Zequality Now

Here are the numbers one to eight in base 2:

1, 10, 11, 100, 101, 110, 111, 1000…

Now see what happens when you count the zeroes:


1, 10[1], 11, 10[2]0[3], 10[4]1, 110[5], 111, 10[6]0[7]0[8]...

In base 2, the numbers one to eight contain exactly eight zeroes, that is, zerocount(1..8,b=2) = 8. But it doesn’t work out so exactly in base 3:


1, 2, 10[1], 11, 12, 20[2], 21, 22, 10[3]0[4], 10[5]1, 10[6]2, 110[7], 111, 112, 120[8], 121, 122, 20[9]0[10], 20[11]1, 20[12]2, 210[13], 211, 212, 220[14], 221, 222, 10[15]0[16]0[17], 10[18]0[19]1, 10[20]0[21]2, 10[22]10[23], 10[24]11, 10[25]12, 10[26]20[27], 10[28]21, 10[29]22, 110[30]0[31], 110[32]1, 110[33]2, 1110[34], 1111, 1112, 1120[35], 1121, 1122, 120[36]0[37], 120[38]1, 120[39]2, 1210[40], 1211, 1212, 1220[41], 1221, 1222, 20[42]0[43]0[44], 20[45]0[46]1, 20[47]0[48]2, 20[49]10[50], 20[51]11, 20[52]12, 20[53]20[54], 20[55]21, 20[56]22, 210[57]0[58], 210[59]1, 210[60]2, 2110[61], 2111, 2112, 2120[62], 2121, 2122, 220[63]0[64], 220[65]1, 220[66]2, 2210[67], 2211, 2212, 2220[68], 2221, 2222, 10[69]0[70]0[71]0[72], 10[73]0[74]0[75]1, 10[76]0[77]0[78]2, 10[79]0[80]10[81], 10[82]0[83]11, 10[84]0[85]12, 10[86]0[87]20[88]...

In base 3, 10020 = 87 and zerocount(1..87,b=3) = 88. And what about base 4? zerocount(1..1068,b=4) = 1069 (n=100,230 in base 4). After that, zerocount(1..16022,b=5) = 16023 (n=1,003,043 in base 5) and zerocount(1..284704,b=6) = 284,705 (n=10,034,024 in base 6).

The numbers are getting bigger fast and it’s becoming increasingly impractible to count the zeroes individually. What you need is an algorithm that will take any given n and work out how many zeroes are required to write the numbers 1 to n. The simplest way to do this is to work out how many times 0 has appeared in each position of the number. The 1s position is easy: you simply divide the number by the base and discard the remainder. For example, in base 10, take the number 25. The 0 must have appeared in the 1s position twice, for 10 and 20, so zerocount(1..25) = 25 \ 10 = 2. In 2017, the 0 must have appeared in the 1s position 201 times = 2017 \ 10. And so on.

It gets a little trickier for the higher positions, the 10s, 100s, 1000s and so on, but the same basic principle applies. And so you can easily create an algorithm that takes a number, n, and produces zerocount(1..n) in a particular base. With this algorithm, you can quickly find zerocount(1..n) >= n in higher bases:


zerocount(1..1000,b=2) = 1,000 (n=8)*
zerocount(1..10020,b=3) = 10,021 (n=87)
zerocount(1..100230,b=4) = 100,231 (n=1,068)
zerocount(1..1003042,b=5) = 1,003,043 (n=16,022)
zerocount(1..10034024,b=6) = 10,034,025 (n=284,704)
zerocount(1..100405550,b=7) = 100,405,551 (n=5,834,024)
zerocount(1..1004500236,b=8) = 1,004,500,237 (n=135,430,302)
zerocount(1..10050705366,b=9) = 10,050,705,367 (n=3,511,116,537)
zerocount(1..100559404366,b=10) = 100,559,404,367
zerocount(1..1006083A68919,b=11) = 1,006,083,A68,919 (n=3,152,738,985,031)*
zerocount(1..10066AA1430568,b=12) = 10,066,AA1,430,569 (n=107,400,330,425,888)
zerocount(1..1007098A8719B81,b=13) = 100,709,8A8,719,B81 (n=3,950,024,143,546,664)*
zerocount(1..10077C39805D81C7,b=14) = 1,007,7C3,980,5D8,1C8 (n=155,996,847,068,247,395)
zerocount(1..10080B0034AA5D16D,b=15) = 10,080,B00,34A,A5D,171 (n=6,584,073,072,068,125,453)
zerocount(1..10088DBE29597A6C77,b=16) = 100,88D,BE2,959,7A6,C77 (n=295,764,262,988,176,583,799)*
zerocount(1..10090C5309AG72CBB3F,b=17) = 1,009,0C5,309,AG7,2CB,B3G (n=14,088,968,131,538,370,019,982)
zerocount(1..10099F39070FC73C1G73,b=18) = 10,099,F39,070,FC7,3C1,G75 (n=709,394,716,006,812,244,474,473)
zerocount(1..100A0DC1258614CA334EB,b=19) = 100,A0D,C12,586,14C,A33,4EC (n=37,644,984,315,968,494,382,106,708)
zerocount(1..100AAGDEEB536IBHE87006,b=20) = 1,00A,AGD,EEB,536,IBH,E87,008 (n=2,099,915,447,874,594,268,014,136,006)

