Summer-Time Twos

by in Arithmetic, Base Behavior, Mathematics and tagged , | Leave a comment

I wondered how often the digits of n2 appeared in sum(n1,n2). For example:

17 → 117 = sum(9,17)
20 → 200 = sum(5,20); 204,4; 207,3; 209,2 (c=4)

As I looked at higher n2, I found that the 2-views continued:

63 → 363 = sum(58,63); 638,53; 1638,28; 1763,23; 1863,18 (c=5)
88 → 1288 = sum(73,88); 2788,48; 2881,46; 3388,33; 3880,9; 3888,8 (c=6)
200 → 20009 = sum(14,200); 20022,13; 20034,12; 20045,11; 20055,10; 20064,9;
20072,8; 20079,7; 20085,6; 20090,5; 20094,4; 20097,3; 20099,2 (c=13)
558 → 39558 = sum(483,558); 55833,448; 95583,348; 105558,318; 125580,247; 126558,243; 143558,158; 152558,83; 155583,28; 155808,18; 155825,17; 155841,16; 155856,15; 155870,14; 155883,13; 155895,12 (c=16)
2000 → 2000010 = sum(45,2000); 2000054,44; 2000097,43; 2000139,42; 2000180,41; 2000220,40; 2000259,39; 2000297,38; 2000334,37; 2000370,36; 2000405,35; 2000439,34; 2000472,33; 2000504,32; 2000535,31; 2000565,30;
2000594,29; 2000622,28; 2000649,27; 2000675,26; 2000700,25; 2000724,24; 2000747,23; 2000769,22; 2000790,21; 2000810,20; 2000829,19; 2000847,18; 2000864,17; 2000880,16; 2000895,15; 2000909,14; 2000922,13; 2000934,
12; 2000945,11; 2000955,10; 2000964,9; 2000972,8; 2000979,7; 2000985,6; 2000990,5; 2000994,4; 2000997,3; 2000999,2 (c=44)

But what about other bases?

Base 9

15 in b9 → 115 = sum(5,15) (n=14 in b10) (c=1)
18 in b9 → 118 = sum(11,17); 180,1 (n=17 in b10) (c=2)
20 in b9 → 203 = sum(4,18); 206,3; 208,2 (n=18 in b10) (c=3)
45 in b9 → 445 = sum(32,41); 745,25; 1045,15; 1145,5 (n=41 in b10) (c=4)
55 in b9 → 555 = sum(41,50); 1055,35; 1355,25; 1555,15; 1655,5 (n=50 in b10) (c=5)
65 in b9 → 665 = sum(50,59); 1265,45; 1665,35; 2065,25; 2265,15; 2365,5 (n=59 in b10) (c=6)
75 in b9 → 775 = sum(59,68); 1475,55; 2075,45; 2475,35; 2750,26; 2775,25; 3075,15; 3175,5 (n=68 in b10) (c=8)
85 in b9 → 885 = sum(68,77); 1685,65; 2385,55; 2885,45; 3385,35; 3685,25; 3853,17; 3885,15; 4085,5 (n=77 in b10) (c=9)
200 in b9 → 20003 = sum(13,162); 20016,13; 20028,12; 20040,11; 20050,10; 20058,8; 20066,7; 20073,6; 20078,5; 20083,4; 20086,3; 20088,2 (n=162 in b10) (c=12)
415 in b9 → 13415 = sum(311,338); 25415,345; 36415,315; 41525,302; 46415,275; 55415,245; 63415,215; 64155,212; 70415,175; 75415,145; 80415,115; 83415,75; 85415,45; 86415,15 (n=338 in b10) (c=14)
[…]
2000 in b9 → 2000028 = sum(38,1458); 2000070,41; 2000120,40; 2000158,38; 2000206,37; 2000243,36; 2000278,35; 2000323,34; 2000356,33; 2000388,32; 2000430,31; 2000460,30; 2000488,28; 2000526,27; 2000553,26; 2000578,25; 2000613,24; 2000636,23; 2000658,22; 2000680,21; 2000710,20; 2000728,18; 2000746,17; 2000763,16; 2000778,15; 2000803,14; 2000816,13; 2000828,12; 2000840,11; 2000850,10; 2000858,8; 2000866,7; 2000873,6; 2000878,5; 2000883,4; 2000886,3; 2000888,2 (n=1458 in b10) (c=37)

