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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 81,428

We could also go with these...

or this

**In mathematics, you don't understand things. You just get used to them.I have the result, but I do not yet know how to get it.All physicists, and a good many quite respectable mathematicians are contemptuous about proof.**

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**eigenguy****Member**- Registered: 2014-03-18
- Posts: 78

eigenguy wrote:

I haven't tried some of the others, but it looks like maybe low values in the continued fraction expression is best, or may near constant, or maybe both.

So much for those ideas.

[0;2,1,1,2,1,1...] = 0.387426

looks very similar to

[0;6,2,1,1,2,1,1, ...] = 0.156558,

but rotated. I also experimented with shifting the pattern, but these did much worse. Apparently a pattern of High, Low is better than Low, High:

[0;1,2,1,1,2,1,1, ...] = 0.720759 opened the middle up

and

[0;1,1,2,1,1,2, ...] = 0.581139 opens it considerably. It only starts to close back up just when the simulation ends.

Based on my ideas that I quoted above, I next tried

[0;2,1,1,1,2,1,1,1,...] = 0.379796

This was significantly looser than [0;2,1,1,2,1,1...], and

[0;3,1,1,3,1,1,...] = 0.280776

leaves large gaps in the center before it closes back up.

As noted already, [0;2,2,2,...] = √2 - 1 = 0.41421356 does well, but not as good the golden ratio [0;1,1,1,...] = 0.61803399

But [0;3,3,3,...] and higher numbers all tend to be very open patterns.

I think it is likely that the continued fraction patterns can help you predict good values, and it appears that maybe 0.156558 is 2nd best after phi. But there is more going on here, or else [0;2,1,1,1,2,1,1,1,...] would do better yet, but it doesn't.

"Having thus refreshed ourselves in the oasis of a proof, we now turn again into the desert of definitions." - Bröcker & Jänich

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,092

Thanks eigenguy,

Great explanation. Now to take it all in .......

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**eigenguy****Member**- Registered: 2014-03-18
- Posts: 78

bobbym wrote:

We could also go with these...

or this

But my experiments with non-quadratic-irrational numbers have not shown any particular success in finding tight patterns, nor suggested any additional avenues of investigation. Quadratic irrationals have, and continued fractions are at least suggestive of possible explanations, though my experiments so far have failed to uncover any definitive patterns.

The particular expression I found for 0.15656 does indeed look fairly arbitrary. But its simple continued fraction form shows otherwise. It is one of the simplest irrational continued fractions to try (other than the leading 6, which mostly seems to rotate the pattern so it could be left off).

"Having thus refreshed ourselves in the oasis of a proof, we now turn again into the desert of definitions." - Bröcker & Jänich

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 81,428

It seems when other programs generate this spiral using these numbers that the graph is different.

**In mathematics, you don't understand things. You just get used to them.I have the result, but I do not yet know how to get it.All physicists, and a good many quite respectable mathematicians are contemptuous about proof.**

**Online**