And you can also easily find the zequal numbers, that is, the numbers n for which, in some base, zerocount(1..n) exactly equals n:


zerocount(1..1000,b=2) = 1,000 (n=8)
zerocount(1..1006083A68919,b=11) = 1,006,083,A68,919 (n=3,152,738,985,031)
zerocount(1..1007098A8719B81,b=13) = 100,709,8A8,719,B81 (n=3,950,024,143,546,664)
zerocount(1..10088DBE29597A6C77,b=16) = 100,88D,BE2,959,7A6,C77 (n=295,764,262,988,176,583,799)
zerocount(1..100CCJFFAD4MI409MI0798CJB3,b=24) = 10,0CC,JFF,AD4,MI4,09M,I07,98C,JB3 (n=32,038,681,563,209,056,709,427,351,442,469,835)
zerocount(1..100DDL38CIO4P9K0AJ7HK74EMI7L,b=26) = 1,00D,DL3,8CI,O4P,9K0,AJ7,HK7,4EM,I7L (n=160,182,333,966,853,031,081,693,091,544,779,177,187)
zerocount(1..100EEMHG6OE8EQKO0BF17LCCIA7GPE,b=28) = 100,EEM,HG6,OE8,EQK,O0B,F17,LCC,IA7,GPE (n=928,688,890,453,756,699,447,122,559,347,771,300,777,482)
zerocount(1..100F0K7MQO6K9R1S616IEEL2JRI73PF,b=29) = 1,00F,0K7,MQO,6K9,R1S,616,IEE,L2J,RI7,3PF (n=74,508,769,042,363,852,559,476,397,161,338,769,391,145,562)
zerocount(1..100G0LIL0OQLF2O0KIFTK1Q4DC24HL7BR,b=31) = 100,G0L,IL0,OQL,F2O,0KI,FTK,1Q4,DC2,4HL,7BR (n=529,428,987,529,739,460,369,842,168,744,635,422,842,585,510,266)
zerocount(1..100H0MUTQU3A0I5005WL2PD7T1ASW7IV7NE,b=33) = 10,0H0,MUT,QU3,A0I,500,5WL,2PD,7T1,ASW,7IV,7NE (n=4,262,649,311,868,962,034,947,877,223,846,561,239,424,294,726,563,632)
zerocount(1..100HHR387RQHK9OP6EDBJEUDAK35N7MN96LB,b=34) = 100,HHR,387,RQH,K9O,P6E,DBJ,EUD,AK3,5N7,MN9,6LB (n=399,903,937,958,473,433,782,862,763,628,747,974,628,490,691,628,136,485)
zerocount(1..100IISLI0CYX2893G9E8T4I7JHKTV41U0BKRHT,b=36) = 10,0II,SLI,0CY,X28,93G,9E8,T4I,7JH,KTV,41U,0BK,RHT (n=3,831,465,379,323,568,772,890,827,210,355,149,992,132,716,389,119,437,755,185)
zerocount(1..100LLX383BPWE[40]ZL0G1M[40]1OX[39]67KOPUD5C[40]RGQ5S6W9[36],b=42) = 10,0LL,X38,3BP,WE[40],ZL0,G1M,[40]1O,X[39]6,7KO,PUD,5C[40],RGQ,5S6,W9[36] (n=6,307,330,799,917,244,669,565,360,008,241,590,852,337,124,982,231,464,556,869,653,913,711,854)
zerocount(1..100MMYPJ[38]14KDV[37]OG[39]4[42]X75BE[39][39]4[43]CK[39]K36H[41]M[37][43]5HIWNJ,b=44) = 1,00M,MYP,J[38]1,4KD,V[37]O,G[39]4,[42]X7,5BE,[39][39]4,[43]CK,[39]K3,6H[41],M[37][43],5HI,WNJ (n=90,257,901,046,284,988,692,468,444,260,851,559,856,553,889,199,511,017,124,021,440,877,333,751,943)
zerocount(1..100NN[36]3813[38][37]16F6MWV[41]UBNF5FQ48N0JRN[40]E76ZOHUNX2[42]3[43],b=46) = 100,NN[36],381,3[38][37],16F,6MW,V[41]U,BNF,5FQ,48N,0JR,N[40]E,76Z,OHU,NX2,[42]3[43] (n=1,411,636,908,622,223,745,851,790,772,948,051,467,006,489,552,352,013,745,000,752,115,904,961,213,172,605)
