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Polygonal Pi Patterns
This little recreation was inspired by realizing that the last digit of the extraordinary Feynmann point in the digits of pi (which consists of the digit sequence 999999) is at the 768th digit. Since 768 is a very interesting number, being three times a 4th power, three times a power of 2, and twelve times a cube (among other things), this led to the idea of arranging the first 768 digits of pi in geometrical patterns based on different ways of representing the number 768. An added requirement is that there be some "sixness" to the arrangement so that the final 999999 is displayed in a prominent way.
My favorite arrangement is based on writing 768 as 12 x 64, and then using the fact that every cube is a sum of consecutive hex numbers. The hex numbers (1, 7, 19, 37, etc.) are those integers that can be arranged in hexagons, like so:
1 7 19 etc. x x x x x x x x x x x x x x x x x x x x x x x x x x xThe resulting arrangment is shown below, consisting of12 copies of the first 4 hex numbers. The digits of pi are to be read from left to right all the way across each line.
3 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 4 6 2 6 4 3 3 8 3 2 7 9 5 0 2 8 8 4 1 9 7 1 6 9 3 9 9 3 7 5 1 0 5 8 2 0 9 7 4 9 4 4 5 9 2 3 0 7 8 1 6 4 0 6 2 8 6 2 0 8 9 9 8 6 2 8 0 3 4 8 2 5 3 4 2 1 1 7 0 6 7 9 8 2 1 4 8 0 8 6 5 1 3 2 8 2 3 0 6 6 4 7 0 9 3 8 4 4 6 0 9 5 5 0 5 8 2 2 3 1 7 2 5 3 5 9 4 0 8 1 2 8 4 8 1 1 1 7 4 5 0 2 8 4 1 0 2 7 0 1 9 3 8 5 2 1 1 0 5 5 5 9 6 4 4 6 2 2 9 4 8 9 5 4 9 3 0 3 8 1 9 6 4 4 2 8 8 1 0 9 7 5 6 6 5 9 3 3 4 4 6 1 2 8 4 7 5 6 4 8 2 3 3 7 8 6 7 8 3 1 6 5 2 7 1 2 0 1 9 0 9 1 4 5 6 4 8 5 6 6 9 2 3 4 6 0 3 4 8 6 1 0 4 5 4 3 2 6 6 4 8 2 1 3 3 9 3 6 0 7 2 6 0 2 4 9 1 4 1 2 7 3 7 2 4 5 8 7 0 0 6 6 0 6 3 1 5 5 8 8 1 7 4 8 8 1 5 2 0 9 2 0 9 6 2 8 2 9 2 5 4 0 9 1 7 1 5 3 6 4 3 6 7 8 9 2 5 9 0 3 6 0 0 1 1 3 3 0 5 3 0 5 4 8 8 2 0 4 6 6 5 2 1 3 8 4 1 4 6 9 5 1 9 4 1 5 1 1 6 0 9 4 3 3 0 5 7 2 7 0 3 6 5 7 5 9 5 9 1 9 5 3 0 9 2 1 8 6 1 1 7 3 8 1 9 3 2 6 1 1 7 9 3 1 0 5 1 1 8 5 4 8 0 7 4 4 6 2 3 7 9 9 6 2 7 4 9 5 6 7 3 5 1 8 8 5 7 5 2 7 2 4 8 9 1 2 2 7 9 3 8 1 8 3 0 1 1 9 4 9 1 2 9 8 3 3 6 7 3 3 6 2 4 4 0 6 5 6 6 4 3 0 8 6 0 2 1 3 9 4 9 4 6 3 9 5 2 2 4 7 3 7 1 9 0 7 0 2 1 7 9 8 6 0 9 4 3 7 0 2 7 7 0 5 3 9 2 1 7 1 7 6 2 9 3 1 7 6 7 5 2 3 8 4 6 7 4 8 1 8 4 6 7 6 6 9 4 0 5 1 3 2 0 0 0 5 6 8 1 2 7 1 4 5 2 6 3 5 6 0 8 2 7 7 8 5 7 7 1 3 4 2 7 5 7 7 8 9 6 0 9 1 7 3 6 3 7 1 7 8 7 2 1 4 6 8 4 4 0 9 0 1 2 2 4 9 5 3 4 3 0 1 4 6 5 4 9 5 8 5 3 7 1 0 5 0 7 9 2 2 7 9 6 8 9 2 5 8 9 2 3 5 4 2 0 1 9 9 5 6 1 1 2 1 2 9 0 2 1 9 6 0 8 6 4 0 3 4 4 1 8 1 5 9 8 1 3 6 2 9 7 7 4 7 7 1 3 0 9 9 6 0 5 1 8 7 0 7 2 1 1 3 4 9 9 9 9 9 9
The next one comes from noting that 768 = 6 x 128, and 128 = (12 x 8) + (6 x 4) + (3 x 2) + (2 x 1), which in turn is a consequence of 128 = 64 + 32 + 16 + 8 + 4 + 2 + 1 + 1. As luck would have it, the rectangles 12x8, etc. appear to be roughly square in shape when printed using ordinary type. Hence the pattern appears to be mostly made up of squares (perhaps a subtle reference to the term "squaring the circle"?!).
