The golden ratio is more than just a number

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The golden ratio is more than just a number. Indeed, as demonstrated by mathematicians, this number appears naturally in geometry and it is also used in the construction of famous buildings. In addition, part of the surgical beauty benefits from this number; because doctors interpret the face of the patient as a partition of several geometric patterns.


Let’s discover the meaning of this number from several angles:


The origin of name golden


Let’s discuss the origin of the name of this number which uses the word “golden”. Obviously, this is because this number has numerous applications in our life. According to the history of mathematics, Luca Pacioli was the first to manage this number. He used geometry and suggested the name “divine proportion”. Several years later, a physicist by the name of Johannes Kepler suggested the name “gem of geometry”. Another famous personality, Leonardo da Vinci, the intuition that this number depends on some of his paintings, and suggested the name “golden section”. The real name “golden rate” dated 1932 is due to the engineer and Prince Mattila Ghyka.


Geometrical meaning of the golden ratio


Let a segment with a length of $\ell$. Assume that we can with $\ell=a+b$ with $a>b$ not null. The value $\frac{a}{b}$ is the golden ratio if it satisfies

\begin{align*}\frac{a}{b}=\frac{a+b}{a}.\end{align*}

In the leturatur, we the symbole $\varphi$ (Phi) for indicates this famous numbder. Observe that we can write

\begin{align*}\varphi&=\frac{a}{b}=\frac{a+b}{a}\cr &= 1+\frac{1}{\frac{a}{b}}.\end{align*}

Hence \begin{align*}\varphi=1+\frac{1}{\varphi}.\end{align*}This means that $\varphi$ is the solution of the following algebraic equation (quadratic equation):

\begin{align*}\varphi^2-\varphi-1=0. \end{align*}The discriminate associated with this equation is $r^2=5$, when then have two solutions. However, as $\varphi$ represent a distance, it is positive. Hence we keep the following solution of $\varphi:$

\begin{align*}\varphi=\frac{1+\sqrt{5}}{2}\approx 1.6180\cdots\end{align*}


How to get this number from the Fibonacci sequence


According to the previous paragraph, we remark that $\varphi$ is a fixed point of the following function


\begin{align*}f(x)=1+\frac{1}{x}.\end{align*}


In this regard, we know that a fixed point of a function is a limit of a certain recurrent sequence (Banach-Picard theorem). So there exists a sequence of real numbers $(u_n)_n$ defined by


\begin{align*}u_0=0,\quad u_{n+1}=f(u_n)= 1+\frac{1}{u_n} \end{align*}

such that $u_n$ tends to $\varphi$ as $n$ tends to $\infty$.


Let us now identify this sequence. In fact, we define the Fibonacci sequence


\begin{align*} &F_0=0,\quad F_1=1\cr & F_{n+2}=F_{n+1}+F_n.\end{align*}

Using this formula, it is clear that $F_n$ is not null. Deviding by $F_{n+1},$ we get

\begin{align*} \frac{F_{n+2}}{F_{n+1}}&=1+\frac{F_n}{F{n+1}}\cr &= 1+\frac{1}{ \frac{F_{n+1}}{F_{n}} }.\end{align*}


Then we have

\begin{align*}\varphi=\lim_{n\to +\infty} \frac{F_{n+1}}{F_{n}} .\end{align*}

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