zerocount(1..100O0WBZO9PU6O29TM8Y0QE3I[37][39]A7E4YN[44][42]70[44]I[46]Z[45][37]Q2WYI6,b=47) = 1,00O,0WB,ZO9,PU6,O29,TM8,Y0Q,E3I,[37][39]A,7E4,YN[44],[42]70,[44]I[46],Z[45][37],Q2W,YI6 (n=182,304,598,281,321,725,937,412,348,242,305,189,665,300,088,639,063,301,010,710,450,793,661,266,208,306,996)
zerocount(1..100PP[39]37[49]NIYMN[43]YFE[44]TDTJ00EAEIP0BIDFAK[46][36]V6V[45]M[42]1M[46]SSZ[40],b=50) = 1,00P,P[39]3,7[49]N,IYM,N[43]Y,FE[44],TDT,J00,EAE,IP0,BID,FAK,[46][36]V,6V[45],M[42]1,M[46]S,SZ[40] (n=444,179,859,561,011,965,929,496,863,186,893,220,413,478,345,535,397,637,990,204,496,296,663,272,376,585,291,071,790)
zerocount(1..100Q0Y[46][44]K[49]CKG[45]A[47]Z[43]SPZKGVRN[37]2[41]ZPP[36]I[49][37]EZ[38]C[44]E[46]00CG[38][40][48]ROV,b=51) = 10,0Q0,Y[46][44],K[49]C,KG[45],A[47]Z,[43]SP,ZKG,VRN,[37]2[41],ZPP,[36]I[49],[37]EZ,[38]C[44],E[46]0,0CG,[38][40][48],ROV (n=62,191,970,278,446,971,531,566,522,791,454,395,351,613,891,150,548,291,266,262,575,754,206,359,828,753,062,692,619,547)
zerocount(1..100QQ[40]TL[39]ZA[49][41]J[41]7Q[46]4[41]66A1E6QHHTM9[44]8Z892FRUL6V[46]1[38][41]C[40][45]KB[39],b=52) = 100,QQ[40],TL[39],ZA[49],41]J[41],7Q[46],4[41]6,6A1,E6Q,HHT,M9[44],8Z8,92F,RUL,6V[46],1[38][41],C[40][45],KB[39] (n=8,876,854,501,927,007,077,802,489,292,131,402,136,556,544,697,945,824,257,389,527,114,587,644,068,732,794,430,403,381,731)
zerocount(1..100S0[37]V[53]Y6G[51]5J[42][38]X[40]XO[38]NSZ[42]XUD[47]1XVKS[52]R[39]JAHH[49][39][50][54]5PBU[42]H3[45][46]DEJ,b=55) = 100,S0[37],V[53]Y,6G[51],5J[42],[38]X[40],XO[38],NSZ,[42]XU,D[47]1,XVK,S[52]R,[39]JA,HH[49],[39][50][54],5PB,U[42]H,3[45][46],DEJ (n=28,865,808,580,366,629,824,612,818,017,012,809,163,332,327,132,687,722,294,521,718,120,736,868,268,650,080,765,802,786,141,387,114)

Anne’s Hans’

Anne Cresacre by Hans Holbein

Anne Cresacre by Hans Holbein (c. 1527)


Prince, n’enquerez de sepmaine
Où elles sont, ne de cest an,
Que ce refrain ne vous remaine:
Mais où sont les neiges d’antan!

Ballade des Dames du temps jadis, François Villon (1431-c.1489)

Oh My Guardian #5

‘We’re stepping out of a binary’ – celebrating the art of marginalized LGBT Muslims

[…] The show features artwork themed around issues of Islamophobia, racism and homophobia to “highlight the struggles common among contemporary Muslim queer, trans and gender non-conforming communities,” said co-curator and activist Yas Ahmed. — ‘We’re stepping out of a binary’, The Guardian, 22/i/2018.


