does better still, with fourteen 2-sumbers. And base-10 does have 3-sumbers, or numbers equal to the sum of the cubes of their digits:
153 = 1^3 + 5^3 + 3^3 = 1 + 125 + 27 = 153
370 = 3^3 + 7^3 + 0^3 = 27 + 343 + 0 = 370
371 = 3^3 + 7^3 + 1^3 = 27 + 343 + 1 = 371
407 = 4^3 + 0^3 + 7^3 = 64 + 0 + 343 = 407
But base-10 disappoints again when it comes to prumbers, or prime sumbers, or numbers that are equal to the sum of the primes whose indices are equal to the digits of the number. The index of a prime number is its position in the list of primes. Here are the first nine primes and their indices (with 0 as a pseudo-prime at position 0):
prime(0) = 0
prime(1) = 2
prime(2) = 3
prime(3) = 5
prime(4) = 7
prime(5) = 11
prime(6) = 13
prime(7) = 17
prime(8) = 19
prime(9) = 23
So the prumber, or prime-sumber, of 1 = prime(1) = 2. The prumber of 104 = prime(1) + prime(0) + prime(4) = 2 + 0 + 7 = 9. The prumber of 186 = 2 + 19 + 13 = 34. But no number in base-10 is equal to its prime sumber. Base-2 and base-3 do better:
Base-2 has 1 prumber:
2 = 10[b=2] = 2 + 0 = 2
Base-3 has 2 prumbers:
4 = 11[b=3] = 2 + 2 = 4
5 = 12[b=3] = 2 + 3 = 5
But prumbers are rare. The next record is set by base-127, with 4 prumbers:
165 = 1[38][b=127] = 2 + 163 = 165
320 = 2[66][b=127] = 3 + 317 = 320
472 = 3[91][b=127] = 5 + 467 = 472
620 = 4[112][b=127] = 7 + 613 = 620
Base-479 has 4 prumbers:
1702 = 3[265] = 5 + 1697 = 1702
2250 = 4[334] = 7 + 2243 = 2250
2800 = 5[405] = 11 + 2789 = 2800
3344 = 6[470] = 13 + 3331 = 3344
Base-637 has 4 prumbers:
1514 = 2[240] = 3 + 1511 = 1514
2244 = 3[333] = 5 + 2239 = 2244
2976 = 4[428] = 7 + 2969 = 2976
4422 = 6[600] = 13 + 4409 = 4422
Base-831 has 4 prumbers:
999 = 1[168] = 2 + 997 = 999
2914 = 3[421] = 5 + 2909 = 2914
3858 = 4[534] = 7 + 3851 = 3858
4798 = 5[643] = 11 + 4787 = 4798
Base-876 has 4 prumbers:
1053 = 1[177] = 2 + 1051 = 1053
3066 = 3[438] = 5 + 3061 = 3066
4064 = 4[560] = 7 + 4057 = 4064
6042 = 6[786] = 13 + 6029 = 6042
Previously pre-posted (please peruse):
Septics vs Dirties
Some interesting patterns at Google’s Ngram Viewer (please follow the links to see the original images with further statistics):
in terms of (American + British English)
in terms of (American English)
in terms of (British English)
issues around (American + British English)
issues around (American English)
issues around (British English)
Previously pre-posted (please peruse):
Performativizing Papyrocentricity #11
Papyrocentric Performativity Presents:
Stellissimus – The Cosmic Gallery: The Most Beautiful Images of the Universe, Giles Sparrow (Quercus 2013)
Eyck’s Eyes – Van Eyck, Simone Ferrari (Prestel 2013)
Dealing Death at a Distance – Sniper: Sniping Skills from the World’s Elite Forces, Martin J. Dougherty (Amber Books 2012)
Serious Stimbulation – Cleaner, Kinder, Caringer: Women’s Wisdom for a Wounded World, edited by Dr Miriam B. Stimbers (University of Nebraska Press 2013)
Keeping It Gweel — Gweel and Other Alterities, Simon Whitechapel (Ideophasis Press 2011) (posted @ Overlord of the Über-Feral)
Ave Aves! — Collins Bird Guide: The Most Complete Guide to the Birds of Britain and Europe (second edition), text and maps by Lars Svensson, illustrations and captions by Killian Mullarney and Dan Zetterström (HarperCollins, 2009) (@ O.o.t.Ü.-F.)
