"Fibonacci Ripples" type 1.

1. Where the circumference of each ripple is a Fibonacci number.

diagram 1: Gazing at the diagram and calculating from tables in previous posting, one sees more relationships.

a) radius of C,F(n) = 10% of C,F(n+1),almost.

F8=21, radius =3.34 ~10%of F9. F9=34. radius=5.4~10% of F10, F10=55.

or 10 x radius F(n) nearly = Circumference F(n+1)

Diagram 2. For higher numbers.

F22=17711, radius=2.8 x 1000~=10% of F23 which is 28657 whose radius is 4.6 1000, ~10% of F24 which is 46368 whose radius is ~7.3 x 1000 which is 10% of F25 which is 75025.

Calculating even higher:- F30=832040, radius = 132 x 1000~=10% of F31 which is 1346269.

b) distance between ripples is close to a Fibonacci number

C,F10=55, r is 8.8. C,F9=34, r is 5.4; difference is 8.8-5.4=3.4=10% of F9.

C,F24=46368, r is 7.3 x 1000. C,F23 is 28657, r is 4.6 x 1000; difference is 7.3-4.6=2.7 x 1000 which is "close" to 10% of F23.
C, F30=832040, r is 132.04 x 1000. F31=1346269, r is 214.77 x 1000; difference is 214.77-132.04=82.73 which is ~=10% of F30.

c) distance between ripples. AB + BC = CD.

d) Circumference Fn/ radius Fn+1 = 21/5.4=3.8888.

( F8=21, r is 3.3, F9=34, r is 5.4.)

F21=10946,F22=17711, r is 2.9 x 1000, C,F21/rF22=10946/2.9 x 1000=3.77448 ~= 3.8

Radius Fn+1/C,Fn = 5.4/31=0.257148 ~=2.6.

e) C1/C2 = 17711/10946=1.618 = the Golden Mean.