Elsewhere other-accessible:

Oh My Guardian #1
Oh My Guardian #2
Oh My Guardian #3
Oh My Guardian #4
Reds under the Thread

Nice Noise

Pre-previously on Overlord-in-terms-of-the-Über-Feral, I looked at how Tolkien used the word “noise” and concluded that he didn’t use it well:

He heard behind his head a creaking and scraping sound. […] There was a shriek and the light vanished. In the dark there was a snarling noise. – “Fog on the Barrowdowns”, Book One, VIII

Now I want to look at a much better writer: Ian Fleming. At first glance, he might seem to be using “noise” badly too in this bit of Live and Let Die (1954):

At about the time he [a treasure-seeking fisherman] should have reached the island the whole village of Shark Bay was awakened by the most horrible drumming noise. It seemed to come from inside the island. It was recognized as the beating of Voodoo drums. It started softly and rose slowly to a thunderous crescendo. Then it died down again and stopped. It lasted about five minutes. – ch. 16, “The Jamaica Version”

Should “drumming noise” not simply have been “drumming”? Well, no: Fleming got it right. The phrase “X noise” or “noise of X” should be used either when a noise resembles X but isn’t X or when there’s some doubt about whether it is X. In the extract above, Fleming’s choice of words captures what must have gone on in the minds of the observers, or rather the auditors: “What is that horrible noise from the island? It sounds like drums. Wait, it is drums. But how on earth could etc.” This is confirmed by what Fleming writes next: “It seemed to come… It was recognized as…”

And once the noise has been recognized, it can be described without qualification. This bit comes later in the chapter:

Strangways described his horror when, an hour after they had left to swim across the three hundred yards of water, the terrible drumming had started up somewhere inside the cliffs of the island.

In the previous chapter, there’s a use of “noise” that I’m not so sure about:

After a quarter of an hour’s meticulous work there was a slight cracking noise and the pane came away attached to the putty knob in his hand. – ch. 15, “Midnight Among the Worms”

Would “slight cracking” have been better? It’s not as clear-cut as “drumming noise”, but I think Fleming got it right again. “Cracking” is ambiguous, because it could have meant that the glass cracked physically but not audibly. Fleming was writing considerately, leaving his readers in no doubt about what he meant.

Now try this from Evelyn Waugh’s Put Out More Flags (1942), as Basil Seal watches one of his girlfriends panicked by an air-raid:

But Poppet was gone, helter-skelter, downstairs, making little moaning noises as she went.

Waugh was an even better writer than Fleming, but did he misuse “noises” there? I don’t think so. These alternatives don’t conjure the scene as effectively:

• But Poppet was gone, helter-skelter, downstairs, emitting little moans as she went.
• But Poppet was gone, helter-skelter, downstairs, uttering little moans as she went.

The noises Poppet was making weren’t real moans and the trailing phrase “making little moaning noises” mimics what Basil would have heard as Poppet fled downstairs.

I conclude that, unlike Tolkien, Fleming and Waugh were making nice noise:

nice, adj. and adv. … Particular, strict, or careful with regard to a specific point or thing. Obs. Fastidious in matters of literary taste or style. Obs.Oxford English Dictionary

Autonomata

“Describe yourself.” You can say it to people. And you can say it to numbers too. For example, here’s the number 3412 describing the positions of its own digits, starting at 1 and working upward:


3412 – the 1 is in the 3rd position, the 2 is in the 4th position, the 3 is in the 1st position, and the 4 is in the 2nd position.

In other words, the positions of the digits 1 to 4 of 3412 recreate its own digits:


3412 → (3,4,1,2) → 3412

The number 3412 describes itself – it’s autonomatic (from Greek auto, “self” + onoma, “name”). So are these numbers:


1
21
132
2143
52341
215634
7243651
68573142
321654798

More precisely, they’re panautonomatic numbers, because they describe the positions of all their own digits (Greek pan or panto, “all”). But what if you use the positions of only, say, the 1s or the 3s in a number? In base ten, only one number describes itself like that: 1. But we’re not confined to base 10. In base 2, the positions of the 1s in 110 (= 6) are 1 and 10 (= 2). So 110 is monautonomatic in binary (Greek mono, “single”). 10 is also monautonomatic in binary, if the digit being described is 0: it’s in 2nd position or position 10 in binary. These numbers are monoautonomatic in binary too:


110100 = 52 (digit = 1)
10100101111 = 1327 (d=0)

