Approximation to Golden Rectangle using Fibonacci Squares

Theorem

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.

$\blacksquare$

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.

$\blacksquare$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $5$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $5$