Base 11

16 in b11 → 116 = sum(6,16) (n=17 in b10) (c=1)
20 in b11 → 201 = sum(5,22); 205,4; 208,3; 20A,2 (n=22 in b10) (c=4)
56 in b11 → 556 = sum(50,61); 956,36; 1156,26; 1356,16; 1456,6 (n=61 in b10) (c=5)
66 in b11 → 666 = sum(61,72); 1066,46; 1466,36; 1669,2A; 1766,26; 1966,16; 1A66,6 (n=72 in b10) (c=7)
86 in b11 → 886 = sum(83,94); 1486,66; 1A86,56; 2486,46; 2886,36; 3086,26; 3286,16; 3386,6 (n=94 in b10) (c=8)
96 in b11 → 996 = sum(94,105); 1696,76; 2296,66; 2896,56; 3296,46; 3696,36; 3996,26; 4096,16; 4196,6 (n=105 in b10) (c=9)
A6 in b11 → AA6 = sum(105,116); 18A6,86; 25A6,76; 31A6,66; 37A6,56; 41A6,46; 45A6,36; 48A6,26; 4AA6,16; 50A6,6 (n=116 in b10) (c=10)
200 in b11 → 1200A = sum(156,242); 20001,15; 20015,14; 20028,13; 2003A,12; 20050,11; 20060,10; 2006A,A; 20078,9; 20085,8; 20091,7; 20097,6; 200A1,5; 200A5,4; 200A8,3; 200AA,2 (n=242 in b10) (c=16)
[…]
A66 in b11 → 1AA66 = sum(1260,1282); A1A66,966; 109A66,946; 182A66,866; 198A66,846; 23A666,786; 253A66,766; 267A66,746; 314A66,666; 326A66,646; 375A66,566; 385A66,546; 416A66,466; 424A66,446; 457A66,366; 463A66,346; 46A666,326; 488A66,266; 492A66,246; 4A6666,186; 4A9A66,166; 501A66,146; 50AA66,66; 510A66,46 (n=1282 in b10) (c=24)
2000 in b11 → 2000005 = sum(52,2662); 2000051,47; 2000097,46; 2000131,45; 2000175,44; 2000208,43; 200024A,42; 2000290,41; 2000320,40; 200035A,3A; 2000398,39; 2000425,38; 2000461,37; 2000497,36; 2000521,35; 2000555,34; 2000588,33; 200060A,32; 2000640,31; 2000670,30; 200069A,2A; 2000718,29; 2000745,28; 2000771,27; 2000797,26; 2000811,25; 2000835,24; 2000858,23; 200087A,22; 20008A0,21; 2000910,20; 200092A,1A; 2000948,19; 2000965,18; 2000981,17; 2000997,16; 2000A01,15; 2000A15,14; 2000A28,13; 2000A3A,12; 2000A50,11; 2000A60,10; 2000A6A,A; 2000A78,9; 2000A85,8; 2000A91,7; 2000A97,6; 2000AA1,5; 2000AA5,4; 2000AA8,3; 2000AAA,2 (n=2662 in b10) (c=51)

Base 3

12 in b3 → 112 = sum(2,12); 120,1 (n=5 in b10) (c=2)
20 in b3 → 120 = sum(4,6); 200,10; 202,2 (n=6 in b10) (c=3)
122 in b3 → 10122 = sum(11,17); 11122,22; 11220,21; 12122,2; 12200,1 (n=17 in b10) (c=5)
1212 in b3 → 121212 = sum(41,50); 1001212,1012; 1101212,212; 1112120,200; 1121212,112; 1201212,12 (n=50 in b10) (c=6)
1222 in b3 → 122222 = sum(44,53); 1101222,1002; 1111222,222; 1112220,221; 1212222,102; 1221222,2; 1222000,1 (n=53 in b10) (c=7)
2000 in b3 → 1112000 = sum(28,54); 1120000,1000; 2000020,21; 2000110,20; 2000122,12; 2000210,11; 2000220,10; 2000222,2 (n=54 in b10) (c=8)
[…]
20000 in b3 → 111120000 = sum(82,162); 111200000,10000; 200000010,111; 200000120,110; 200000222,102; 200001100,101; 200001200,100; 200001222,22; 200002020,21; 200002110,20; 200002122,12; 200002210,11; 200002220,10; 200002222,2 (n=162 in b10) (c=14)

