A mathematician is a blind man in a dark room looking for a black cat which isn’t there – Charles Darwin
So, before we begin what really is the golden ratio?
If we were to divide a line into two parts, then if the ratio of the sum of length of the two parts to the longer part should be equal to the ratio of the longer path to the shorter one.
It has been denoted by different names too such as the divine proportion, golden mean, golden section, golden cut, golden number so on and so forth but mathematically it has been denoted as ‘phi’ Φ.
Also, Mathematical Φ’s value is,
Φ=(1+_/5)/2 = 1.6180339887…….etc.
So that is to say it’s not a whole number but an irrational no, but as it is with ‘pi’ π it is usually rounded off to 1.618.
The golden ratio has been known to have held a special fascination of about 2400 years, phew!!!! That is a pretty big time huh…. But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its perceived omnipresence and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics.
The golden ratio has been known to have turned up frequently in fields of Aesthetics, Architecture, Paintings, Book Designs, Music and even Nature.
Due to its common appearances in nature, the Greeks considered the golden ratio to be something divine even Euclid have been known to have debarred his students from using Φ until and unless necessary.
This rectangle has been made using the golden ratio which according to some scientists and people popular from other fields assumes to be aesthetically pleasing. Don’t you find this uncannily similar to a classical painting frame?? Well if you will find it aesthetic or not depends totally upon you!!
Have you ever noticed that why the monitors or TV’s we use is in the ratio of 4:3 or 16:10 or 16:9.Try finding out these given ratios and compare their value to the golden ratio. Does it not seem to slowly approach the value of 1.618??For example 16:9 is 1.777 is it not close right.
So if we go into the field of architecture one of the most iconic buildings of the past the Parthenon has been known to have incorporated the golden ratio in its construction. It is observed that the façade as well as the elements of the façade are known to be circumscribed with golden rectangles. It is observed even in the great pyramids of Giza, the length of each side of the base is 756 feet with a height of 481 feet and if the proportion of its base to height is taken then it comes to be 1.5717 which is quite close to the golden ratio.
The façade of Parthenon The Pyramid of Giza
How golden ratio has been applicated in The Pyramid of Giza
A 2004 geometrical analysis of the great mosque of Kairouan reveals a consistent application of the golden ratio in the design, the overall proportioning of the plan and the dimensioning of the prayer hall, the court and even the minaret.
The Overall design of the mosque is uncannily similar ain’t it.
If we move over to art the golden ratio has been even found their too!!! The golden ratio has been used to achieve balance and beauty in many Renaissance paintings and sculptures. Leonard Da Vinci himself has used the golden ratio to define all the proportions in his Last Supper, including the dimension of the table even the proportion of the walls and backgrounds. The golden ratios have been present in Da Vinci’s Vitruvian man and even Mona Lisa. Other artists who have been known to have incorporated the golden ratio in their works are Michelangelo, Raphael, Rembrandt, Seurat, and Salvador Dali.
Mona Lisa The sacrament of Last Supper by Salvador Deli
The Vitruvian Man
Before we go into the realm of nature related with the golden ratio let me show you its connection with the Fibonacci series.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55……….
If we were to take the ratio of 2 consecutive (one after the other) Fibonacci numbers, their ratio gets very closer to the golden ratio. In fact the bigger the number in the Fibonacci series the closer it gets to the approximation.
A | B | B/A | |
2 | 3 | 1.5 | |
3 | 5 | 1.666666666… | |
5 | 8 | 1.6 | |
8 | 13 | 1.625 | |
… | … | … | |
144 | 233 | 1.618055556… | |
233 | 377 | 1.618025751… |
Amazing, isn’t it?
I think I have exceeded the attention span of an average human being (approx. 7 min). So I am going to stop for now But we will see you next part and Do read it! I swear on the river Styx (Percy Jackson fans will get my reference),its much more interesting.
Vishal Kumar
(Mathematics at Utkarshini)