"Fibonacci ripples" type 2.

1. Where the diameter is a fibonacci number. diagram 3





a) F11=89, radius=44.5, circumference=279.7

F12=144, r=72, C=452.6; F13=233, r=116.5, C=732.3; F14=377, r=188.5, C=1184.9

10 x radius F11=445 ="close to" C of F12, =452.6

10 x radius F12=720 ="close to" C of F13, =732.3

10 x 10 x radius F13=1165="close to" C of F14, =1184.9

So 10 x radius = "close to" Circumference of Fn+1 when the diameter is a Fibonacci number.

b) Intervals between ripples;- 188.5 - 116.5=72

c) 27.5+44.5=72 etc

d)Radius Fn+1/C, Fn. 144/279.7143=0.5148, 233/452.47=0.5148

Circumference Fn/radius Fn+1. 452.57/233=1.94236, 732.2857/377=1.9424

e) diam 1/diam 2. 233/144=1.618,

C1/C2=452.5714/279.7143=1.61797 ~1.618, r1/r2=116.5/72=1.618

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2. Where radius is a Fibonacci number. diagram 4

a) F14----radius 377----circumference 2369.7
---F15----610-----------3834.3
---F16----987-----------6204
---F17---1597-----------10038.3
---F18---2584-----------16242.3
---F19---4181-----------26280.6
---F20---6765-----------26280.6
---F21--10946-----------68803.4
---F22--17711------------111326.3

Circumference Fn+1------radius Fn x 10---------difference----------ratio
3834.3--------------------3770------------------64.3
6204----------------------6100-----------------104----------------104/64.3=1.6174183
10038.3-------------------9870-----------------168.3-------------168.3/104=1.618269216235.7------------------15970-----------------272.3------------272.3/168.3=1.617944126280.6------------------25840----------------440.6------------440.6/272.3=1.6180683
42522.9------------------41810-----------------712.9------------712.9/440.6=1.6180208
68803.4------------------67650----------------1153.4-----------1153.4/712.9=1.6178987111326.3----------------109460----------------1868.3=--------1868.3/1153.4=1.6198196

This is an extraordinary discovery!
It means that
an increment of "close to" is actually governed by The Golden Mean, 1.618 !

more:- Fn x 44/7 = circumference; higher numbers:-

F22--17711-----x44/7 =11326.3
F23--28657-----"-----=180129.7, -177110 = difference =3019.3,

F24--46368--------------291456, -286570 =4886------ratio 4886/3019.3=1.6182558
F25--75025------------471585.7, -463680 =7905.7-----------7905.7/4886=1.6180311
F26--21393------------763041.7, -750250 =12791.7-------12791.7/7905.7=1.6180335
F27--196418-----------123462.4, -1213930 =20697.4---20697.4/12791.7=1.6180335

Here is The Golden Mean , 1.618, playing a part.

***large fibonacci numbers are too much for calculator to handle.

b) intervals between ripples
1597-987=610, etc

c) 233+377=610, etc

d) r1/r2=1597/987=1.6180344
C1/C2=10038.25/6204=1.6180286


There are quite a few websites concerned with ripples and numbers.
Some are technical; some are religious, to do with Fibonacci numbers. I shall list some asap.