Geometrical π-digit patterns Mike Keith

This little recreation was inspired by realizing that the last digit of the extraordinary Feynman point in the digits of π (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 π 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 x```

The resulting arrangment is shown below, consisting of 12 copies of the first 4 hex numbers. The digits of π 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 arrangement 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 (oh, no - have we "squared 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 π 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```