The relationship between nature and mathematics is relatively close. The golden ratio reflects the unique relationship between nature and mathematics. The ratio is golden just like its name. It is represented by the Greek letter Φ (Phi).
Definition: Two numbers a and b are said to be in the golden ratio when the ratio of a + b and b is equal to the ratio of a and b, where a> b> 0. That is, (a+b): a = a: b = Φ.
The value of Φ is 1.618033 ... . This is an irrational number.
Evidence of Φ:
From this, the value of Φ is proved to be ((1 + √5) / 2 = 1.618033 .. (because Φ > 0)
Example of Φ:
1. Fibonacci series: Fn = nth Fibonacci number, F0 = 0, F1 = 1, F2 = 2
2. A Fibonacci spiral (golden spiral) is formed based on the Fibonacci numbers.
First draw a 1x1 square. Then create an arc by combining any of its two opposite corners. This is the first arc of the spiral. Draw another 1x1 square on the previous square. Expand the previous arc into this new square. Now draw a 2x2 square including the sides of other two squares.
As we continue this process, the spiral becomes bigger. In nature, this spiral can be seen in the tail of chameleon, conch shell, shells of snails, etc.
3. The seeds in a sunflower are unique examples of Fibonacci numbers and golden spiral. Starting from the center, some seeds look like a spiral. If we look closely, we can see that many spirals are clockwise. Similarly, a number of spirals are anticlockwise. Surprisingly, the number of clockwise spirals is less than the number of anticlockwise spirals, and these two numbers are the value of any two consecutive Fibonacci numbers. If there are 21 clockwise spirals, then there must be 34 anticlockwise spirals. It is found in all sunflowers.
4. Leonardo Da Vinci's Mona Lisa painting is beautiful indeed, but it is also a wonderful example of the golden spiral. The golden spiral appears from the nose of Mona Lisa to her left hand. Besides in the picture, many people also believe that the human face contains golden spiral. Even if we try to make a spiral with the help of different squares like the golden spiral in the face of any human being and the ratio of the length of two consecutive squares is taken, then the face, in which this ratio is closer to Φ, is the more beautiful face.
Examples of Fibonacci numbers and golden ratios can be seen in all areas of art, music, etc. Many musicians have composed their songs based on Φ. There are also examples of Φ in the pyramid of Egypt. So the golden ratio is definitely a unique and beautiful number.
Somnath Jena, IIT, Kharagpur
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