Base 4

13 in b4 → 130 = sum(1,13) (n=7 in b10) (c=1)
20 in b4 → 201 = sum(3,8); 203,2 (n=8 in b10) (c=2)
200 in b4 → 20001 = sum(6,32); 20012,11; 20022,10; 20031,3; 20033,2 (n=32 in b10) (c=5)
2000 in b4 → 2000021 = sum(11,128); 2000103,22; 2000130,21; 2000210,20; 2000223,13; 2000301,12; 2000312,11; 2000322,10; 2000331,3; 2000333,2 (n=128 in b10) (c=10)
20000 in b4 → 200000003 = sum(23,512); 200000121,112; 200000232,111; 200001002,110; 200001111,103; 200001213,102; 200001320,101; 200002020,100; 200002113,33; 200002211,32; 200002302,31; 200002332,30; 200003021,23; 200003103,22; 200003130,21; 200003210,20; 200003223,13; 200003301,12; 200003312,11; 200003322,10; 200003331,3; 200003333,2 (n=512 in b10) (c=22)

Base 8

17 in b8 → 170 = sum(1,17) (n=15 in b10) (c=1)
20 in b8 → 202 = sum(4,16); 205,3; 207,2 (n=16 in b10) (c=3)
200 in b8 → 20011 = sum(11,128); 20023,12; 20034,11; 20044,10; 20053,7; 20061,6; 20066,5; 20072,4; 20075,3; 20077,2 (n=128 in b10) (c=10)
2000 in b8 → 2000020 = sum(32,1024); 2000057,37; 2000115,36; 2000152,35; 2000206,34; 2000241,33; 2000273,32; 2000324,31; 2000354,30; 2000403,27; 2000431,26; 2000456,25; 2000502,24; 2000525,23; 2000547,22; 200057
0,21; 2000610,20; 2000627,17; 2000645,16; 2000662,15; 2000676,14; 2000711,13; 2000723,12; 2000734,11; 2000744,10; 2000753,7; 2000761,6; 2000766,5; 2000772,4; 2000775,3; 2000777,2 (n=1024 in b10) (c=31)

