What if we divide the numbers of the Fibonacci Series by NINE?
4181/9 = 464.55555*rpt
2584/9 = 287.11111*
1597/9 = 177.44444*
..........
34/9 = 3.77777*
21/9 = 2.33333*
13/9 = 1.44444*
A Pattern emerges esp from the last three:-
Fn / 9 = F(n + 5) / 100 plus "extra"...eg .004444* or .003333*
F14 is 377, divide by 9 = 41.88888*
F14 + 5 = F19 = 4181; F19 / 100 = 41.81000- extra is .0788888*
F13 is 233, divide by 9 = 25.88888*
F13 + 5 = F18 = 2584; F18 / 100 = 25.84 plus .04888*
.07888 / .04888 = 1.6137479
Higher number ratios of "extras" = closer to 1.618, the "Golden Mean"
eg F19/9 = 464.555*. F24/100 = 463.68; difference is 0.87555*
F18/9 = 287.111*. F23/100 = 286.57; difference is 0.54111I
ratio of differences is 1.618069
................................................................................
One can compile tables of the Fibonacci numbers divided by 9; also by 2, 3, 4, 5, 6, 7, 8.
Only one number to F25 divides evenly by all single digits except for 5.
ie number F24 which is 46 368
............................................................
What if we multiply the first 25 Fibonacci Numbers by NINE?
I shall only do a few:-
17711 x 9 = 159 399
10946 x 9 = 98 514
6765 x 9 = 60 885
4181 x 9 = 37 629
2584 x 9 = 14 373
987 x 9 = 8 883
610 x 9 = 5 490
377 x 9 = 3 393
233 x 9 = 2 097
144 x 9 = 1 296
89 x 9 = 801
55 x 9 = 495
An Equation emerges:- 9 x Fn = 1/2F(n+6) + 1/2F(n-6) !
eg. 987 x 9 = 1/2 x 17711 + 1/2 x 55
=8855.5 + 27.5 = 8 883 = 987 x 9 !
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Multiply by 18,
then the Equation is 18 x Fn = F(n+6) + F(n-6)
eg. 4181 x 18 = 75258 = 75025 + 233 !
Try it for other numbers, even huge ones.....
*************************************
Multiply by 27,
then the Equation is 27 x Fn = 10 x F(n+2) + 1/2F(n+1) + 1/2F(n-8)
eg. 27 x 10946 = 10 x 28657 + 1/2 x 17711 + 1/2 x 233
.........................= 286570 + 8855.5 + 116.5 = 295542
.........................= 27 x 10946 !
*************************************
Multiply by 36, (4 x 9).
The Equations might be 36 x Fn = 100F(n-3) + 10 x Fn + F(n+1) + F(n-2) + F(n-4) + F(n-9).....OR......
36 x Fn = 100Fn-3) + 10F(n-3) + F(n+4) + F(n+2) + F(n-1) + F(n-4) + F(n-6) + F(n-10) - F(n-12)
try 233 or 610; I will check on this sometime.....
*************************************
What if we Multiply by 45, (5 x 9), ?
The Equations are:-
45 x Fn = F(n+8) - F(n+1) - F(n-3) - F(n-5) - F(n-7)......and.....
45 x Fn = F(n+7) + F(n+5) + F(n+3) + F(n-1) + F(n-8)......then.....
F(n+8) - F(n-8) = F(n+7) + F(n+5) + F(n+3) + F(n+1) + F(n-1) + F(n-3) + F(n-5) + F(n-7) !
try 610.....
*************************************
Something might happen with x54, x63, x72, x81, etc; I have not tried it yet...
What use is it? It is just a buzz to see Patterns, " palindromes" almost!
There might be a particular model which one can make with these patterns.....
*************************************
*************************************
What if we multiply by 19?
An Equation that emerges is:-
19 x Fn = F(n+6) + Fn + F(n_6) !
eg. 19 x 610 = 11 590 = 10946 + 610 + 34, correct.
Try other numbers; see how far up the series one can go with this!
************************************
What if we multiply by 29?
