The Golden Ratio

Also called as the Divine Ratio, the Golden Ratio has been a fascination for Engineers, architects, Mathematicians, scientists, and many more. This ratio also comes in many names like Golden Proportion, Golden Mean, Golden Section and Devine Proportion. According to Goldnumber.net, the Golden Ratio and the Fibonacci sequence are mathematical cousins.

golden ratioThe derivation of such constant is very easy to derive. Using the line above partitioned in a and b. The ratio of b to a and the whole line segment to b such that b>a  is the golden ratio.

Derivation:

Ratio of the whole line segment to b

\displaystyle\frac{a+b}{b}=\displaystyle\frac{b}{a}=\varphi

We can rewrite \displaystyle\frac{a+b}{b} \to \displaystyle\frac{a}{b}+1

Since \varphi=\displaystyle\frac{b}{a}.

\displaystyle\frac{a}{b}= \displaystyle\frac{1}{\varphi}

The resulting equation will become,

\displaystyle\frac{1}{\varphi}+1=\varphi

1+\varphi=\varphi^2

\varphi^2-\varphi-1=0

Now, this is a quadratic equation in terms of \varphi  and we can solve the value of \varphi using the quadratic formula.

\varphi=\displaystyle\frac{-b\pm\sqrt{b^2-4ac}}{2a}

\varphi=\displaystyle\frac{1\pm\sqrt{(-1)^2-4(1)(-1)}}{2(1)}

\varphi=\displaystyle\frac{1\pm\sqrt{5}}{2}

Since a,b>0 Their ratio must be greater than zero as well.

Thus,

\boxed{\varphi=\displaystyle\frac{1+\sqrt{5}}{2}\sim 1.618}

parthenonIt was said that the façade of the Parthenon as well as the elements of its façade and elsewhere was carefully made to Golden Rectangles. According to studies, Phi is anywhere in our body parts. From the shape of our face, to ratio of our arms, fingers as well as our beauty, Phi is embedded.

 

 

Dan

Dan

Blogger and a Math enthusiast. Has no interest in Mathematics until MMC came. Aside from doing math, he also loves to travel and watch movies.
Dan

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