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): $5$