Fibonacci and Golden Ratio

One of the ways to compute the golden ration

[Andrew Newell]

One way is to Google it (Google returns the answer).

[Jan]

Do you have a Matlab related question?

[Jan]

@Ashley: Don't give up: As soon as you edit the question and add any details - and a question! - you will get meaningful answers.

[Khan Muhammad Babar]

Is there any way to quantify the Golden mean of Image in MATLAB. Please help.

Answer 1

[Clemens]

Actually the Golden Ratio is exactly:

( 1 + sqrt(5) ) / 2

so no need for iteration. Proof is easy through z-transform.

[Walter Roberson]

But that gets back to my original answer, "The Golden Ratio is an irrational number, and thus an infinite number. It is not possible to compute its decimal expansion in a finite amount of time."

[Jan]

Fortunately the universe is finite. Therefore I do not believe, that an infinite number will match into it. While there is a minimal Planck length and a minimal Plank time, I propose a Planck eps for irrational numbers. According to Rupert Sheldrake, I claim that PI has as many numbers as has been calculated already. And after reading http://scientopia.org/blogs/goodmath/2010/12/08/really-is-wrong/ I'm not sure at all anymore about this fuzzy digits stuff.

Answer 2

[Walter Roberson]

The Golden Ratio is an irrational number, and thus an infinite number. It is not possible to compute its decimal expansion in a finite amount of time.

[Jan]

But if you have a sharp knife and a firm sausage, you can apply the golden section in a finite amount of time.

[Matt Tearle]

But the OP was asking for the golden ration, not the golden ratio. So I think Jan's right -- sausages might be necessary.

[Walter Roberson]

Not an option here, Jan -- I'm a vegetarian.

[Walter Roberson]

Matt: "Golden Ration" is clearly a Pale Ale or Bitter, not a sausage!

[Sean de Wolski]

@Walter: you should make that an answer so we can vote for it!

[Jan]

Consider soya sausages. They are in the new Veggi-Toolbox of Matlab 2011a.

[Sean de Wolski]

Soya sausages? That's like one term in the Taylor-series expansion of sausages.

[Walter Roberson]

Jan, Soya Beans used for the production of soya products are the dried fruit of the soya plant, and thus were not covered by the Veggi-Toolbox in R2011a (which, I understand, is still withheld from production due to legal battles over whether Tomatoes are fruits or vegetables....)

Answer 3

[Walter Roberson]

Let F(t) be Fibonacci number #t. Then

y = 100;  %initial guess
x = (F(t+2) * y + F(t+1)) / (F(t+1) * y + F(t));
while x ~= y;
  y = x;
  x = (F(t+2) * y + F(t+1)) / (F(t+1) * y + F(t));
end

When the loop finishes (no more than a few centuries later, I'm sure), x and y will be the Golden ratio.

[Jan]

Are you sure that x converges to a single value, or can it oscillate between two values due to floating point arithmetics?

[Walter Roberson]

Not completely certain. It worked for the F() values that I tried.

[Jack Lê]

Thanks

Answer 4

[Kishore]

fib=[0 1];
i=3;
while(i<=21)
    fib(i)=fib(i-1)+fib(i-2);
    gr=fib(i)/fib(i-1)
    i=i+1;
end

gr = 1

gr = 2

gr = 1.5000

gr = 1.6667

gr = 1.6000

gr = 1.6250

gr = 1.6154

gr = 1.6190

gr = 1.6176

gr = 1.6182

gr = 1.6180

gr = 1.6181

gr = 1.6180

gr = 1.6180

gr = 1.6180

gr = 1.6180

gr = 1.6180

gr = 1.6180

gr = 1.6180

disp(fib)

  Columns 1 through 16

           0           1           1           2           3           5           8          13          21          34          55          89         144         233         377         610

  Columns 17 through 21

         987        1597        2584        4181        6765

Answer 5

[Guna]

% Function to calculate Fibonacci sequence up to a certain number of terms
function fib_sequence = fibonacci(n)
    fib_sequence = zeros(1, n);
    fib_sequence(1) = 0;
    fib_sequence(2) = 1;
    for i = 3:n
        fib_sequence(i) = fib_sequence(i-1) + fib_sequence(i-2);
    end
end

% Calculate the golden ratio using Fibonacci sequence
n = 20; % Number of Fibonacci terms to generate
fib_seq = fibonacci(n);

% Calculate the ratio of consecutive Fibonacci numbers
golden_ratio_approximations = fib_seq(3:end) ./ fib_seq(2:end-1);

% Display the approximations of the golden ratio
disp('Approximations of the golden ratio using Fibonacci sequence:');

Approximations of the golden ratio using Fibonacci sequence:

disp(golden_ratio_approximations);

    1.0000    2.0000    1.5000    1.6667    1.6000    1.6250    1.6154    1.6190    1.6176    1.6182    1.6180    1.6181    1.6180    1.6180    1.6180    1.6180    1.6180    1.6180