Here are some which I have figured out for myself.
I do not know if anyone else has discovered them.
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1/2F(n) = 9F(n-6) - 1/2F(n-12)
Take a look at the following:-
F21.......10946.....half=5473
9 x 610, ieF(21-6) ,ieF15 =5490
difference is 17 "too many"
=1/2 x 34, (1/2F9=1/2F(21-12)
F22........17711........8855.5..... 9 x 987 = 8883.........diff=28
F23........28657........14328.5....9 x 1597 = 14373.....diff=45
F24........46368........23184.......9 x 2584 = 23256.....diff=72
F25........75025........37512.......9 x 4181 = 37629.....diff=117
F26.......121393.......60696.5....9 x 6765 = 60885.....diff=189
F27.......196418......98209........9 x 10946 = 98514....diff=305
As you can see, "diff" = 1/2F(n-12)
Moreover, 1/2F(n) / Diff =~320 point something, stabilising to 321.... all along the series.
even:-
F59.....956 722 026 041........half is 478 361 013 020.5
9 x 53 316 291 173 = 479 846 620 557
diff is 1 485 607 536.5.......=1/2 F47 = 1/2 x 2 971 215 073 = 1 485 607 536.5
Moreover, 1/2F(n) / diff =
47 836 101 / 148560.75 = 321.9969
(Divide this by 9:- we get 35.6666666)
What does all this mean?
If we want to crochet the Fibonacci Series , half shell
we can use this equation to vaguely evenly create a vortex, cone, etc.
I imagine it could make a triangular spiral, such as in diagram 3... on previous post dated 26/5/2011,giant clam.... reproduced here:-
For F(n-6) substitute the appropriate "Fn=2F(n-2) + F(n-3)" Pattern for the increase stitches in crocheting or knitting. I have not done this yet, apart from the simple model previously made with 9x bivalve pattern, above.
The 1/2Fn / F(n-12) is always 321. I think this means to decrease 1 every 321 stitches all along the row. This takes some concentration to decrease stitches, and one needs to mark each spot with a coloured thread, to keep up with pattern.
One has to micro adjust the final stitches to get the exact Fibonacci numbers in each row.
One could of course never knit a model with stitches numbering in 12 digits.
It just illustrates how Nature might do it.
Maurice Cotterell, author of "The Mayan Prophecies" mentioned the Maya used number 20 as a base (fingers and toes add up to 20)......and the number Nine featured in their calculations. I wondered HOW?
He noted that 1/9 =0.1111111rpt, 2/9 =0.2222222rpt, 3/9 =0.33333333rpt, etc
Amazingly, he decoded the meaning of the picture on the lid at Palenque, Temple of Jaguar, Tikal, by realising the Maya could visualise in reverse, so he used a piece of perspex with a bit of pattern drawn on it and reversed- to fit the jigsaw! He figured out 2012 and much astronomy. He'd been an officer on British Navy submarines for his working life and had time to study the effects of the Heavens on people.
I try to remember as much as I can; maybe some of my memory is not all correct - I gave the book to someone who might understand. Mr Cotterill also decoded more mysteries- eg the Chinese(Entombed Warriors); also some Celtic mysteries....check out his website. He has also written "Future Science- Forbidden Science". I would like to read these all but there is so much I want to do, unable to fit it all in!
Meanwhile, I publish this bit, and there are "9 - equations" more to come.
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