Flesh and Fear — Understanding Owls: Biology, Management, Breeding, Training, Jemima Parry-Jones (David & Charles, 1998) (@ O.o.t.Ü.-F.)
Hit and Smith — Songs that Saved Your Life: The Art of The Smiths 1982-87, Simon Goddard (Titan Books 2013) (@ O.o.t.Ü.-F.)
Or Read a Review at Random: RaRaR
The Term Turns dot dot dot
In Titus Graun, I interrogated issues around the Grauniness, or Guardianisticity, of two keyly committed core components of the counter-cultural community: the semiotician Stewart Home and the æsthetician John Coulthart. Seeking to utilizate their usage-metrics for the core/epicentral Guardianista phrase “in terms of” (i.t.o.), I interrogated their personal websites like this in terms of January 2013:
site:http://www.johncoulthart.com “in terms of”
About 2,180 resultssite:http://www.johncoulthart.com “the”
About 8,860 resultssite:http://www.johncoulthart.com “and”
About 8,150 results
site:http://www.stewarthomesociety.org “in terms of”
About 123 resultssite:http://www.stewarthomesociety.org “the”
About 602 resultssite:http://www.stewarthomesociety.org “and”
About 599 results
Noting that Coulthart’s site used “the/and” approximately 14 times more often than Home’s, I adjusted Home’s raw i.t.o.-score accordingly: 123 x 14 = 1722. I concluded that Coulthart, with an i.t.o.-score of 2180, was approximately 26·59% Graunier than Home – exactly as one might have hoped, given that Coulthart is not merely a Guardianista (good), but a gay Guardianista (doubleplusgood). But that was in terms of January. When I re-interrogated their websites in terms of June 2013, I discovered that the semiotic situation had transitioned in a most disturbing and disquieting way:
site:http://www.johncoulthart.com “in terms of”
About 1,080 resultssite:http://www.johncoulthart.com “the”
About 8,680 resultssite:http://www.johncoulthart.com “and”
About 8,010 results
site:http://www.stewarthomesociety.org “in terms of”
About 119 resultssite:http://www.stewarthomesociety.org “the”
About 541 resultssite:http://www.stewarthomesociety.org “and”
About 536 results
I was aghast (literally) to see that Coulthart’s i.t.o.-metrics have spiked (in reverse). Other lexicostatistical metrics have transitioned relatively little: his site now seems to use “the/and” approximately 15·5 times more often than Home’s. Home’s raw i.t.o.-score is 119 and 119 x 15·5 = 1844·5. So it is now Home who is approximately 70·78% Graunier than Coulthart.
This can only be described as highly suspicious. What has Coulthart been up to? Has he been spraying his site with verbicide? Has he donned a black Savoy nihilinja-suit™, crept out under cover of darkness and clubbed innocent i.t.o.’s as they lay basking in the feral radiance of Manchester’s Most Maverick Messiahs? If so, this is “‘Pushing the Transgressive Envelope Too Far’ Too Far” too far. Even M.M.M.M. must look askance at behaviour like that. Surely.
Previously pre-posted (please peruse):
The World as Worm
In “Hymn to Herm”, I wrote about a religion based on √2, or the square root of two, the number that, multiplied by itself, equals 2. In the religion, neophytes learn the mystery and majesty of this momentous number when they try to calculate its exact value. The calculation involves adding and subtracting fractions based on powers of two. The first step is this: 1 x 1 = 1. So that’s too small. Add 1/2^1 = ½ and re-multiply: 1½ x 1½ = 2¼. Too big. So subtract 1/2^2 = ¼, and re-multiply. 1¼ x 1¼ = 1+9/16. Too small. Add 1/8 and re-multiply. 1+3/8 x 1+3/8 = 1+57/64. Too small again. Add 1/16 and re-multiply. And so on.
