Golden Ratio and Fibonacci Numbers

The quadratic equation

\( \varphi^2 =\varphi + 1 \)

has two (algebraic irrational) real solutions: The Golden Ratio

\(\varphi = 2 \cos(\pi/5) = (1+\sqrt{5})/2 \approx 1.618\),

as well as its inverse

\( \omega = \varphi -1  = 1 /\omega = 2 \cos(\pi/5)  – 1 =  (\sqrt{5} -1) / 2 \approx 0.618\).

The Golden Ratio has the property  that it is the limit of the ratio of two successive Fibonacci numbers. The name “Fibonacci“, corresponds to “filius Bonacii”, or the son of Bonacci, and was given to him posthumously. The sequence \(F_n\) of Fibonacci numbers is defined by the recurrence relation

\(F_n = F_{n-1} + F_{n-2},\)

for \(n \ge 2\), with seed values \(F_0 = 0\), \( F_1 = 1\).

A Fibonacci spiral created by drawing circular arcs connecting the opposite corners of squares in the Fibonacci tiling; this one uses squares of sizes 1, 1, 2, 3, 5, 8, 13, 21, and 34.

Known to Euler, Bernoulli, and de Moivre, attributed to Binet, the Fibonnaci numbers can be explicitely solved via a recurrence relation to obtain a closed-form expression:

\(F_n = (\varphi^n – (-1)^n \omega^n)/\sqrt{5}\),


\(\omega = 1 / \varphi = \varphi – 1 = 2 \cos(2 \pi/5) = (\sqrt{5}-1)/2\).

It is straightforward to verify that \(F_0 = 0\), \(F_1 = 1\). Furthermore, since \(|\varphi|>1\), \(|\omega |= |1/\varphi | <1\),

\(\lim_{n\rightarrow +\infty} F_{n+1}/F_n = \lim_{n\rightarrow +\infty} \varphi^{n+1}/\varphi^n = \varphi.\)

For a historical overiew and miconceptions of the usage of the golden ratio, please read the article “Misconceptions about the Golden Ratio” by George Markowsky. Another worthwhile, more accessible publication on the topic: “The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number“, by Mario Livio.

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