Golden Ratio

November 19, 2020 Shreyas Comments 10 comments

Calculating value of the Golden Ratio

Introduction:

The Golden Ratio, represented by Greek letter “phi” ( $\phi$ ) exists when a line is divided into two parts and the longer part (b) divided by the smaller part (a) is equal to the sum of (a) + (b) divided by (b), which both equal 1.618. See diagram below where

(1) $\begin{equation*} \frac{b}{a} = \frac{a + b}{b} \end{equation*}$

Proof:

Suppose $\phi$ is the ratio $\frac{b}{a}$ such that $\frac{b}{a} = \frac{a + b}{b}$ , where b > a

For convenience, let’s set a = 1, so solving for b will give you the ratio, since $\frac{b}{a}$ will become $\frac{b}{1}$ , which equals to b. So equation (1) becomes:

$\begin{flalign*} &b = \frac{1 + b}{b}& \\ &b^2 = 1 + b& \\ &b^2 - b - 1 = 0& \\ &(b^2 - b + \frac{1}{4}) - \frac{1}{4} - 1 = 0& \\ &(b - \frac{1}{2})^2 - \frac{1}{4} - 1 = 0& \\ &(b - \frac{1}{2})^2 = \frac{5}{4}& \\ &b - \frac{1}{2} = \sqrt{\frac{5}{4}}& \\ &b - \frac{1}{2} = \frac{\sqrt{5}}{2}& \\ &b = \frac{\sqrt{5}}{2} + \frac{1}{2} = \frac{\sqrt{5} + 1}{2}& \end{flalign*}$

Thus, the Golden Ratio, $\phi$ , equals $\frac{\sqrt{5} + 1}{2}$ , which is approximately 1.618

Interesting Facts about Golden Ratio:

Golden Ratio is closely related to Fibonacci numbers. When you take any two successive Fibonacci numbers, their ratio is very close to Golden ratio. Golden Ratio exists in abundance in nature also. Number of petals in flowers consistently follows Fibonacci sequence. Similarly, seed heads and petals of flowers are placed in spiral pattern which follow Golden Ratio for maximum sunlight exposure and minimum shadow. Spirals on seashells, cauliflowers, pinecones, hurricanes etc. also follow similar pattern. Golden Ratio also plays an important role in something as small as double helix of DNA molecule to as big as spiral galaxies, from as miniscule as atomic radius of hydrogen in methane to as gargantuan as Sagittarius $A^*$ .

One of the special properties of the Golden Ratio is that it can be defined in terms of itself. For example:

$\begin{equation*} \phi = 1 + \frac{1}{\phi} \end{equation*}$

You can expand this equation by replacing value of $\phi$ with its right hand side value to obtain a continued fraction like:

$\begin{equation*} \phi = 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{\frac{1}{1 + ...}}}}} \end{equation*}$

I hope you enjoyed learning about Golden Ratio.

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