Base 16

1F in b16 → 1F0 = sum(1,1F) (n=31 in b10) (c=1)
20 in b16 → 201 = sum(6,32); 206,5; 20A,4; 20D,3; 20F,2 (n=32 in b10) (c=5)
200 in b16 → 20003 = sum(23,512); 20019,16; 2002E,15; 20042,14; 20055,13; 20067,12; 20078,11; 20088,10; 20097,F; 200A5,E; 200B2,D; 200BE,C; 200C9,B; 200D3,A; 200DC,9; 200E4,8; 200EB,7; 200F1,6; 20
0F6,5; 200FA,4; 200FD,3; 200FF,2 (n=512 in b10) (c=22)
[…]
EE4 in b16 → 42EE4A = sum(961,EE4); 6EE413,16; 6EE428,15; 6EE43C,14; 6EE44F,13; 6EE461,12; 6EE472,11; 6EE482,10; 6EE491,F; 6EE49F,E; 6EE4AC,D; 6EE4B8,C; 6EE4C3,B; 6EE4CD,A; 6EE4D6,9; 6EE4DE,8; 6EE4E5,7; 6EE4EB,6; 6EE4F0,5; 6EE4F4,4; 6EE4F7,3; 6EE4F9,2; 6EE4FA,1 (n=3812 in b10) (c=23)
2000 in b16 → 2000001 = sum(5B,2000); 200005B,5A; 20000B4,59; 200010C,58; 2000163,57; 20001B9,56; 200020E,55; 2000262,54; 20002B5,53; 2000307,52; 2000358,51; 20003A8,50; 20003F7,4F; 2000445,4E; 2000492,4D; 20004DE,4C; 2000529,4B; 2000573,4A; 20005BC,49; 2000604,48; 200064B,47; 2000691,46; 20006D6,45; 200071A,44; 200075D,43; 200079F,42; 20007E0,41; 2000820,40; 200085F,3F; 200089D,3E; 20008DA,3D; 2000916,3C; 2000951,3B; 200098B,3A; 20009C4,39; 20009FC,38; 2000A33,37; 2000A69,36; 2000A9E,35; 2000AD2,34; 2000B05,33; 2000B37,32; 2000B68,31; 2000B98,30; 2000BC7,2F; 2000BF5,2E; 2000C22,2D; 2000C4E,2C; 2000C79,2B; 2000CA3,2A; 2000CCC,29; 2000CF4,28; 2000D1B,27; 2000D41,26; 2000D66,25; 2000D8A,24; 2000DAD,23; 2000DCF,22; 2000DF0,21; 2000E10,20; 2000E2F,1F; 2000E4D,1E; 2000E6A,1D; 2000E86,1C; 2000EA1,1B; 2000EBB,1A; 2000ED4,19; 2000EEC,18; 2000F03,17; 2000F19,16; 2000F2E,15; 2000F42,14; 2000F55,13; 2000F67,12; 2000F78,11; 2000F88,10; 2000F97,F; 2000FA5,E; 2000FB2,D; 2000FBE,C; 2000FC9,B; 2000FD3,A; 2000FDC,9; 2000FE4,8; 2000FEB,7; 2000FF1,6; 2000FF6,5; 2000FFA,4; 2000FFD,3; 2000FFF,2 (n=8192 in b10) (c=90)

Previously Pre-Posted (Please Peruse)

• Summer-Time Hues
• Summer-Climb Views
• Summult-Time Hues

Apostrophizing Andy

by in English Usage, In Terms Of, Language, Linguistics and tagged , , , , , , , , , | Leave a comment

If you get it, you’ll laugh. If you don’t, you won’t:

Eee, I know what you mean. Shine’s gone off this government faster than gravy off chips, as we say up here in the North, where I authentically am. What t’party needs is a leader who’s reet proper connected with t’working man. In terms of names, we’ll see to that when dog’s in t’barn, as Northerners like me say up here in the North. — “Mandelson: Let’s chat about Keir…”, 29ix25

Well, I laughed anyway. That’s Robert Hutton in The Critic joking about the prime-ministerial ambitions of Andy Burnham, the mayor of Manchester. And I’m wondering about the “In terms of names…” I think it’s there for deliberate contrast. As I’ve endlessly adumbrated in terms of Overlord-of-theÜber-Feral, “in terms of” is an ugly, pretentious piece of bureaucratese that’s keyly characteristic of politicians, lawyers, academics in the humanities, and other core communities of windbags. In short, it’s highly bourgeois.

And I reckon that’s why Hutton put it into his mockery of Burnham, who’s pretending to be reet down-to-earth but can’t help letting his true nature coom through. Whatever his roots, he’s a bourgeois bureaucrat now. If I’m right, then Hutton recognizes the rebarbativity of “in terms of”. Good on ya, Bob. But bad on ya for getting the northern accent wrong. The apostrophe’s in the wrong place: “t’party” and “t’working man” should be “’t party” and “’t working man”, because that northern form of the definite article doesn’t represent the → t’ but that → ’t (in Old English þæt was the neuter form of the definite article, while the masculine and feminine forms were sē and sēo). You can hear the truth in the glottal stop, which is sometimes all that’s left of the original “that”. In fact, that’s what “t’” is generally a bad transcription of — a glottal stop, “ʔ” in phonetic transcription. But in some dialects of northern English, the glottal stop disappears too, so there’s no definite article and English weirdly seems like Latin or Russian or some other language that doesn’t use definite articles.