In effect, what the neophytes are doing is calculate the digits of √2 in binary, or base two. When the multiplication is too small, put a 1; when it’s too big, put a 0. Like this:
1 x 1 = 1 < 2, so √2 ≈ 1·…
1½ x 1½ = 2¼ > 2, so √2 ≈ 1·0…
1¼ x 1¼ = 1+9/16 < 2, so √2 ≈ 1·01…
(1+3/8) x (1+3/8) = 1+57/64 < 2, so √2 ≈ 1·011…
(1+7/16) x (1+7/16) = 2+17/256 > 2, so √2 ≈ 1·0110…
(1+13/32) x (1+13/32) = 1+1001/1024 < 2, so √2 ≈ 1.01101…
(1+27/64) x (1+27/64) = 2+89/4096 > 2, so √2 ≈ 1.011010…
(1+53/128) x (1+53/128) = 1+16377/16384 < 2, so √2 ≈ 1·0110101…
(1+107/256) x (1+107/256) = 2+697/65536 > 2, so √2 ≈ 1·01101010…
(1+213/512) x (1+213/512) = 2+1337/262144 > 2, so √2 ≈ 1·011010100…
(1+425/1024) x (1+425/1024) = 2+2449/1048576 > 2, so √2 ≈ 1·0110101000…
(1+849/2048) x (1+849/2048) = 2+4001/4194304 > 2, so √2 ≈ 1·01101010000…
(1+1697/4096) x (1+1697/4096) = 2+4417/16777216 > 2, so √2 ≈ 1·011010100000…
(1+3393/8192) x (1+3393/8192) = 1+67103361/67108864 < 2, so √2 ≈ 1·0110101000001…
Mathematically naïve neophytes, seeing the process miss 2 by smaller and smaller amounts on either side, might imagine that eventually the exact root will appear and the calculations end. But they would be wrong. They could work a year or a million years: they would never calculate the exact square root of two. There is no ratio of whole numbers, a/b, such that a^2/b^2 = 2. In other words, √2 is an irrational number, or number that can’t be represented as a ratio of integers (please see appendix for the proof).
This discovery, made by Greek mathematicians more than two millennia ago, is both mind-boggling and world-shattering. In fact, it’s mind-boggling in part because it’s world-shattering. √2 shatters the world because the world is too small to contain it: in the words of the Cult of Infinite Hermaphrodites, “Were the sky all parchment, the seas all ink, and gulls all plucked for quills”, the square root of two could not be recorded in full. This is far more certain than tomorrow’s sunrise, because predicting tomorrow’s sunrise depends on fallible scientific reasoning from incomplete knowledge. Proving the irrationality of √2 depends on infallible mathematical reasoning.
At least, it’s as close to infallible as human beings can get. But that’s another part of what is mind-boggling about √2. A finite, feeble human being, with a speck of soon-decaying brain, can prove the existence of things larger than the universe. A few binary digits of √2 are shown above. Here are a few more:
1· 0110101000001001111001100110011111110011101111001100100100001000 1011001011111011000100110110011011101010100101010111110100111110 0011101011011110110000010111010100010010011101110101000010011001 1101101000101111010110010000101100000110011001110011001000101010 1001010111111001000001100000100001110101011100010100010110000111 0101000101100011111111001101111110111001000001111011011001110010 0001111011101001010100001011110010000111001110001111011010010100 1111000000001001000011100110110001111011111101000100111011010001 1010010001000000010111010000111010000101010111100011111010011100 1010011000001011001110001100000000100011011110000110011011110111 1001010101100011011110010010001000101101000100001000101100010100 1000110000010101011110001110010001011110111110001001110001100111 1000110110101011010100010100011100010111011011111101001110111001 1001011001010100110001101000011001100011111001111001000010011011 1110101001011110001001000001111100000110110111001011000001011101 1101010101001001010000010001001100100000100000011001010010010101 0000001001110010100101010110110110110001111110100001110111111011 1110100110100111010000000101100111010111100100100111110000011000 1000010011001001101101010111100110101010010100010110110010100011 0111000110011110011010000011011011011111000001000110110110001110 0000001000001001101110000000001111111100011001000110101001011110 0110011001010100101111010011111011110111101101000011110101111111 1110110101000011011111000111111110010100010001000010011000001111 1011110101000000110001001000001111101111010101010000001110000101 1000001111111001011110111011110101000101111011111011100001100110 0011000100000111000101000101110101011111111010111110011101100101 1010010010011110100101001110110001111111010110010111000100000101 1111101111111100001011100001111110100111011000111110111100000001 1111001101011001100111001000001011110010111111100101000000001011 