........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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