How is the Golden Ratio related to the Golden Spiral?

The golden ratio is simply condensing, or expanding that ratio. If you take a rectangle (with this proportion) and cut a square out of it, than you are left with a congruent rectangle, that can be divided indefinitely by the same method.

So... as the spiral becomes larger, or smaller, it is defined primarily in this ratio. Congruent to the latter or previous.

** What ever orpheus-sword said is wrong... This number cannot be caculated. It's like PI, never ending.

Do it on a calculator, and see for yourself.

Not exactly astronomy, but what the heck.

What you said about the Fibonacci numbers is true, but that's not how you need too look at them.  They are defined by an iterative sum:

so if the first number is 1
then the second number is 1
then the third is the sum of first and second: 2
then the fourth is the sum of the 2nd and 3rd: 3
then the fifth is the sum of the 3rd and 4th: 5
and the sixth is the sum of the 4th and 5th: 8, and so on:

so the sequence is:
0,1,1,2,3,5,8,13,21,......

Now their relationship to phi (golden ratio is simple).  Instead of adding the successive numbers together, compute their ratios:

2/1 = 2
3/2 =1.5
5/3 = 1.666666
and so on.

Therefore, if the nth fibanocci number is F(n), then phi is given by
limit as n goes to infinity of F(n+1)/F(n).

YOu can check it.  Get the list of Fib numbers:

http://en.wikipedia.org/wiki/Fibonacci_number

and just take the ratios of two large, neighboring numbers
for instance:
F_20/F_19 = 6765/4181 = 1.61803, which is pretty close to phi:

http://en.wikipedia.org/wiki/Golden_ratio

but you need to take n to be infinitely large for it to be exact.

Hope this helps!



WHAT?  This is exactly true.  What you said about pi, is also accurate, both numbers are transcendental, but that doesn't make this wrong!  Both numbers pi and phi can be calculated by infinite sums and products.  It can be proven by very basic calculus.

So exactly what part did I say is wrong?  The fact that they are the sum of the preceding values?  Nope, it's true.  That's how Fibonacci defined them.  About that ratio of successive values?  Nope, it was mathematically proven.

you know what the fibonacci is:
1,1,2,3,5,8,13,21,34......
you know what the ratio is: 1:1.618

as the fibonacci sequence is going further up in greater numbers....the PHI ratio becomes more perfect.

The Golden Mean spiral improves the PHI ratio as the spiral gets larger.....more perfect.

how are they related? Through geometry. The pentagram describes both things at the same time.
The numbers are a way to work back to the geometry, but mainly it is the geometry which is prevailent.
The geometry describes all. It is physical and it is nature.