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
Explanation
We start with the Fibonacci sequence
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.
HistoryThe Italian mathematician Fibonacci (1180 - 1241) described this sequence. |