Surprising Pattern (Or is it)

[Otis]

I was goofing around, looking at deteriorating patterns in decimal forms of very small Irrational numbers. I wanted to explore a ridiculously small number because super-small numbers seem to contain more patterns, so I chose

10^(-1000) - sin[10^(-1000)]

In the first 15,000 digits (roughly), I'd seen good examples. After 15,000, the digits appeared to be totally random. Yet, after a closer look at those random digits (using an ASCII editor that displays 1,024 characters per row), I'd found what I'd decided was a final pattern before actual randomness: 868 digits, followed by 266 digits, followed by the same 868 digits (2,002 digits total).

That repeating string didn't surprise me because the earlier examples all involved repeating strings (although, none of those had random digits between the repeating strings).

Then something surprising appeared. After noticing some 3-digit blocks (eg, 444, 777, 999) that seemed to be evenly distributed, I'd arranged the 2,002 digits on three lines in various ways looking for alignments. Eventually, I had the following arrangement.

14 blank spaces, then the first 868 digits (i.e., 882 characters in the first row)

The 266 "random" digits followed by the first 616 of the repeated 868 digits (i.e., 882 characters in the second row)

The remaining 252 digits (third row)

Here's an arbitrary section of those three rows:

1455236904972354707804443254178
2566348016083465818915554365289
3677459127194576930026665476400

Consider columns, and a second pattern appears. Where the pattern breaks, repair it by rounding digits (according to the next digit in their row). That sequence pattern spans the entire width of the rows.

Those 2,002 digits aren't so random after all, and I was very surprised at the sequence pattern in the columns because I've never seen anything like it before.

Maybe someone can explain, someday.

 

[Cubist]

I don't know if this post will help or not. Maybe it will give someone an idea, or stop someone from going down the wrong path!

Here's the series expansion of x-sin(x)

[math]\sum_{n=0}^{\infty}{(-1)^n\frac{x^{\left(3+2n\right)}}{\left(3+2n\right)!}}[/math]
If x=10^(-1000), and we restrict to the first 15,000 digits of the answer, then only the terms up to n=5 matter.

I looked for the point where the digits of each term start to repeat (and the length of that repeat)...

n     starts repeating at     length of repeat
==============================================
0         3002                      1
1         5004                      1
2         7007                      6
3         9008                     18
4        11008                     18
5        13011                     54

 

[Otis]

Hi Cubist. Yes, I'd seen the blocks of repeating strings. They didn't surprise me because those types of patterns are what led me to goof off in the first place -- I was curious to watch them deteriorate, only to start up again and again, before a final "last gasp", ha.

(I laugh when characterizing Irrationals as trying to act Rational. But, in the end, I'm not sure patterns could be much more than just a trivial attribute of a number's decimal form, even if there were a detailed explanation.)

The surprising part for me is the sequence pattern in the column digits we see by arranging just so (in three rows) a string of 2,002 digits that appears shortly after the 54-digit repeating blocks (i.e., arranging the block containing a repeat of length 868 -- not listed in your chart because it starts at the 15,011th digit). That string is different in at least four ways from the six prior blocks containing repeating strings you've listed. I've never seen such a thing as that sequence pattern.

Anyway, sometimes it's fun to be surprised, even when there's no payoff in sight.

?