You can see Mancunian English moving towards no-definite-article with “Shine’s gone off this government…” But the most natural way to read that line is with a glottal stop: “ʔShine’s gone off this government…” If Hutton meant it to be read like that, he’s implicitly recognizing that “t’” is a bad transcription. “T’shine’s gone off…” would sound like “Chine’s gone off…” But no Mancunian would say it like that. Something else that no Mancunian would say is that the Fat Slags are from Newcastle. But that’s a story for another day.

Peri-Performative Post-Scriptum

As is usual with sociology or biology, the story of the northern definite article is much more complicated than a short discussion can cover. And I can’t remember where I read about its true origins and can’t find anything online at the moment. But this supports what I’m saying:

The phenomenon of Definite Article Reduction (DAR) is the realization of the definite article in northern British English dialects in a range of vowel-less forms, usually written t’ in literature. The origin of DAR is assumed to be the assimilation of the initial fricative of the Middle English definite article þe to produce a te form, a sound change recorded for many dialects of Middle English. This article examines the validity of this hypothesis by analysing the distribution of fricative allomorphs in the modern dialects in comparison with the details of the Middle English change. The predicted distribution of fricative forms is not found at most localities, indicating that the development hypothesis is incorrect, but the available data are too scanty to suggest an alternative model. — “The origin of Definite Article Reduction in northern English dialects: evidence from dialect allomorphy, Mark J. Jones in English Language and Linguistics, November 2002

Punctuated Pairimeters

by in Arithmetic, Circles, Geometry, Mathematics and tagged , , , , | Leave a comment

Imagine using the digits of n in two different bases to generate two fractions, a/b and c/d, where a/b < 1 and c/d < 1 (see Appendix for a sample program). Now use the fractions to find a pair of points on the perimeter of a circle, (x1, y1) and (x2, y2), then calculate and mark the midpoint of (x1, y1) and (x2, y2). If the bases have a prime factor in common, pretty patterns will appear from this punctuated pairimetry:

b1 = 2; b2 = 6

b1 = 2; b2 = 10

b1 = 2; b2 = 14

b1 = 4; b2 = 10

b1 = 4; b2 = 20

b1 = 4; b2 = 28

b1 = 6; b2 = 42

b1 = 12; b2 = 39

b1 = 24; b2 = 28

b1 = 28; b2 = 40

b1 = 32; b2 = 36

b1 = 42; b2 = 78

Appendix: Sample Program for Pairimetry

GetXY(xyi)=

fr = 0
recip = 1
bs = base[xyi]
for gi = 1 to di[xyi]
recip = recip/bs
fr += d[xyi,gi] * recip
next gi

x[xyi] = xcenter + sin(pi2 * fr) * radius
y[xyi] = ycenter + cos(pi2 * fr) * radius

endproc

Dinc(i1) =

d[i1,1]++;
if d[i1,1] == base[i1] then

i2 = 1

while d[i1,i2] == base[i1]

d[i1,i2] = 0
i2++;
d[i1,i2]++;

endwhile

if i2 > di[i1] then di[i1] = i2 endif

endif

endproc

Drawfigure =

base = x = y = di = array(2)
d = array(2,100)
radius = 100
pi2 = pi * 2
base[1] = 2
base[2] = 6
di[1] = 1
di[2] = 1

while true

for i = 1 to 2
call Dinc(i)
call GetXY(i)
next i

plot (x[1]+x[2]) / 2, (y[1] + y[2]) / 2

endwhile

endproc

call drawfigure

Summult-Time Hues

by in Arithmetic, Integer Sequences, Mathematics and tagged , , , , | Leave a comment

sum(3,6) = 3 * 6 = 18
• 3 * 2.3 = 2.3^2
sum(15,35) = 15 * 35 = 525
• 3.5 * 5.7 = 3.5^2.7
sum(85,204) = 85 * 204 = 17340
• 5.17 * 2^2.3.17 = 2^2.3.5.17^2
sum(493,1189) = 493 * 1189 = 586177
• 17.29 * 29.41 = 17.29^2.41
sum(2871,6930) = 2871 * 6930 = 19896030
• 3^2.11.29 * 2.3^2.5.7.11 = 2.3^4.5.7.11^2.29
sum(16731,40391) = 16731 * 40391 = 675781821
• 3^2.11.13^2 * 13^2.239 = 3^2.11.13^4.239
[…]