The Equation is 29 x Fn = F(n+7) - F(n-7) !
eg. 29 x F18 = F25 - F11
......29 x 2584 = 74936 = 75025 - 89, = correct.
*************************************
Multiply by 39:-
Equation seems to be 39 x Fn = 1/2 F(n+9) + Fn - 1/2 F(n-9) !
eg. 39 x 610 = 46368/2 + 610 - 8/4
.....................=23184 + 610 - 4 = 23790. = correct.
eg. 39 x 144 = 10946/2 +144 - 2/2
.....................=5473 + 144 - 1 = 5616, = correct.
************************************
Multiply by 59 or 99:-
there could well be Patterns; I am working on it.....
*************************************
At least I have got it all down on paper and online
There could well be corrections to any mistakes...
At least it satisfies my curiosity.
What does the number nine signify indeed?!
For the Maya, the number nine was to be worshipped.
Nine "Lords of the Night" were painted on the tomb walls of the Temple of Inscriptions.
1/9 = 0.1111*rpt
2/9 = 0.2222
3/9 = 0.3333
....
8/9 = 0.8888
....
10/9 = 1.1111
20/9 = 2.2222rpt
nine based decimal system is half as complex as today's 10 based system,
This is from page 324 of " The Mayan Prophecies"
by Adrian G. Gilbert and Maurice M Cotterell publ 1995 Element Books, Inc.
***************************
You might like to play with the first 135 Fibonacci numbers:-
See post 5/4/2013
*********************************
------------------------------------------------
Thursday, June 16, 2011
It looks like a clam...**
There is a model of two halves of a bivalve, shown open before sewing, and after forming into shape. It was made May 2010 and only a few days ago I sewed it up and padded it.
.............................
Here are photos:-
It looks like a Pattern is needed with 1/4 F(n) stitches each row. To be continued....
.............................
Sunday, June 5, 2011
Fibonacci Equations having the Number Nine.
Apparently there are thousands of Fibonacci equations.
Here are some which I have figured out for myself.
I do not know if anyone else has discovered them.
-------------------
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.
Here are some which I have figured out for myself.
I do not know if anyone else has discovered them.
-------------------
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.
Friday, June 3, 2011
BACK INDEX
First 100 Fibonacci numbers 16/4/2011.,
Patterns for crochet and knitting Fibonacci Seashells
29/4/2010, 19/5,25/5, 21/6, 23/6, 24/6, 3/5/2011, 25/5, etc.,
***A brand new Pattern to knit and crochet will soon be published;
similar to those on recent posts, 17/05/2012
Singularity 21/4/2011 etc.,
Pyramids 24/4, 2/5/2011.,
Vortex 28/4/2011.,
Tree, cones, ripples 21/12/2010, 21/4/2011.,
Equations 5/12/2010, 21/11/2010 etc.,
Astronomy 9/5/2011, 24/4.,
Colour pencil drawings 16,17/4 etc*asterisked...
These have been removed 17/05/2012
Similar posts are on my new website http://www.acolourfulpost.blogspot.com/
Quite a few sections of some posts have been removed and
similar ideas may be found on my http://www.awhatifworld.blogspot.com/
Ancestors 15/8/2010, 24, 28/6/2010 etc . More to come later.
Patterns for crochet and knitting Fibonacci Seashells
29/4/2010, 19/5,25/5, 21/6, 23/6, 24/6, 3/5/2011, 25/5, etc.,
***A brand new Pattern to knit and crochet will soon be published;
similar to those on recent posts, 17/05/2012
Singularity 21/4/2011 etc.,
Pyramids 24/4, 2/5/2011.,
Vortex 28/4/2011.,
Tree, cones, ripples 21/12/2010, 21/4/2011.,
Equations 5/12/2010, 21/11/2010 etc.,
Astronomy 9/5/2011, 24/4.,
Colour pencil drawings 16,17/4 etc*asterisked...
These have been removed 17/05/2012
Similar posts are on my new website http://www.acolourfulpost.blogspot.com/
Quite a few sections of some posts have been removed and
similar ideas may be found on my http://www.awhatifworld.blogspot.com/
Ancestors 15/8/2010, 24, 28/6/2010 etc . More to come later.
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