1000010010001100111100001011110100100101001010101110000001000110 1011111110011111000111101111011110010100011111010100011001110110 1001101011111000110000010100101111001100011001111100011111000010 1001000010111110011101101001001010011011000001010111100011000001 0000101101011000010011111011010010000111110010010010010011110101 1011011100011111100000101101110011010010100100000011011000001001 1101111011101000100100010010100110000011110101001110101010101101 0000111011101010001100100001111101110100100010011111010001101010 0111111010010000001100001011111000100000111110110111011010010100 1110111110110101100011001001100110000100110011011101011100001010 0001110110101001000001000101110000111101000100110011101000000110 1000010000100011110101101110001110000011000000111101100100000001 1011101010011101101000110100011101100110100001000111100101101100 0101110011010101100101110010110111000000111111110011010101000000 1100001101000001001010010100001011010110010000000110000100000001 1110111101101111110001101101111010010001000101001010001010110100 1111001001001000110001101000100111000110000000001011101101000000 1010100010110101011010110000010000011111110101011101111001101110 0000110111010000110001100110110101001000001100011111111001111111 1111111101010111010101111110010001110001000010011000000011001101 1011110101011100001001101000010010000101110110100101111010010001 1011001111100010111100100000010110110111001001110010010110111001 0111000111010110000010100001111110001000100011110000100010100000 1010011011100001000000001100110011101101110000101100111001011011 1101100110001010111011100111000111100100001011100010011010001101 0011011110100110000001110010111100100010000000100011010001100001 0011111111111100001000100100010100110100001110011110101010010111 1010100110011001101101101100100111100011110011100111000111111001 0100110101100000100100101010110011100001001000001010101110001110 0101010100001110000011010101010100010001011010001000011000110001 0111011110001100111101100000001101010000110100000010111111101000 0101111100101001111011001000101111100101110001110010101110000000 0111101011110101011101110001101110000010010110110011000010100000 1110011110000011011101101010100100011100000010001100011010100111 1111000011111000111100110010001110110011011000101000000111010010 0010010101101000100111000000101101011010100000100000010001111101 1011100110001001111101100011101010001010011001001110100001010001 1001101111000000110100001100000111100010001000101000000001001000 0100110110010100111101001111100110111011001111010100101100110001 1101010010001001110101110101001000110001101101011100011000110011 1100010010000000110010010110101111100101010010011011111101011101 1001011001100111100010110100110100101100010011011101101010000110 0111101111011000111001001000000000101001111111101010100011001000 0001011100110101011001111100001010111010001111010011110011101001 1101111111100000101111010001101101001101110101110111000100010111 1000000001010111101101101001010110110111111010101111000110110000 0101110000100010110010001101010111111110111010111101000001110111 1111111011001001011011011011100011110111011110001111110000011100 0010101110111011110011100001101101001001111010111111010110101111 0100010001100000100010000010100101011000101011011101000000011100 1010011111110001101101101011110000001011011111101100000110111100 0110111000001010011011101101101111000110011111111000010110110010 0111010011100000100001100001101100111010000100110111010101110001 1011000101100101010010011000011100111101001000010001110001101010 1010111001101001110110000000000111100101011110010100010001011011 1100011000001010001111100000101001001111110110001001011010001110 1011011110010100101111011100011100000010110101101001010001011101 1010100101001011000001001010010001000000110010101011110010010100 0011100001111100001111010010011011111101000011110011101111101000 1010111101100011000001011010010100111010000101110111001010001000 1010110010100001001111111011010000000110110010011000001010010001 0101110110000011101110100000110100110101010110001101100000011101 1101000100010101100111101001011001000011111010101010001001111110 1011011101110101011110100010000001010010100101110101101101101111 0100101010001000100111100011110100001001001010111011000111000110 1000010101001000000011011100001011101001100110010100011110110011 0111001011011110110110100000010111100010000110010010111110010101 1011000110111001001001100101000100101011010000000100110000110011 0001100000011101011...