Elsewhere Other-Accessible

1, 18, 525, 17340, 586177, 19896030, 675781821, 22956120408, 779829016225, 26491211221770, 899921240562957, 30570830315362260, 1038508305678375841, 35278711540581704598, 1198437683944896688125, 40711602541832856049200, 1382996048733983114022337 — A011906 at the Online Encyclopedia of Integer Sequences

Sky-Guy for the Strayed Eye

by in Poetry, The Sea and tagged , , , , , , , | Leave a comment

The sky is in the sand,
That blend of sea and land,
Where ribbled pools
Make optic fools
Of eyes that stray or strand.

Peri-Performative Post-Scriptum

This poem is my sub-Housmanesque attempt to capture the sight of sky reflected in pools between wave-ribbed sand, so that there seemed to be another world floating there. I don’t like “optic fools”, where the adjective is obtrusively un-Anglish. But I also considered “photic fools”, for the alliteration. In the end, I might have used “eyeish fools”, if it hadn’t meant I couldn’t use “eyes” in the final line. Alternatives like “Of all that…” or “Of those that…” didn’t seem good. Oh, and “Sky-Guy” uses guy in the sense of “trick” or “hoax”, not as it’s used in the title of the TV program paronomasized in the title of this post.

The Sumber of the B’s

by in Arithmetic, Mathematics and tagged , , , , , , , , , , | Leave a comment

First a bit of a boredom. Then a bit of beauty. These are the triangular numbers, including 666, the Number of the Beast:

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, …

You can createthem as sumbers, that is, as numbers made by summing the whole numbers:

tri(1) = 1 = 1
tri(2) = 3 = 2+1
tri(3) = 6 = 3+2+1
tri(4) = 10 = 4+3+2+1
tri(5) = 15 = 5+4+3+2+1
tri(6) = 21 = 6+5+4+3+2+1
tri(7) = 28 = 7+6+5+4+3+2+1
tri(8) = 36 = 8+7+6+5+4+3+2+1
tri(9) = 45 = 9+8+7+6+5+4+3+2+1
tri(10) = 55 = 10+9+8+7+6+5+4+3+2+1

And here are the square numbers:

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500, …

You can create square numbers in various ways. Most obviously, by multiplying each whole number by itself:

sq(1) = 1*1 = 1
sq(2) = 2*2 = 4
sq(3) = 3*3 = 9
sq(4) = 4*4 = 16
sq(5) = 5*5 = 25
sq(6) = 6*6 = 36
sq(7) = 7*7 = 49
sq(8) = 8*8 = 64
sq(9) = 9*9 = 81
sq(10) = 10*10 = 100

Less obviously, by summing consecutive odd numbers:

sq(1) = 1 = 1
sq(2) = 1+3 = 4
sq(3) = 1+3+5 = 9
sq(4) = 1+3+5+7 = 16
sq(5) = 1+3+5+7+9 = 25
sq(6) = 1+3+5+7+9+11 = 36
sq(7) = 1+3+5+7+9+11+13 = 49
sq(8) = 1+3+5+7+9+11+13+15 = 64
sq(9) = 1+3+5+7+9+11+13+15+17 = 81
sq(10) = 1+3+5+7+9+11+13+15+17+19 = 100

And by summing pairs of consecutive triangular numbers (note that tri(0) = 0):

sq(1) = tri(0) + tri(1) = 0 + 1 = 1
sq(2) = tri(1) + tri(2) = 1 + 3 = 4
sq(3) = tri(2) + tri(3) = 3 + 6 = 9
sq(4) = tri(3) + tri(4) = 6 + 10 = 16
sq(5) = tri(4) + tri(5) = 10 + 15 = 25
sq(6) = tri(5) + tri(6) = 15 + 21 = 36
sq(7) = tri(6) + tri(7) = 21 + 28 = 49
sq(8) = tri(7) + tri(8) = 28 + 36 = 64
sq(9) = tri(8) + tri(9) = 36 + 45 = 81
sq(10) = tri(9) + tri(10) = 45 + 55 = 100

