Cover of ASG’s Survive Sunrise (2018) by Malleus Rock Art Lab
Hersute Nubility
Author Archives: Krilling for Company
Hersute Nubility
Prime Times
The factorial of an integer is equal to that that integer multiplied by all the integers smaller than it. For example, this is factorial(7) or 7!:
7! = 7 * 6 * 5 * 4 * 3 * 2 * 1 = 5040
The primorial of a prime is equal to that that prime multiplied by all the primes smaller than it. For example, this is primorial(7):
primorial(7) = 7 * 5 * 3 * 2 = 210 = 4# (the product of the first four primes)
Here’s an interesting set of primorials incremented-by-one:
primorial(2) + 1 = 2 + 1 = 3 (prime)
primorial(3) + 1 = 2*3 + 1 = 7 (prime)
primorial(5) + 1 = 2*3*5 + 1 = 31 (prime)
primorial(7) + 1 = 2*3*5*7 + 1 = 211 (prime)
primorial(11) + 1 = 2*3*5*7*11 + 1 = 2311 (prime)
primorial(31) + 1 = 2*3*5*7*11*13*17*19*23*29*31 + 1 = 200560490131 (prime)
primorial(379) + 1 = 1,719,620,105,458,406,433,483,340,568,317,543,019,584,575,635,895,742,560,438,771,105,058,321,655,238,562,613,083,979,651,479,555,788,009,994,557,822,024,565,226,932,906,295,208,262,756,822,275,663,694,111 (prime)
primorial(1019) + 1 = 20,404,068,993,016,374,194,542,464,172,774,607,695,659,797,117,423,121,913,227,131,032,339,026,169,175,929,902,244,453,757,410,468,728,842,929,862,271,605,567,818,821,685,490,676,661,985,389,839,958,622,802,465,986,881,376,139,404,138,376,153,096,103,140,834,665,563,646,740,160,279,755,212,317,501,356,863,003,638,612,390,661,668,406,235,422,311,783,742,390,510,526,587,257,026,500,302,696,834,793,248,526,734,305,801,634,165,948,702,506,367,176,701,233,298,064,616,663,553,716,975,429,048,751,575,597,150,417,381,063,934,255,689,124,486,029,492,908,966,644,747,931 (prime)
primorial(1021) + 1 = 20,832,554,441,869,718,052,627,855,920,402,874,457,268,652,856,889,007,473,404,900,784,018,145,718,728,624,430,191,587,286,316,088,572,148,631,389,379,309,284,743,016,940,885,980,871,887,083,026,597,753,881,317,772,605,885,038,331,625,282,052,311,121,306,792,193,540,483,321,703,645,630,071,776,168,885,357,126,715,023,250,865,563,442,766,366,180,331,200,980,711,247,645,589,424,056,809,053,468,323,906,745,795,726,223,468,483,433,625,259,000,887,411,959,197,323,973,613,488,345,031,913,058,775,358,684,690,576,146,066,276,875,058,596,100,236,112,260,054,944,287,636,531 (prime)
primorial(2657) + 1 = 78,244,737,296,323,701,708,091,142,569,062,680,832,012,147,734,404,650,078,590,391,114,054,859,290,061,421,837,516,998,655,549,776,972,299,461,276,876,623,748,922,539,131,984,799,803,433,363,562,299,977,701,808,549,255,204,262,920,151,723,624,296,938,777,341,738,751,806,450,993,015,446,712,522,509,989,316,673,420,506,749,359,414,629,957,842,716,112,900,306,643,009,542,215,969,000,431,330,219,583,111,410,996,807,066,475,261,560,303,182,609,636,056,108,367,412,324,508,444,341,178,028,289,201,803,518,093,842,982,877,662,621,552,756,279,669,241,303,362,152,895,160,479,720,040,128,335,518,247,125,849,521,099,841,272,983,588,935,580,888,630,036,283,712,163,901,558,436,498,481,482,160,712,530,124,868,714,141,094,634,892,999,056,865,426,200,254,647,241,979,548,935,087,621,308,526,547,138,125,987,102,062,688,568,486,250,939,447,065,798,353,626,745,169,380,579,442,233,006,898
,444,700,264,240,321,482,823,859,842,044,524,114,576,784,795,294,818,755,525,169,192,652,108,755,230,262,128,210,258,672,754,900,845,837,728,345,782,457,465,793,874,408,469,588,052,577,208,643,754,019,053,756,394,151,041,512,099,598,925,557,724,343,099,264,685,155,934,891,439,161,866,250,113,047,185,553,511,797,406,764,115,907,248,713,405,817,594,729,550,600,082,808,324,331,387,143,679,800,355,356,811,873,430,669,962,333,651,282,822,030,473,702,042,073,141,618,450,021,084,993,659,382,646,598,194,115,178,864,433,545,186,250,667,775,794,249,961,932,761,063,071,117,967,553,887,984,011,652,643,245,393,971 (prime)
