The Golden Ratio and Fibonacci Numbers

We give here some brief mathematical notes on the connection between the Golden ratio and Fibonacci numbers.

The Golden Ratio

The Golden ratio has long been thought to be aesthetically pleasing to architects and artists.

Definition

Consider two numbers ${a}$, $b$, such that $a>b$.

The Golden ratio

\begin{displaymath} R_{\rm G}=\frac{b}{a}=\frac{a}{a+b} \end{displaymath} (1)

is the ratio of the smaller number $b$ to the larger number $a$, such that this is identical to the ratio of $a$ to the sum $a+b$.

Numerical Values

Eq. (1) implies that $R_{\rm G}^{-1}=1+R_{\rm G}$. That is
\begin{displaymath} R_{\rm G}^{2}+R_{\rm G}-1=0, \end{displaymath} (2)

with solutions
\begin{displaymath} R_{\rm G}= \left\{ \begin{array}{r@{\qquad}l} (1/2)(-1+\... ...9,\\ (1/2)(-1-\sqrt{5})&=-1.6180339. \end{array} \right. \end{displaymath} (3)

It is interesting to consider the quadratic equation
\begin{displaymath} \left(\frac{1}{R_{\rm G}}\right)^{2}- \left(\frac{1}{R_{\rm G}}\right)-1=0 \end{displaymath} (4)

for $1/R_{\rm G}$. The solutions
\begin{displaymath} \frac{1}{R_{\rm G}}= \left\{ \begin{array}{r@{\qquad}l} ... ...9,\\ (1/2)(+1-\sqrt{5})&=-0.6180339. \end{array} \right. \end{displaymath} (5)

of (4) are symmetrical to (3).

Golden Rectangle

It is a consequence of the definition of the golden ratio that if one were to draw a rectangle with sides in the golden ratio (a golden rectangle) and remove from it a square, the rectangle that remains is also a golden rectangle. If this process were to be repeated then the successive points of division lie on a logarithmic spiral.

Continued Fraction

The inverse
\begin{displaymath} \frac{1}{R_{\rm G}}= \frac{1}{1+\frac{1}{1+\frac{1}{1+\cdots}}} \end{displaymath} (6)

of the golden ratio is also the limit of a continued fraction.

Fibonacci Numbers

Leonardo Fibonacci (circa 1170-1250), also known as Leonardo of Pisa, was a number theorist who introduced Arabic numbers into Europe. He is credited with the sequence of numbers that bear his name.

Definition

The Fibonacci sequence may be defined as
\begin{displaymath} F_{n}=F_{n-1}+F_{n-2}, \end{displaymath} (7)

where $n=1$, $2$, $3$, etc. are integers and, by definition,
\begin{displaymath} F_{0}=F_{1}=1. \end{displaymath} (8)

This leads to the construction of the Fibonacci numbers $1$, $1$, $2$, $3$, $5$, $8$, $13$ ...etc.

Lucas Numbers

By defining $F_{0}=2$, $F_{1}=1$, the same recursive series (7) generates the Lucas numbers $2$, $1$, $3$, $4$, $7$, $11$ ...etc.

Connection

The ratio of successive Fibonacci numbers in the limit of diverging $n$ is equal to the Golden ratio (1).

This may be proved by first using Eq. (7) to write

\begin{displaymath} \frac{F_{n}}{F_{n-1}}= 1+\frac{F_{n-2}}{F_{n-1}}. \end{displaymath} (9)

In the limit of diverging $n$,
\begin{displaymath} \lim_{n\to\infty}\frac{F_{n}}{F_{n-1}}= \lim_{n\to\infty}\left\{\frac{F_{n-1}}{F_{n-2}}\right\}^{-1}. \end{displaymath} (10)

Therefore,
\begin{displaymath} \lim_{n\to\infty}\frac{F_{n}}{F_{n-1}}=R_{\rm G}^{-1} \end{displaymath} (11)

and
\begin{displaymath} \lim_{n\to\infty}\frac{F_{n-1}}{F_{n}}=R_{\rm G}. \end{displaymath} (12)

Eqs. (11) and (12) are the required results.

 

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