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Squares of Fibonacci numbers

From squares with sides corresponding to Fibonacci numbers, you can form rectangles with which you can calculate the golden ratio as

φ = 1,61803398874989…

 


Explanation

We start with the Fibonacci sequence

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, ...

In a representation you can show squares with sides that correspond to these numbers. This results in rectangles where the ratio between the long side and the short side develop as 3 : 2, 5 : 3, 8 : 5, 13 : 8, 21 : 13, ...

1 1 3 8 21
2
5
13

If you divide each number in the Fibonacci sequence by its direct predecessor you get

  1 / 1 = 1,000
  2 / 1 = 2,000
  3 / 2 = 1,500
  5 / 3 = 1,666
  8 / 5 = 1,600
  13 / 8 = 1,625
  21 / 13 = 1,615
  34 / 21 = 1,619
  55 / 34 = 1,617
  89 / 55 = 1,618
  144 / 89 = 1,617
  233 / 144 = 1,618
  377 / 233 = 1,618

The divisions stabilize so that the golden number 1.61803398874989… is created.

 


Additional information

The Italian mathematician Leonardo Fibonacci (1180 - 1241) described this sequence.

 


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