But sometimes squares are the sum of two triangular numbers that aren’t consecutive:

sq(4) = tri(1) + tri(5) = 1+15 = 16
sq(9) = tri(2) + tri(12) = 3+78 = 81
sq(16) = tri(2) + tri(22) = 3+253 = 256
sq(52) = tri(2) + tri(73) = 3+2701 = 2704
sq(14) = tri(3) + tri(19) = 6+190 = 196
sq(21) = tri(3) + tri(29) = 6+435 = 441
sq(44) = tri(9) + tri(61) = 45+1891 = 1936
sq(51) = tri(9) + tri(71) = 45+2556 = 2601
sq(49) = tri(10) + tri(68) = 55+2346 = 2401
sq(56) = tri(10) + tri(78) = 55+3081 = 3136
sq(16) = tri(11) + tri(19) = 66+190 = 256
sq(38) = tri(11) + tri(52) = 66+1378 = 1444
sq(54) = tri(11) + tri(75) = 66+2850 = 2916
sq(87) = tri(47) + tri(113) = 1128+6441 = 7569
sq(77) = tri(48) + tri(97) = 1176+4753 = 5929
sq(121) = tri(64) + tri(158) = 2080+12561 = 14641
sq(141) = tri(96) + tri(174) = 4656+15225 = 19881
sq(121) = tri(100) + tri(138) = 5050+9591 = 14641

Here’s a graph of squares that are the sum of any two triangular numbers, that is, is_square(tri(k1)+tri(k2)). The x axis is 1..k1 and the y axis is 1..k2, so the graph is symmetrical:

tri(k1) + tri(k2) = square(k3)

The (double) line at 45° represents squares that are the sum of consecutive triangulars. Other lines represent similarly regular patterns. Now for a bit of beauty. Things get more visually interesting when you test for squares that are the sums of any integer and a triangular number:

k1 + tri(k2) = square(k3)

The curves are optical oddities: where do they begin and end? The upper ones become lost to the eye in the lower ones. And vice versa. But you can force your eye to trace them further that it wants to.

Now try sums of integers and other polygonal numbers:

k1 + tri(k2) = pentagonal(k3)

k1 + square(k2) = pentagonal(k3)

k1 + pentagonal(k2) = square(k3)

k1 + hexagonal(k2) = pentagonal(k3)

And try other number sequences, like multiples of 4 with polygonals:

k1*4 + pentagonal(k2) = tri(k3)

k1*4 + square(k2) = tri(k3)

k1*4 + heptagonal(k2) = tri(k3)

And primes with polygonals:

tri(k1) + prime(k2) = tri(k3)

prime(k1) + tri(k2) = square(k3)

prime(k1) + octagonal(k2) = square(k3)

prime(k1) + pentagonal(k2) = square(k3)

prime(k1) + square(k2) = decagonal(k3)

prime(k1) + tri(k2) = hendecagonal(k3)

Partitional Pulchritude

by in Arithmetic, Geometry, Integer Sequences, Mathematics and tagged , , , , , , | Leave a comment

If you want a good example of how, in math, something very simple can quickly get very deep, just look at partitions. Here are the partitions of 1 to 5, that is, the ways 1 to 5 can be expressed as a sum of integers smaller than or equal to themselves:

1 = 1

numbpart(1) = 1

2 = 2
1 + 1 = 2

numbpart(2) = 2

3 = 3
1 + 2 = 3
1 + 1 + 1 = 3

numbpart(3) = 3

4 = 4
1 + 3 = 4
2 + 2 = 4
1 + 1 + 2 = 4
1 + 1 + 1 + 1 = 4

numbpart(4) = 5

5 = 5
1 + 4 = 5
2 + 3 = 5
1 + 1 + 3 = 5
1 + 2 + 2 = 5
1 + 1 + 1 + 2 = 5
1 + 1 + 1 + 1 + 1 = 5

numbpart(5) = 7

It’s very easy to understand the concept of partitions, but very difficult to understand how partitions behave. For example, here is numbpart(n), the count of partitions for 1, 2, 3,…