314159265358 979323846264 338327950288 419716939937 510582097494 459230781640 628620899862 803482534211 706798214808 651328230664 709384460955 058223172535 940812848111 745028410270 193852110555 964462294895 493038196442 881097566593 344612847564 823378678316 527120190914 564856692346 034861045432 664821339360 726024914127 372458700660 631558817488 152092096282 925409171536 436789259036 001133053054 882046652138 414695194151 160943305727 036575959195 309218611738 193261179310 511854807446 237996274956 735188575272 489122793818 301194912983 367336244065 664308602139 494639522473 719070217986 094370277053 921717629317 675238 467481 846766 940513 200056 812714 526356 082778 577134 275778 960917 363717 872146 844090 122495 343014 654958 537105 079227 968925 892354 201995 611212 902196 086 403 441 815 981 362 977 477 130 996 051 870 7 2 1 1 3 4 9 9 9 9 9 9
The final two examples are less related to patterns of integers but still somewhat geometrical. Since pi is so related to the circle, I wondered if I could format the 768 digits in a circle with exactly 6 digits on the top and bottom lines (so that the Feynmann sequence would again be prominent). It turns out that there are many ways to do this, one of the reasons for this being that text characters aren't exactly square (they are taller than wide), so the arrangement actually has to be an ellipse, and there are many ellipses of slightly different sizes possible. Here is one:
314159 2653589793238462 6433832795028841971693 99375105820974944592307816 4062862089986280348253421170 67982148086513282306647093844609 5505822317253594081284811174502841 0270193852110555964462294895493038 196442881097566593344612847564823378 678316527120190914564856692346034861 04543266482133936072602491412737245870 06606315588174881520920962829254091715 36436789259036001133053054882046652138 41469519415116094330572703657595919530 92186117381932611793105118548074462379 96274956735188575272489122793818301194 912983367336244065664308602139494639 522473719070217986094370277053921717 6293176752384674818467669405132000 5681271452635608277857713427577896 09173637178721468440901224953430 1465495853710507922796892589 23542019956112129021960864 0344181598136297747713 0996051870721134 999999
The final arrangement is a triangle, whose edges are perfect (grid-constrained) straight lines, and whose bottom two lines consists of exactly 6 characters (all 9's, of course).
3141592653589793238462643383279502884197169399 37510582097494459230781640628620899862803482 534211706798214808651328230664709384460955 058223172535940812848111745028410270193852 1105559644622948954930381964428810975665 93344612847564823378678316527120190914 56485669234603486104543266482133936072 602491412737245870066063155881748815 2092096282925409171536436789259036 0011330530548820466521384146951941 51160943305727036575959195309218 611738193261179310511854807446 2379962749567351885752724891 2279381830119491298336733624 40656643086021394946395224 737190702179860943702770 539217176293176752384674 8184676694051320005681 27145263560827785771 34275778960917363717 872146844090122495 3430146549585371 05079227968925 89235420199561 121290219608 6403441815 9813629774 77130996 051870 721134 9999 99