The distribution of 1’s and 0’s seems effectively random, as though the God of Mathematics were endlessly tossing a coin, putting 1 for heads, 0 for tails. Yet √2 is the opposite of a random number. Change a single digit anywhere and it ceases to be √2. Every 1 and every 0 is rigidly determined by “unalterable law”. So are the position and magnitude of the digits of √2 in every other base. Here, for example, is √2 in base 4:
1· 112220021321212133303233030210020230233230103121232222111133 103320322313230011311010213131100212131220233112100230012121 303020222211133210012002013111...
Another word for base-4 is DNA: genes are in fact written in a base-4 code based on the chemicals guanine, adenine, thymine and cytosine, or G, A, T, C for short. If the digits of √2 are truly random, in the statistical sense, then all genomes, actual and potential, occur somewhere along its length: yours, mine, the Emperor Heliogabalus’s, Bilbo Baggins’, the sabre-toothed tiger’s, the dodo’s, and so on. But almost all the “DNA” of √2 in base-4 will be meaningless: although √2 is the opposite of random, it is effectively a typing chimpanzee. Or a typing worm – a type-worm. √2 is like an endless worm that types out its own segments on a typewriter with two keys (for binary numbers) or four keys (for quaternary numbers) or ten keys (for decimal numbers) and so on.
But √2 doesn’t just encode the genomes of individual people, animals and plants: it encodes everything they do throughout their lives. In fact, it encodes the entire universe. And perhaps the universe is √2 or some number like it. Perhaps, in some sense, everything exists within the digits of an irrational number, or a sufficiently large rational number. If so, then √2 has become aware of itself through human beings: the World as Worm has bitten its own tail.
Appendix: Proof of the irrationality of √2
1. Suppose that there is some ratio, a/b, such that
2. a and b have no factors in common and
3. a^2/b^2 = 2.
4. It follows that a^2 = 2b^2.
5. Therefore a is even and there is some number, c, such that 2c = a.
6. Substituting c in #4, we derive (2c)^2 = 4c^2 = 2b^2.
7. Therefore 2c^2 = b^2 and b is also even.
8. But #7 contradicts #2 and the supposition that a and b have no factors in common.
9. Therefore, by reductio ad absurdum, there is no ratio, a/b, such that a^2/b^2 = 2. Q.E.D.
Stories and Stars
A story is stranger than a star. Stronger too. What do I mean? I mean that the story has more secrets than a star and holds its secrets more tightly. A full scientific description of a star is easier than a full scientific description of a story. Stars are much more primitive, much closer to the fundamentals of the universe. They’re huge and impressive, but they’re relatively simple things: giant spheres of flaming gas. Mathematically speaking, they’re more compressible: you have to put fewer numbers into fewer formulae to model their behaviour. A universe with just stars in it isn’t very complex, as you would expect from the evolution of our own universe. There were stars in it long before there were stories.
A universe with stories in it, by contrast, is definitely complex. This is because stories depend on language and language is the scientific mother-lode, the most difficult and important problem of all. Or rather, the human brain is. The human brain understands a lot about stars, despite their distance, but relatively little about itself, despite brains being right on the spot. Consciousness is a tough nut to crack, for example. Perhaps it’s uncrackable. Language looks easier, but linguistics is still more like stamp-collecting than science. We can describe the structure of language in detail – use labels like “pluperfect subjunctive”, “synecdoche”, “bilabial fricative” and so on – but we don’t know how that structure is instantiated in the brain or where language came from. How did it evolve? How is it coded in the human genome? How does meaning get into and out of sounds and shapes, into and out of speech and writing? These are big, important and very interesting questions, but we’ve barely begun to answer them.