primorial(3229) + 1 = 689,481,240,122,180,255,681,227,812,346,871,771,457,221,628,238,467,511,261,402,638,443,056,696,165,896,544,725,098,860,107,293,247,422,610,010,824,870,599,655,026,129,367,004,672,337,297,193,288,816,463,520,704,235,722,580,204,218,943,598,425,089,855,869,341,564,771,022,924,163,236,141,415,235,947,085,902,422,536,824,665,765,244,189,167,643,048,218,572,769,125,400,511,177,245,717,452,516,267,205,786,258,497,574,258,715,214,994,129,786,103,824,740,384,634,788,909,041,221,747,073,062,941,769,355,745,272,170,421,584,636,198,911,899,164,272,930,590,704,655,882,680,817,754,473,306,122,122,423,384,160,639,995,940,152,584,830,810,911,265,680,382,263,051,658,031,509,463,010,733,595,465,426,943,956,643,445,876,702,680,730,987,739,513,538,299,069,540,636,616,098,525,527,546,435,002,783,615,353,417,794,625,251,129,892,373,849,727,119,530,335,366,131,575,986,221,685,088,118,143,088,371,896,087,248,659,669,154,564,925,048,225,211,644,681,303,874,490,648,860,319,990,785,185,350,796,853,298,548,942,407,689,617,641,587,755,314,125,485,345,107,782,298,938,892,240,282,038,605,672,241,010,302,874,153,509,795,545,077,305,234,459,038,983,235,361,138,814,897,166,376,363,090,128,647,084,552,385,969,054,439,430,382,421,762,883,708,894,899,853,286,109,068,224,980,793,075,241,538,872,287,253,835,877,394,821,667,363,465,425,187,353,453,157,415,169,810,167,271,517,665,273,484,442,461,468,031,313,956,356,871,467,191,959,110,440,864,194,544,244,079,053,955,897,287,010,339,385,419,923,838,571,256,564,818,350,769,518,898,003,780,557,167,344,272,499,224,580,817,920,441,512,610,104,625,622,872,289,967,615,843,092,782,763,554,732,404,239,287,463,466,833,602,966,629,613,502,579,134,371,295,289,680,374,088,987,611,189,907,873,072,122,808,833,765,972,650,050,982,877,578,244,899,073,193,043,546,490,795,625,023,568,563,926,988,371 (prime)
Elsewhere Other-Accessible
• A005234 at the Online Encylopedia of Integer Sequences — “Primorial plus 1 primes: primes p such that 1 + product of primes up to p is prime”.
मेरा मॉज़ महान
Post-Performative Post-Scriptum
The title of this incendiary intervention is again self-explanatory.
Gleet the Beatles
The Guardian incisively interrogates issues around the Scouse Superstars:
Just in terms of pure sales they still dominate. In the first half of the year in the US – half a century on from Ed Sullivan, screaming fans, the olds just not getting it – they sold more albums than anyone else; the only group that came close over that period were BTS, a group who are regularly compared to the Beatles in terms of their planet-straddling massiveness. — The Guide #10: the enduring appeal of the Beatles, The Guardian, 26xi21
Elsewhere other-accessible
• Ex-Term-In-Ate! — interrogating issues around why “in terms of” is so teratographically toxic…
• All posts interrogating issues around “in terms of”…
• All posts interrogating issues around the Guardian-reading community and its affiliates…
Lichen Likened
Pixie-cup lichen, Cladonia pyxidata (possibly)
The Whisper of the Stars
• Le record de froid peut atteindre -77°C, alors que l’été le thermomètre peut monter jusqu’à 30°C. Les températures hivernales causent des phénomènes étonnants. Par exemple, ce que les Yakoutes appellent « le chuchotement des étoiles » : lorsqu’il gèle, l’homme entend en permanence le doux bruissement de sa respiration qui gèle dès qu’il expire.