In 110100, the 1s are in 1st, 2nd and 4th position, or positions 1, 10, 100 in binary. In 10100101111, the 0s are in 2nd, 4th, 5th and 7th position, or positions 10, 100, 101, 111 in binary. Here are more monautonomatic numbers in other bases:


21011 in base 4 = 581 (digit = 1)
11122122 in base 3 = 3392 (d=2)
131011 in base 5 = 5131 (d=1)
2101112 in base 4 = 9302 (d=1)
11122122102 in base 3 = 91595 (d=2)
13101112 in base 5 = 128282 (d=1)
210111221 in base 4 = 148841 (d=1)

For example, in 131011 the 1s are in 1st, 3rd, 5th and 6th position, or positions 1, 3, 10 and 11 in quinary. But these numbers run out quickly and the only monautonomatic number in bases 6 and higher is 1. However, there are infinitely long monoautonomatic integer sequences in all bases. For example, in binary this sequence at the Online Encyclopedia of Integer Sequences describes itself using the positions of its 1s:


A167502: 1, 10, 100, 111, 1000, 1001, 1010, 1110, 10001, 10010, 10100, 10110, 10111, 11000, 11010, 11110, 11111, 100010, 100100, 100110, 101001, 101011, 101100, 101110, 110000, 110001, 110010, 110011, 110100, 111000, 111001, 111011, 111101, 11111, …

In base 10, it looks like this:


A167500: 1, 2, 4, 7, 8, 9, 10, 14, 17, 18, 20, 22, 23, 24, 26, 30, 31, 34, 36, 38, 41, 43, 44, 46, 48, 49, 50, 51, 52, 56, 57, 59, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 75, 77, 80, 83, 86, 87, 89, 91, 94, 95, 97, 99, 100, 101, 103, 104, 107, 109, 110, 111, 113, 114, 119, 120, 124, … (see A287515 for a similar sequence using 0s)

In any base, you can find some sequence of integers describing the positions of any of the digits in that base – for example, the 1s or the 7s. But the numbers in the sequence get very large very quickly in higher bases. For example, here are some opening sequences for the digits 0 to 9 in base 10:


3, 10, 1111110, … (d=0)
1, 3, 10, 200001, … (d=1)
3, 12, 100000002, … (d=2)
2, 3, 30, 10000000000000000000000003, … (d=3)
2, 4, 14, 1000000004, … (d=4)
2, 5, 105, … (d=5)
2, 6, 1006, … (d=6)
2, 7, 10007, … (d=7)
2, 8, 100008, … (d=8)
2, 9, 1000009, … (d=9)

In the sequence for d=0, the first 0 is in the 3rd position, the second 0 is in the 10th position, and the third 0 is in the 1111110th position. That’s why I’ve haven’t written the next number – it’s 1,111,100 digits long (= 1111110 – 10). But it’s theoretically possible to write the number. In the sequence for d=3, the next number is utterly impossible to write, because it’s 9,999,999,999,999,999,999,999,973 digits long (= 10000000000000000000000003 – 30). In the sequence for d=5, the next number is this:


1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000005 (100 digits long = 105 – 5).

And in fact there are an infinite number of such sequences for any digit in any base – except for d=1 in binary. Why is binary different? Because 1 is the only digit that can start a number in that base. With 0, you can invent a sequence starting like this:


111, 1110, 1111110, …

Or like this:


1111, 11111111110, …

Or like this:


11111, 1111111111111111111111111111110, …

And so on. But with 1, there’s no room for manoeuvre.

Russell in Your Head-Roe

Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry. What is best in mathematics deserves not merely to be learnt as a task, but to be assimilated as a part of daily thought, and brought again and again before the mind with ever-renewed encouragement. Real life is, to most men, a long second-best, a perpetual compromise between the ideal and the possible; but the world of pure reason knows no compromise, no practical limitations, no barrier to the creative activity embodying in splendid edifices the passionate aspiration after the perfect from which all great work springs. Remote from human passions, remote even from the pitiful facts of nature, the generations have gradually created an ordered cosmos, where pure thought can dwell as in its natural home, and where one, at least, of our nobler impulses can escape from the dreary exile of the actual world. — Bertrand Russell, “The Study Of Mathematics” (1902)


The title of this incendiary intervention is of course a paronomasia on these lines from Led Zeppelin’s magisterial “Stairway to Heaven”:

“If there’s a bustle in your hedgerow, don’t be alarmed now:
It’s just a spring-clean for the May Queen…”

And “head-roe” is a kenning for “brain”.