The Golden Ratio, Fibonacci series and their applications.

The Golden Number is a magical phenomenon in Mathematics and finds application in nature and our life in a big way. Read on to find out more. WHAT has mathematics got to do with beauty? A lot if you just look around. The most beautiful things that catch our eye are the ones that are the most symmetrical. Nature is replete with such beauty— whether trees, flowers, seeds, vegetables, fruits, and so on.

Leonardo da Vinci - The Annunciation

Leonardo da Vinci – The Annunciation

We are more likely to notice a symmetrical body and a proportional face. We also find certain buildings and monuments more attractive than others because they have mathematical symmetry built into them.

Things that appeal to us seem to conform to a Golden Ratio. Now, what is this Golden Ratio and where do we find them in nature and life? Before we actually go on to ferret out the beautiful things in nature that conform to the Golden Ratio let us first get familiar with the Golden Number.

φ (phi), the Golden Number

Golden ratio line

Golden ratio line

Can you tell a number, adding 1 to which will give you the square of it? The number is 1.61803398….  (1.6.. +1 = 1.6.. * 1.6..  ) This number, mathematically called  φ (phi), is known as the Golden Number. The ratio, 1:1.61803398…is known as the Golden Ratio or ‘Extreme and Mean Ratio’. In mathematics and the arts, two quantities are in the Golden Ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller.

Euclid of Alexandria (ca. 300 BC) defined this Golden Proportion in Elements, one of the most important textbooks of mathematics ever written, as:

φ = 1.618…So, if BC is 1, then AB is 1.618…This gives us a simplified derivation of the Golden Number and the Golden Ratio. We will gradually discover the goldmine underlying this Golden Number and Ratio.

Golden Number & Fibonacci Series

Now consider the following series-


Do you find anything interesting? Well, any successive number of a particular number in the series is the summation of that number and its previous number. For instance, 1+0 = 1, 1+1 = 2, 2+1 = 3,3+2 = 5 and so on.

Interestingly, as the series progresses, the ratio of a particular number and its successive number is approximately the golden ratio. For example, 13/8 = 1.625,21/13 = 1.615385, 34/21 = 1.619048 and so on. Leonardo of Pisa, also known as Fibonacci, in the year 1202, first discovered the series. Fibonacci discovered the series while solving a mathematical problem regarding breeding of rabbits. He was trying to find out the number of rabbit pairs after one year, starting from a single pair of rabbits, assuming the following ideal set of considerations:

■     Rabbits gain sexual maturity at the age of one month

■     After gaining maturity, a female rabbit always gives birth to one male and one female rabbit

■     Gestation period of rabbit is one month and

■     No rabbits die during the period of investigation, that is, one year.

Surprisingly, the number of rabbit pairs after each month, starting from the first month, is exactly the number series that we have discussed as you can see in the following table:

 Fibonacci Rabbits

Fibonacci Rabbits

So, we observe the reflection of the Golden Ratio in the population growth of rabbits! We will soon be acquainted with some really mysterious correlations of life and golden number in our forthcoming discussion. Prior to that, let’s discover two major concepts in this regard—the Golden Rectangle and the Golden Spiral.

Golden Rectangle & Golden Spiral


 Fibonacci Squares - Golden Ratio

Fibonacci Squares – Golden Ratio

Taking the Fibonacci series 0,1,1,2,3,5,8,13…, if we start drawing squares with 1cm, 2cm, 3cm, 5cm, 8cm sides, they can be arranged in a particular way as shown in the following figure. The overall structure is a rectangle. Naturally, the length:width ratio of this rectangle is simply the Golden Ratio. Such a rectangle is known as the Golden Rectangle. If you deduct a square from the golden rectangle, the residual rectangle will again be a golden rectangle.

The Golden Spiral

The Golden Spiral

Now, let’s draw quarter circles inside each square of the above shown golden rectangle, in such a manner that the arms of the squares become tangent of the concerned circle. The spiral structure generated in this way is known as the Golden Spiral. Mathematically it is nearly a logarithmic spiral. We find this spiral structure in various living forms in nature.

Mathemagical Beauty

The Golden Ratio is also believed to be related with music. When a song is in accord with the Fibonacci numbers and the golden ratio, it becomes more catchy. ‘Born in the USA1, a famous song by Springsteen is 377 (a Fibonacci number) seconds long and has its instrumental climax after 144 (another Fibonacci number) seconds. Such a song is known as a Fibonacci song. Another famous piece of work is ‘One Way Out’ by Allman Brothers. It is 420 seconds long and its climax appears after 260 seconds. The ratio 420/260= 1.615 is very close to the golden ratio. Such a song is known as a Golden Section song.

Our universe is also an example of the golden ratio. The structure of galaxies is very close to the structure of the golden spiral. More interestingly, if we consider the rotational time of earth around the sun to be 1 year, then Mercury takes φ-1(0.24) years,Venus φ-1(0.62) years, Jupiter approximately φ5 (11.1; the actual time is 11.9) years and Saturn approximately φ 7 (29.0; the actual time is 29.5) years to complete a full rotation around the sun.

The Fibonacci series is found even in poetry. The Fibonacci style of poetry aims at a typical pattern, where the lines of the poem have syllables in Fibonacci series. For example, 1 st line with 1 syllable, 2nd with 1 syllable, 3rd with 2, 4th with 3, 5th with 8 syllable and so on. In some other cases another fantastic pattern is found. Here the lines of the poem, which fit in the Fibonacci number series (like 3rd line, 5th line, 8th line, etc.), are attributed with some special features. Benjamin Moon finds a similar example in case of the poem Golden. The poem is written in the context of golden numbers. The 6th line of the poem is shortest, 10th line is the longest and the poem is of total 16 lines. Just divide the numbers by 2 and you will get the 3, 5, and 8…relationship (Fibonacci series)!

It is now time to discover the golden ratio in nature and life. The mathemagics of the golden number is distributed throughout in nature and our life. The Golden Number can be found in nearly every aspect of life, starting from ancient art, architecture, biology and even in human body systems.

Further Read :

It is now time to discover the golden ratio in nature and life. The mathemagics of the golden number is distributed throughout in nature and our life. The Golden Number can be found in nearly every aspect of life, starting from ancient art, architecture, biology and even in human body systems.

Read further :

Applications of Golden ratio in Biology and Human Body

Applications of golden Ratio in Ancient Art and Architecture