• At its worst the cold can reach -77°C, while in summer the thermometer can climb to 30°C. Winter temperatures cause some astonishing phenomena. For example, there is what the Yakuts call “the whisper of the stars”: when it’s freezing, you constantly hear the soft rustle of your own breath, which is turning into ice-crystals even as you exhale.
Elsewhere other-engageable
• Cry’ Me A Shiver — an interview with French avant-gardistes Cryogénie, les Rois du Froid and Kings of Cold…
Silicircle
Diatoms with siliceous shells arranged in a filled circle using a microscope
• Photograph of diatoms collected in Russia and arranged on a microscope slide in 1952 by A.L. Brigger
Fib and Let Tri
It’s a simple sequence with hidden depths:
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, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155... — A000045 at OEIS
That’s the Fibonacci sequence, probably the most famous of all integer sequences after the integers themselves (1, 2, 3, 4, 5…) and the primes (2, 3, 5, 7, 11…). It has a very simple definition: if fib(fi) is the fi-th number in the Fibonacci sequence, then fib(fi) = fib(fi-1) + fib(fi-2). By definition, fib(1) = fib(2) = 1. After that, it’s easy to generate new numbers:
2 = fib(3) = fib(1) + fib(2) = 1 + 1
3 = fib(4) = fib(2) + fib(3) = 1 + 2
5 = fib(5) = fib(3) + fib(4) = 2 + 3
8 = fib(6) = fib(4) + fib(5) = 3 + 5
13 = fib(7) = fib(5) + fib(6) = 5 + 8
21 = fib(8) = fib(6) + fib(7) = 8 + 13
34 = fib(9) = fib(7) + fib(8) = 13 + 21
55 = fib(10) = fib(8) + fib(9) = 21 + 34
89 = fib(11) = fib(9) + fib(10) = 34 + 55
144 = fib(12) = fib(10) + fib(11) = 55 + 89
233 = fib(13) = fib(11) + fib(12) = 89 + 144
377 = fib(14) = fib(12) + fib(13) = 144 + 233
610 = fib(15) = fib(13) + fib(14) = 233 + 377
987 = fib(16) = fib(14) + fib(15) = 377 + 610
[...]
How to create the Fibonacci sequence is obvious. But it’s not obvious that fib(fi) / fib(fi-1) gives you ever-better approximations to a fascinating constant called φ, the golden ratio, which is 1.618033988749894…:
1/1 = 1
2/1 = 2
3/2 = 1.5
5/3 = 1.66666...
8/5 = 1.6
13/8 = 1.625
21/13 = 1.615384...
34/21 = 1.619047...
55/34 = 1.6176470588235294117647058823...
89/55 = 1.618181818...
144/89 = 1.617977528089887640...
233/144 = 1.6180555555...
377/233 = 1.618025751072961...
610/377 = 1.618037135278514...
987/610 = 1.618032786885245...
[...]
And that’s just the start of the hidden depths in the Fibonacci sequence. I stumbled across another interesting pattern for myself a few days ago. I was looking at the sequence and one of the numbers caught my eye:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597...
55 is a palindrome, reading the same forward and backwards. I wondered whether there were any other palindromes in the sequence (apart from the trivial single-digit palindromes 1, 1, 2, 3…). I couldn’t find any more. Nor can anyone else, apparently. But that’s in base 10. Other bases are more productive. For example, in bases 2, 3 and 4, you get this:
11 in b2 = 3
101 in b2 = 5
10101 in b2 = 21
22 in b3 = 8
111 in b3 = 13
22122 in b3 = 233
11 in b4 = 5
111 in b4 = 21
202 in b4 = 34
313 in b4 = 55
I decided to concentrate on tripals, or palindromes with three digits. I started looking at bases that set records for the greatest number of tripals. And there are some interesting patterns in the digits of the tripals in these bases (when a digit > 9, the digit is represented inside square brackets — see base-29 and higher). See how quickly you can spot the patterns:
Palindromic Fibonacci numbers in base-4
111 in b4 (fib=21, fi=8)
202 in b4 (fib=34, fi=9)
313 in b4 (fib=55, fi=10)
4 = 2^2 (pal=3)
Palindromic Fibonacci numbers in base-11
121 in b11 (fib=144, fi=12)
313 in b11 (fib=377, fi=14)
505 in b11 (fib=610, fi=15)
818 in b11 (fib=987, fi=16)
11 is prime (pal=4)
Palindromic Fibonacci numbers in base-29
151 in b29 (fib=987, fi=16)
323 in b29 (fib=2584, fi=18)
818 in b29 (fib=6765, fi=20)
[13]0[13] in b29 (fib=10946, fi=21)
[21]1[21] in b29 (fib=17711, fi=22)
29 is prime (pal=5)
Palindromic Fibonacci numbers in base-76
1[13]1 in b76 (fib=6765, fi=20)
353 in b76 (fib=17711, fi=22)
828 in b76 (fib=46368, fi=24)
[21]1[21] in b76 (fib=121393, fi=26)
[34]0[34] in b76 (fib=196418, fi=27)
[55]1[55] in b76 (fib=317811, fi=28)
76 = 2^2 * 19 (pal=6)
Palindromic Fibonacci numbers in base-199
1[34]1 in b199 (fib=46368, fi=24)
3[13]3 in b199 (fib=121393, fi=26)
858 in b199 (fib=317811, fi=28)
[21]2[21] in b199 (fib=832040, fi=30)
[55]1[55] in b199 (fib=2178309, fi=32)
[89]0[89] in b199 (fib=3524578, fi=33)
[144]1[144] in b199 (fib=5702887, fi=34)
199 is prime (pal=7)
Palindromic Fibonacci numbers in base-521
1[89]1 in b521 (fib=317811, fi=28)
3[34]3 in b521 (fib=832040, fi=30)
8[13]8 in b521 (fib=2178309, fi=32)
[21]5[21] in b521 (fib=5702887, fi=34)
[55]2[55] in b521 (fib=14930352, fi=36)
[144]1[144] in b521 (fib=39088169, fi=38)
[233]0[233] in b521 (fib=63245986, fi=39)
[377]1[377] in b521 (fib=102334155, fi=40)
521 is prime (pal=8)
Palindromic Fibonacci numbers in base-1364
1[233]1 in b1364 (fib=2178309, fi=32)
3[89]3 in b1364 (fib=5702887, fi=34)
8[34]8 in b1364 (fib=14930352, fi=36)
[21][13][21] in b1364 (fib=39088169, fi=38)
[55]5[55] in b1364 (fib=102334155, fi=40)
[144]2[144] in b1364 (fib=267914296, fi=42)
[377]1[377] in b1364 (fib=701408733, fi=44)
[610]0[610] in b1364 (fib=1134903170, fi=45)
[987]1[987] in b1364 (fib=1836311903, fi=46)
1364 = 2^2 * 11 * 31 (pal=9)
Two patterns are quickly obvious. Every digit in the tripals is a Fibonacci number. And the middle digit of one Fibonacci tripal, fib(fi), becomes fib(fi-2) in the next tripal, while fib(fi), the first and last digits (which are identical), becomes fib(fi+2) in the next tripal.
But what about the bases? If you’re an expert in the Fibonacci sequence, you’ll spot the pattern at work straight away. I’m not an expert, but I spotted it in the end. Here are the first few bases setting records for the numbers of Fibonacci tripals:
4, 11, 29, 76, 199, 521, 1364, 3571, 9349, 24476, 64079, 167761, 439204, 1149851, 3010349, 7881196...
These numbers come from the Lucas sequence, which is closely related to the Fibonacci sequence. But where fib(1) = fib(2) = 1, luc(1) = 1 and luc(2) = 3. After that, luc(li) = luc(li-2) + luc(li-1):
1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, 15127, 24476, 39603, 64079, 103682, 167761, 271443, 439204, 710647, 1149851, 1860498, 3010349, 4870847, 7881196... — A000204 at OEIS
It seems that every second number from 4 in the Lucas sequence supplies a base in which 1) the number of Fibonacci tripals sets a new record; 2) every digit of the Fibonacci tripals is itself a Fibonacci number.
