Approximation to Golden Rectangle using Fibonacci Squares

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An approximation to a golden rectangle can be obtained by placing adjacent to one another squares with side lengths corresponding to consecutive Fibonacci numbers in the following manner:


It can also be noted, as from Sequence of Golden Rectangles, that an equiangular spiral can be approximated by constructing quarter circles as indicated.

Proof 1

Let the last two squares to be added have side lengths of $F_{n - 1}$ and $F_n$.

Then from the method of construction, the sides of the rectangle generated will be $F_n$ and $F_{n + 1}$.

From Continued Fraction Expansion of Golden Mean it follows that the limit of the ratio of the side lengths of the rectangle, as $n$ tends to infinity, is the golden section $\phi$.

Hence the result.


Proof 2

From Sum of Sequence of Squares of Fibonacci Numbers:

$\forall n \ge 1: \ds \sum_{j \mathop = 1}^n {F_j}^2 = F_n F_{n + 1}$

Hence the result.