1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, 6842, 8349, 10143, 12310, 14883, 17977, 21637, 26015, 31185, 37338, 44583, 53174, 63261, 75175, 89134, 105558, 124754, 147273, 173525, 204226, … A000041 at the Online Encyclopedia of Integer Sequences, “a(n) is the number of partitions of n (the partition numbers)”

What’s the formula for numbpart(n)? That’s a tricky question. And what’s the formula for the curves produced by counting the various lengths of partitions(n)? That’s another tricky question, but one thing is easy to see. As n gets bigger, the graph of countlen(partitions(n)) acquires a strange, lopsided beauty. Here are the partitions of 8, with the count of how many partitions of a particular length there are:

8 = 8 (1 partition of length 1)
1 + 7 = 8
2 + 6 = 8
3 + 5 = 8
4 + 4 = 8 (4 partitions of length 2)
1 + 1 + 6 = 8
1 + 2 + 5 = 8
1 + 3 + 4 = 8
2 + 2 + 4 = 8
2 + 3 + 3 = 8 (5 of length 3)
1 + 1 + 1 + 5 = 8
1 + 1 + 2 + 4 = 8
1 + 1 + 3 + 3 = 8
1 + 2 + 2 + 3 = 8
2 + 2 + 2 + 2 = 8 (5 of length 4)
1 + 1 + 1 + 1 + 4 = 8
1 + 1 + 1 + 2 + 3 = 8
1 + 1 + 2 + 2 + 2 = 8 (3 of length 5)
1 + 1 + 1 + 1 + 1 + 3 = 8
1 + 1 + 1 + 1 + 2 + 2 = 8 (2 of length 6)
1 + 1 + 1 + 1 + 1 + 1 + 2 = 8 (1 of length 7)
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 8 (1 of length 8)

When counts like that are shown as a graph, the graphs look like this (maximum counts are normalized to the same height):


graph of countlen(partitions(2))


countlen(partitions(3))


countlen(partitions(4))


countlen(partitions(5))


countlen(partitions(6))


countlen(partitions(7))


countlen(partitions(8))


countlen(partitions(9))


countlen(partitions(10))


countlen(partitions(15))


countlen(partitions(20))


countlen(partitions(30))


countlen(partitions(40))


countlen(partitions(50))


countlen(partitions(60))


countlen(partitions(70))


countlen(partitions(80))


countlen(partitions(90))


countlen(partitions(100))


Animated gif of partlen graphs (courtesy EZgif)

The graphs have a long, low right tail because the counts rise to great heights very quick, then fall away again, as you can see with partitions(100):

1 = count(partitions(10),len=1)
50 = count(partitions(10),len=2)
833 = count(partitions(10),len=3)
7153 = count(partitions(10),len=4)
38225 = count(partitions(10),len=5)
143247 = count(partitions(10),len=6)

[…]

10643083 = count(partitions(10),len=16)
11022546 = count(partitions(10),len=17)
11087828 = count(partitions(10),len=18)
10885999 = count(partitions(10),len=19)
10474462 = count(partitions(10),len=20)

[…]

30 = count(partitions(10),len=91)
22 = count(partitions(10),len=92)
15 = count(partitions(10),len=93)
11 = count(partitions(10),len=94)
7 = count(partitions(10),len=95)
5 = count(partitions(10),len=96)
3 = count(partitions(10),len=97)
2 = count(partitions(10),len=98)
1 = count(partitions(10),len=99)
1 = count(partitions(10),len=100)

Papillons de Papier

by in Literature, Poetry, Quotations, Translation and tagged , , , , , , , , , , , | Leave a comment

Tsavudz’ gvdjo
Hmorksa ržmju:
Í hmístaghjo,
Í hmůldzva lšju! — Franček Zymosjő (1883-1941)

White butterflies,
On paper wings,
Are mystagogues,
Enchanted things!

• Translation by Elena Nebotsaya in On Paper Wings: Selected Poems and Prose of Franček Zymosjő (Symban Press 1986)