Distribution of dental fricatives and the O blood-group in Europe (from David Crystal’s Cambridge Encyclopedia of Language)
But certain things seem clear already. Language-genes must differ in important ways between different groups, influencing their linguistic skills and their preferences in phonetics and grammar. For example, there are some interesting correlations between blood-groups and use of dental fricatives in Europe. The invention of writing has exerted evolutionary pressures in Europe and Asia in ways it hasn’t in Africa, Australasia and the Americas. Glossogenetics, or the study of language and genes, will find important differences between races and within them, running parallel with differences in psychology and physiology. Language is a human universal, but that doesn’t mean one set of identical genes underlies the linguistic behaviour of all human groups. Skin, bones and blood are human universals too, but they differ between groups for genetic reasons.
Understanding the evolution and effects of these genetic differences is ultimately a mathematical exercise, and understanding language will be too. So will understanding the brain. For one thing, the brain must, at bottom, be a maths-engine or math-engine: a mechanism receiving, processing and sending information according to rules. But that’s a bit like saying fish are wet. Fish can’t escape water and human beings can’t escape mathematics. Nothing can: to exist is to stand in relation to other entities, to influence and be influenced by them, and mathematics is about that inter-play of entities. Or rather, that inter-play is Mathematics, with a big “M”, and nothing escapes it. Human beings have invented a way of modelling that fundamental micro- and macroscopic inter-play, which is mathematics with a small “m”. When they use this model, human beings can make mistakes. But when they do go wrong, they can do so in ways detectable to other human beings using the same model:
In 1853 William Shanks published his calculations of π to 707 decimal places. He used the same formula as [John] Machin and calculated in the process several logarithms to 137 decimal places, and the exact value of 2^721. A Victorian commentator asserted: “These tremendous stretches of calculation… prove more than the capacity of this or that computer for labor and accuracy; they show that there is in the community an increase in skill and courage…”
Augustus de Morgan thought he saw something else in Shanks’s labours. The digit 7 appeared suspiciously less often than the other digits, only 44 times against an average expected frequency of 61 for each digit. De Morgan calculated that the odds against such a low frequency were 45 to 1. De Morgan, or rather William Shanks, was wrong. In 1945, using a desk calculator, Ferguson found that Shanks had made an error; his calculation was wrong from place 528 onwards. Shanks, fortunately, was long dead. (The Penguin Dictionary of Curious and Interesting Numbers, 1986, David Wells, entry for π, pg. 51)
Unlike theology or politics, mathematics is not merely self-correcting, but multiply so: there are different routes to the same truths and different ways of testing a result. Science too is self-correcting and can test its results by different means, partly because science is a mathematical activity and partly because it is studying a mathematical artifact: the gigantic structure of space, matter and energy known as the Universe. Some scientists and philosophers have puzzled over what the physicist Eugene Wigner (1902-95) called “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”. In his essay on the topic, Wigner tried to make two points:
The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it. Second, it is just this uncanny usefulness of mathematical concepts that raises the question of the uniqueness of our physical theories. (Op. cit., in Communications in Pure and Applied Mathematics, vol. 13, No. I, February 1960)
I disagree with Wigner: it is not mysterious or uncanny and there is a rational explanation for it. The “effectiveness” of small-m maths for scientists is just as reasonable as the effectiveness of fins for fish or of wings for birds. The sea is water and the sky is air. The universe contains both sea and sky: and the universe is maths. Fins and wings are mechanisms that allow fish and birds to operate effectively in their water- and air-filled environments. Maths is a mechanism that allows scientists to operate effectively in their maths-filled environment. Scientists have, in a sense, evolved towards using maths just as fish and birds have evolved towards using fins and wings. Men have always used language to model the universe, but language is not “unreasonably effective” for understanding the universe. It isn’t effective at all.
