Monday, November 9, 2015

An Eternal Protocol For Building One Fibonacci Number into the Next, Part 2.

1.  Building the Fibonacci Series in two stages, namely by making it gradually in the "form" of:-

........13,  17,  21,  27.5,  34,  44.5,  55,  72,  89,  116.5,  144,  
188.5,  233,  305,  377,  493.5,  610,  798.5,  987,  1292,  1597,  etc





Interesting.  This shows that the best way is to round off the numbers to a whole number  ie the lower value   (not  .5 ). 
 Increases (or decreases) may be made 1)  every 3 stitches  ie 3 to 4 stitches;  2)  every 4 stitches  ie 4 to 5 stitches.  Adjust the remainder of stitches needed at the end of the row.



Please click on Tables to enlarge image.  You can then right click and save, copy onto Word, or save link as...a download or favourite.







































































This is new.  An Equation:-

please click on image to enlarge it, to see it better.

































I have been most excited at these discoveries!
Previously it was trial by error to get the numbers right,  but this is exact!  except for the very small numbers at the beginning of the series where we must figure it out simply.

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Imaginary Fibonacci Ripples,   The blue is a Fibonacci number,  the white is the half Fibonacci number !


So,  the Pattern is to increase every 3rd stitch to make 4,  for the half Fibonacci numbers.  Increase every 4th stitch to make 5 for the real Fibonacci numbers.  Here we have calculated the radius of each circle with circumference equal to the Numbers  and the distance between each band.  
At first I crocheted the last stitch by "hook into loop, crochet 2 chains,  Begin new row.  Single crochet was used  ie  hook into loop, wool around hook, pull wool or cotton through,  wool around hook, pull through all stitches.   There was an ugly join.  A piece of colored yarn was used to keep track of the end of the row.  Often I had to count to check if it was ok.
To make a clean join i just continued on with the spiral, and only the colored yarn marked the end of the row.



*** Please note that UK Europe Australia crochet instructions and gauges are different from American usage.  Please see post of 12/26/2014, "Information for US and UK Crocheters"
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Calculations for Concentric Circles (Ripples)
iaw the Fibonacci Series and half Fibonacci Series.

Fn                    radius              r1 – r2              rows
2584                411.3               78.5                 79
2091                332.8               78.6                 79
1597                254.2               48.6                 49
1292                205.6               48.5                 49
987                  157.1               29.9                 30
799                  127.2               30.1                 30
610                  97.1                 18.5                 19
494                  78.6                 18.6                 19
377                  60                    11.5                 12
305                  48.5                 11.4                 11
233                  37.1                 7.2                   7                     
188                  29.9                 7                      7                                             
144                  22.9                 4.3                   4
117                  18.6                 4.4                   4
89                    14.2                 2.7                   3
72                    11.5                 2.7                   3
55                    8.8                   1.7                   2
45                    7.2                   1.7                   2
34                    5.4                   0.9                   1
28                    4.5                   1.2                   1
21                    3.3                   0.6                   1
17                    2.7                   0.6                   1
13                    2.1                   0.5                   1
10                    1.6                   0.3                   0
8                      1.3                                           

I have tried another way to upload the Table-  control c to copy document, the control v to paste here.
You can see on the blue and white model, that the last white band has 11 white rows,  this shows it has circumference ...305.
I am unable to see it right!   Very stressful;  I  am working at a friend's place to upload all of this;  no electricity or Internet at home in my caravan.   I shall go away and check on it. 
Something is wrong somewhere.  Maybe the Table needs fixing.  **  Fixed 23/11/2015

A story of me in High School.  Maths class.  Geometry.  early 1960s.  All those of us who got a Theorem wrong in a Test were humiliated;  forced to line up in the middle of the gymnasium at lunch time, for the full hour.  I was having a rough time, unable to see the way the geometry worked.  For the life of me, I could not get it.  Usually I came top in Biology and such,  but this day I had problems.  No wonder I am fated to do blogs on Geometry!
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More 23/11/2015
















You can click on the images to see them more clearly.

Right click to save or print.....

From this info you can crochet a flat model of ripples in this fashion,  increasing every 3rd stitch  and then no increase for the required number to attain the half Fibonacci number iaw the Tables.  Increase every 4th stitch  to make 5  etc to make the Fibonacci number.
eg last band, blue is 377 from 305 white.   column 3.
base is 305  ie 72 x 4 = 288 + 17;
increase is       72 x 5 = 360 +17.
Therefore increase every 4th stitch 72 times until 17 loops remain,  and crochet normally the last 17.
So the same for all the Series,  as above.  Good luck.  Any mistakes you might find on my blog, please let me know.  I am always editing and correcting!

*********************a photo still to come...



insert 19/11/2015


This Table is a correct one. 
Some previous info has been rounded off to whole numbers, to make it easier to knit or crochet.



















































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