[I think I found something?] - A New Kind of Science: The NKS Forum

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I think I found something?

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Posted by: Mark McAndrew

Just as the Fibonacci-represented-as-binary diagram finishes (lines 502 - 511), a line of fractal triangles is emerging from the (apparent) chaos above.

http://mathworld.wolfram.com/FibonacciNumber.html

It's NOT the appended binary indicator that appears on:

http://www.wolframscience.com/confe...acciBinary.html

although the emerging pattern of white triangles obviously correlates with the binary one below it.

Your thoughts, folks?


Posted by: Jason Cawley

I was skeptical - the bulk of the pattern looks very much class 3 random to me, but there does seem to be something to it. Here is what Ed Pegg and Oyvind Tajford of our science group said after checking your claim again -

Ed -

"It's particularly clear if you look at
http://www.maa.org/editorial/mathga...s_12_08_03.html

It has a strong correlation with the binary numbers. There is always a large white triangle directly above Fibonacci[2^(n + 1) + 2^n]. Illustrative code:

Table[Take[IntegerDigits[Fibonacci[2^(n + 1) + 2^n], 2, 30], -30], {n, 1, 20}]
"

Oyvind -

"In other words, this is saying that Fibonacci[2^(n+1)+2^n] (e.g.
Fibonacci[3*2^n]) has exactly (n+2) powers of 2 in its factorization. Similar statements seems to be true for other primes (e.g., change all 2's to 3's in the above code). There's probably a nice simple proof for this."


Posted by: Tony Smith

The successive binary expansions are shown vertically from left to right, the pattern along the bottom being due to Mod[Fibonacci[i], 2*2**n] having a cycle of period 3*2**n.


Posted by: Mark McAndrew

Many thanks, everyone. Ed Pegg and Eric (surname unknown) also got back to me about it.

I'm wondering now if any other series also shows order periodically re-emerge from (apparent) chaos like the Fibonacci?

(Probably an infinite number of them do, but you know what I mean...!)




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