'The book of nature is written in the language of Mathematics'. Know the 'Golden Ratio' and its significance

What is the value of 'X'? Read the question in my Maths paper. I gazed at my wrist-watch kept at the front of the desk. The time was running fast, really fast. It was already 2 hours into the exam but I was done with just the 60% of the paper. One more hour and 40% of the paper still remaining, I had to go full throttle or else risk loosing the race. 

'Don't worry Prakhar, you will complete the last territory in time', assured my inner voice. I wiped the sweat on my brow and started working on the question, to solve for X.

X has been a favourite 'unknown' known worldwide. Its elusiveness has been tormenting one and all in the schools and colleges. Though I quite liked Mathematics and was at ease with solving the puzzles, my answer sheet was never bereft of one or the other silly mistake. During my 12th exam, instead of reading A.B, I read it as A-B and lost 5 precious marks due to it.☹️ Unforced errors, I like to call them, relating to sports.

There are so many areas in mathematics such as Calculus, Trigonometry, Geometry, Mensuration … But Mensuration was the one which I found quite easy. It was all about creating a picture in mind. As has been aptly put by the great scientist Galileo Galilei - 'The book of nature is written in the language of Mathematics', you find a pattern and move towards an answer. Once you are able to form one, half of the question gets solved. 

Well, this blog isn't about the formulae or shortcuts in Mathematics but is about one specific number/ratio called 'Phi', which though not part of my curriculum (at least I don’t recollect studying it), holds high importance in the area of Mathematics. 'Phi' or the 'Golden Ratio', can be seen in multiple facets of our human life and also applies to flora and fauna.   

What is Phi or the Golden Ratio?

'Without mathematics, there's nothing you can do. Everything around you is mathematics. Everything around you is numbers'. These were the words of a world renowned Indian - Shakuntala Devi, also known by the sobriquet 'Human Computer'. And the Golden ratio provides weight to these words of wisdom.

Golden ratio has been of prime importance since the very earlier days. Also known as Divine Proportion, it is formed by bifurcating a line such that the longer part divided by the smaller is equal to the total length divided by the longer part. In equation form, it can be summed up as: 

(x+y)/x= x/y ; 

x=longer part and y= shorter part

Solving it, we get the value of 'phi' (denoted by 'φ') as 1.618033988… 

Just like its famous similar sounding cousin PI, it is also an irrational number, i.e. it has endless numbers post the decimal and can't be represented accurately as a simple fraction. But Phi is farthest from being close to any simple fraction and that is what makes it special.

What is so special about the Golden ratio?

There are many applications of the Golden ratio or rather, the ratio's presence can be seen in multiple natural things such as:

Fibonacci sequence:
13 – 3 – 2 – 21 – 1 – 1 – 8 – 5

"O, Draconian devil! Oh, lame saint!"

These are the words by Robert Langdon from the famous fiction Da Vinci Code by Dan Brown, when he sees the numbers scribbled on the floor by a dying man. These numbers are (I am sure you too can make out) the first eight numbers of Fibonacci series, though not arranged in order.

Fibonacci sequence is a sequence of numbers such that a number in the series is equal to the sum of preceding two numbers, i.e. 1,1,2,3,5,8,13,21,34....
The ratio of the successive numbers of the Fibonacci series approaches the Phi value, as we go higher in the sequence.

Human body:
The navel in the human body divides it roughly in the Golden ratio. The ratio of the length of the lower part of body to the upper part of body is equal to 'Phi'.

Though definitely arguable, the so called 'beautiful face' is said to align to the Golden ratio. Much for this reason, many facial plastic surgeries strive towards achieving the number.

Even the spirals of Human DNA incorporate the 'Phi ratio'. Each turn of DNA double helix measures 21 Angstrom wide and 34 Angstrom long. Thereby following Fibonacci sequence and hence the Golden ratio.

DNA helix exhibits Golden ratio

Ancient marvels:
Great architectures such as Pyramids of Egypt, the Parthenon are believed to have Golden ratio used in their design, though it isn't conclusively proved that these structures were built keeping the ratio in mind or are just co-incidentally based on the ratio.

The Parthenon - Greece

Flowers and Trees: 

The sunflower seeds are filled up in such a way that if you count the spirals, they are usually a Fibonacci number such as 21, 34, 55 or 87 (Fibonacci sequence). The seeds start from the centre and the subsequent seeds  turn at an angle to form a spiral. The angle so chosen by the seeds aptly makes for a circular compact binding, without any gaps. And you will be surprised that the angle is 1.618 or the Phi. Nature surprises us, doesn't it?

Spirals in Sunflower

As an activity, try finding out the best possible ratio for yourself using the calculator in this page.

Petals in flowers are too placed in a similar fashion, forming a cohesive unit with the unique turn of Phi value. Also, the petals can be observed following the Fibonacci sequence. Example: Petals of Buttercups and Daisy are 5 & 34 in number.

Nerd Notes:

• (x+y)/x= x/y can be solved by the Quadratic as  φ = 1 + 1/φ
• Phi is the only number having its square equal to number plus one i.e. Φ² = Φ + 1 = 2.618
• Shakuntala Devi multiplied two randomly picked 13 digit numbers in just 29 seconds to create a Guinness record

Some people really play with numbers with quite ease. While definitely there are a few shortcuts to finding roots and multiplying numbers, the challenge to do them repeatedly sans any mistake is really special. Are you aware of any such tricks that you may have learnt for the competitive exams ? If yes, do let us know in the comments.

And as always, thanks for reading.

- Prakhar