It is effective, however, in manipulating and controlling other human beings, which explains its importance in politics and theology. In politics, language is used to manipulate; in science, language is used to explain. That is why mathematics is so important in science and so carefully avoided in politics. And in certain academic disciplines. But the paradox is that physics is much more intellectually demanding than, say, literary theory because the raw stuff of physics is actually much simpler than literature. To understand the paradox, imagine that two kinds of boulder are strewn on a plain. One kind is huge and made of black granite. The other kind is relatively small and made of chalk. Two tribes of academic live on the plain, one devoted to studying the black granite boulders, the other devoted to studying the chalk boulders.
The granite academics, being unable to lift or cut into their boulders, will have no need of physical strength or tool-making ability. Instead, they will justify their existence by sitting on their boulders and telling stories about them or describing their bumps and contours in minute detail. The chalk academics, by contrast, will be lifting and cutting into their boulders and will know far more about them. So the chalk academics will need physical strength and tool-making ability. In other words, physics, being inherently simpler than literature, is within the grasp of a sufficiently powerful human intellect in a way literature is not. Appreciating literature depends on intuition rather than intellect. And so strong intellects are able to lift and cut into the problems of physics as they aren’t able to lift and cut into the problems of literature, because the problems of literature depend on consciousness and on the hugely complex mechanisms of language, society and psychology.
Intuition is extremely powerful, but isn’t under conscious control like intellect and isn’t transparent to consciousness in the same way. In the fullest sense, it includes the senses, but who can control his own vision and hearing or understand how they turn the raw stuff of the sense-organs into the magic tapestry of conscious experience? Flickering nerve impulses create a world of sight, sound, scent, taste and touch and human beings are able to turn that world into the symbols of language, then extract it again from the symbols. This linguifaction is a far more complex process than the ignifaction that drives a star. At present it’s beyond the grasp of our intellects, so the people who study it don’t need and don’t build intellectual muscle in the way that physicists do.
Or one could say that literature is at a higher level of physics. In theory, it is ultimately and entirely reducible to physics, but the mathematics governing its emergence from physics are complex and not well-understood. It’s like the difference between a caterpillar and a butterfly. They are two aspects of one creature, but it’s difficult to understand how one becomes the other, as a caterpillar dissolves into chemical soup inside a chrysalis and turns into something entirely different in appearance and behaviour. Modelling the behaviour of a caterpillar is simpler than modelling the behaviour of a butterfly. A caterpillar’s brain has less to cope with than a butterfly’s. Caterpillars crawl and butterflies fly. Caterpillars eat and butterflies mate. And so on.
Stars can be compared to caterpillars, stories to butterflies. It’s easier to explain stars than to explain stories. And one of the things we don’t understand about stories is how we understand stories.
2:1 Now when Jesus was born in Bethlehem of Judaea in the days of Herod the king, behold, there came wise men from the east to Jerusalem, 2:2 Saying, Where is he that is born King of the Jews? for we have seen his star in the east, and are come to worship him. 2:3 When Herod the king had heard these things, he was troubled, and all Jerusalem with him. 2:4 And when he had gathered all the chief priests and scribes of the people together, he demanded of them where Christ should be born. 2:5 And they said unto him, In Bethlehem of Judaea: for thus it is written by the prophet, 2:6 And thou Bethlehem, in the land of Juda, art not the least among the princes of Juda: for out of thee shall come a Governor, that shall rule my people Israel. 2:7 Then Herod, when he had privily called the wise men, enquired of them diligently what time the star appeared. 2:8 And he sent them to Bethlehem, and said, Go and search diligently for the young child; and when ye have found him, bring me word again, that I may come and worship him also. 2:9 When they had heard the king, they departed; and, lo, the star, which they saw in the east, went before them, till it came and stood over where the young child was. 2:10 When they saw the star, they rejoiced with exceeding great joy. 2:11 And when they were come into the house, they saw the young child with Mary his mother, and fell down, and worshipped him: and when they had opened their treasures, they presented unto him gifts; gold, and frankincense and myrrh. – From The Gospel According to Saint Matthew.