But can I prove that this is always true? No. And do I understand why these patterns exist? No. My simple search for palindromes in the Fibonacci sequence soon took me far out of my mathematical depth. But it’s been fun to find huge bases like this in which every digit of every Fibonacci tripal is itself a Fibonacci number:
Palindromic Fibonacci numbers in base-817138163596
1[139583862445]1 in b817138163596 (fib=781774079430987230203437, fi=116)
3[53316291173]3 in b817138163596 (fib=2046711111473984623691759, fi=118)
8[20365011074]8 in b817138163596 (fib=5358359254990966640871840, fi=120)
[21][7778742049][21] in b817138163596 (fib=14028366653498915298923761, fi=122)
[55][2971215073][55] in b817138163596 (fib=36726740705505779255899443, fi=124)
[144][1134903170][144] in b817138163596 (fib=96151855463018422468774568, fi=126)
[377][433494437][377] in b817138163596 (fib=251728825683549488150424261, fi=128)
[987][165580141][987] in b817138163596 (fib=659034621587630041982498215, fi=130)
[2584][63245986][2584] in b817138163596 (fib=1725375039079340637797070384, fi=132)
[6765][24157817][6765] in b817138163596 (fib=4517090495650391871408712937, fi=134)
[17711][9227465][17711] in b817138163596 (fib=11825896447871834976429068427, fi=136)
[46368][3524578][46368] in b817138163596 (fib=30960598847965113057878492344, fi=138)
[121393][1346269][121393] in b817138163596 (fib=81055900096023504197206408605, fi=140)
[317811][514229][317811] in b817138163596 (fib=212207101440105399533740733471, fi=142)
[832040][196418][832040] in b817138163596 (fib=555565404224292694404015791808, fi=144)
[2178309][75025][2178309] in b817138163596 (fib=1454489111232772683678306641953, fi=146)
[5702887][28657][5702887] in b817138163596 (fib=3807901929474025356630904134051, fi=148)
[14930352][10946][14930352] in b817138163596 (fib=9969216677189303386214405760200, fi=150)
[39088169][4181][39088169] in b817138163596 (fib=26099748102093884802012313146549, fi=152)
[102334155][1597][102334155] in b817138163596 (fib=68330027629092351019822533679447, fi=154)
[267914296][610][267914296] in b817138163596 (fib=178890334785183168257455287891792, fi=156)
[701408733][233][701408733] in b817138163596 (fib=468340976726457153752543329995929, fi=158)
[1836311903][89][1836311903] in b817138163596 (fib=1226132595394188293000174702095995, fi=160)
[4807526976][34][4807526976] in b817138163596 (fib=3210056809456107725247980776292056, fi=162)
[12586269025][13][12586269025] in b817138163596 (fib=8404037832974134882743767626780173, fi=164)
[32951280099]5[32951280099] in b817138163596 (fib=22002056689466296922983322104048463, fi=166)
[86267571272]2[86267571272] in b817138163596 (fib=57602132235424755886206198685365216, fi=168)
[225851433717]1[225851433717] in b817138163596 (fib=150804340016807970735635273952047185, fi=170)
[365435296162]0[365435296162] in b817138163596 (fib=244006547798191185585064349218729154, fi=171)
[591286729879]1[591286729879] in b817138163596 (fib=394810887814999156320699623170776339, fi=172)
817138163596 = 2^2 * 229 * 9349 * 95419 (pal=30)
Flaubert le Flaubard du Flaubeau
«Je ne suis rien qu’un lézard littéraire qui se chauffe toute la journée au grand soleil du beau» — Gustave Flaubert, Croisset, 17 octobre 1846
• “I am nothing but a literary lizard basking all day in the great sun of beauty.”
Six Mix Trix
Here’s an equilateral triangle divided into six smaller triangles:
Equilateral triangle divided into six irregular triangles (Stage #1)
Now keep on dividing:
Stage #2
Stage #3
Stage #4
Stage #5
Equilateral triangle dividing into six irregular triangles (animated)
But what happens if you divide the triangle, then discard some of the sub-triangles, then repeat? You get a self-similar shape called a fractal:
Divide-and-discard stage #1
Stage #2
Stage #3
Stage #4
Stage #5
Stage #6
Triangle fractal (animated)
Here’s another example:
Divide-and-discard stage #1
Stage #2
Stage #3
Stage #4
Stage #5
Stage #6
Stage #7
Triangle fractal (animated)
You can also delay the divide-and-discard to create a more symmetrical fractal, like this:
Delayed divide-and-discard stage #1
Stage #2
Stage #3
Stage #4
Stage #5
Stage #6
Stage #7
Triangle fractal (animated)
What next? You can use trigonometry to turn the cramped triangle into a circle:
Triangular fractal
⇓
Circular fractal
(Open in new window for full image)
Triangle-to-circle (animated)
Here’s another example:
Triangular fractal
⇓
Circular fractal
Triangle-to-circle (animated)
And below are some more circular fractals converted from triangular fractals. Some of them look like distorted skulls or transdimensional Lovecraftian monsters:
(Open in new window for full image)
Previous Pre-Posted
• Circus Trix — an earlier look at sextally-divided-equilateral-triangle fractals
Overlord In Terms of Core Issues Around Maximal Engagement with Key Notions of the Über-Feral 