Roo’s Who
11 is a prime number, divisible by only itself and 1. If you add its digits, 1 + 1, you get 2. 11 + 2 = 13, another prime number. And 13 + (1 + 3) = 17, a third prime number. And there it ends, because 17 + (1 + 7) = 25 = 5 x 5. I call (11, 13, 17) kangaroo primes, because one jumps to another. In base 10, the record for numbers below 1,000,000 is this:
6 primes: 516493 + 28 = 516521 + 20 = 516541 + 22 = 516563 + 26 = 516589 + 34 = 516623.
In base 16, the record is this:
8 primes: 97397 = 17,C75[b=16] + 32 = 97429 = 17,C95[b=16] + 34 = 97463 = 17,CB7[b=16] + 38 = 97501 = 17,CDD[b=16] + 46 = 97547 = 17,D0B[b=16] + 32 = 97579 = 17,D2B[b=16] + 34 = 97613 = 17,D4D[b=16] + 38 = 97651 = 17,D73[b=16].
Another kind of kangaroo prime is found not by adding the sum of digits, but by adding their product, i.e., the result of multiplying the digits (except 0). 23 + (2 x 3) = 29. 29 + (2 x 9) = 47. But 47 + (4 x 7) = 75 = 3 x 5 x 5. So (23, 29, 47) are kangaroo primes too. In base 10, the record for numbers below 1,000,000 is this:
9 primes: 524219 + 720 = 524939 + 9720 = 534659 + 16200 = 550859 + 9000 = 559859 + 81000 = 640859 + 8640 = 649499 + 69984 = 719483 + 6048 = 725531.
But what about subtraction? For a reason I don’t understand, subtracting the digit-sum doesn’t seem to create any kangaroo-primes in base 10. But 11 in base 8 is 13 = 1 x 8^1 + 3 x 8^0 and 13[b=8] – (1 + 3) = 7. In base 2, this sequence appears:
1619 = 11,001,010,011[b=2] – 6 = 1613 = 11,001,001,101[b=2] – 6 = 1607 = 11,001,000,111[b=2] – 6 = 1601 = 11,001,000,001[b=2] – 4 = 1597.
However, subtracting the digit-product creates kangaroo-primes in base 10. For example, 23 – (2 x 3) = 17. The record below 1,000,000 is this (when 0 is found in the digits of a number, it is not included in the multiplication):
7 primes: 64037 – 504 = 63533 – 810 = 62723 – 504 = 62219 – 216 = 62003 – 36 = 61967 – 2268 = 59699.
Base 2 also provides examples of addition/subtraction pairs of kangaroo-primes, like this:
3 = 11[b=2] + 2 = 5 = 101[b=2] | 5 = 101[b=2] – 2 = 3
277 = 100,010,101[b=2] + 4 = 281 = 100,011,001[b=2] | 281 – 4 = 277
311 = 100,110,111[b=2] + 6 = 317 = 100,111,101[b=2] | 317 – 6 = 311
In base 10, addition/subtraction pairs are created by the digit-product, like this:
239 + 54 = 293 | 293 – 54 = 239
563 + 90 = 653 | 653 – 90 = 563
613 + 18 = 631 | 631 – 18 = 613
2791 + 126 = 2917 | 2917 – 126 = 2791
3259 + 270 = 3529 | 3529 – 270 = 3259
5233 + 90 = 5323 | 5323 – 90 = 5233
5297 + 630 = 5927 | 5927 – 630 = 5297
6113 + 18 = 6131 | 6131 – 18 = 6113
10613 + 18 = 10631 | 10631 – 18 = 10613
12791 + 126 = 12917 | 12917 – 126 = 12791
You could call these boxing primes, like boxing kangaroos. The two primes in the pair usually have the same digits in different arrangements, but there are also pairs like these:
24527 + 560 = 25087 | 25087 – 560 = 24527
25183 + 240 = 25423 | 25423 – 240 = 25183
50849 + 1440 = 52289 | 52289 – 1440 = 50849
Overlord In Terms of Core Issues Around Maximal Engagement with Key Notions of the Über-Feral 