In the world of art, architecture, and design, the golden ratio has earned a tremendous reputation. Greats like Le Corbusier and Salvador Dalí have used the number in their work. The Parthenon, the Pyramids at Giza, the paintings of Michelangelo, the Mona Lisa, even the Apple logo are all said to incorporate it.

The golden rectangle is produced mathematically when the side DC of a square ABCD is bisected at X and an arc with radius XB is swung on to DC at F. The resultant rectangles AEFD and BEFC are both golden rectangles whose sides conform to the golden ratio, thus: DF/AD = BC/CF = 1.618. The Parthenon is an interesting example of a mathematical approach to art. Once its ruined triangular pediment is restored, the ancient temple fits almost precisely into a golden rectangle. The height (in meters) of the building follows golden ratio proportions . That is: AD/AB = AB/BD…19.87 / 12.28 = 12.28 / 7.59 ≈ 1.618 = Φ. In addition the facade of the Parthenon was designed around the proportions of two large and four small golden ratio, or √5, rectangles. Was the the Great Pyramid designed so that the ratio of the slant height of the pyramid to half the length of the base would be O? For example if h represents the height, b half the base, and s the slant height of the Great Pyramid, hence from the Pythagorean theorem s ~ 612.01 feet. This gives a ratio of 612.01/377.90 ~ 1.62 which differs from O by only 0.1%. Thus, we must examine the claims put forward for the presence of the golden ratio in the dimensions of the Great Pyramid. Furthermore, Herodotus's figures about the dimensions of the Great Pyramid are wildly off. The Great Pyramid neither is nor was (it has lost some height over the years) anywhere near 800 feet tall nor 800 feet square at the base. Finally, we should note that Herodotus wrote roughly two millennia after the Great Pyramid was constructed. The Parthenon at Athens fits into a golden rectangle almost precisely once its ruined triangular pediment is drawn in. Though it incorporates many geometric balances, its builders in the fifth century b.c. probably had no conscious knowledge of the golden ratio.

Thank you Pradeep Paraman for the nice answer

Essentially, it is true that whenever we notice an exceptional beauty and harmony, we will usually reveal the presence of golden ratio so one should not wonder why this concept, which connects mathematics, nature, science, engineering and art in a very unusual and interesting way, is present in all aspects of human life. Human aspiration is to be surrounded by structures and works pleasant to the eye, so it is logical to expect the magic of golden ratio to be found in the pores of mathematics, architecture, painting, sculpture, music and many other scientific disciplines...

Huntley, H. E. The Divine Proportion. New York: Dover, 1970.

Thank you Anton Vrdoljak for the answer

You're welcome Sourangshu Ghosh, and thanks for recommendations!!

Once someone said: Nature is written in mathematical language... and Math is fun:

https://www.mathsisfun.com/numbers/golden-ratio.html

The ancient mathematics had a fascination with numbers and patterns. Starting out with the natural numbers, then to the concept of zero to the concept of appending 0 to the natural numbers which led to the negative numbers and then the integers and the rational numbers, the algebraic numbers and then the transcendentals, e.g., pi, e, etc. This led to the concept of groups, rings, fields, extensions of fields and other mathematical constructs. In parallel questions in geometry were an active area of research. Over time some numbers took on almost a mystical and spiritual meaning which gets more into numerology than mathematics. There is no better example of this venturing into numerology than the golden ratio and the mystic nature that has grown up around it, https://www.youtube.com/watch?v=tBBvki9NqUY .

The golden ratio is not mysterious or spiritual (although DiVinci called it the "divine proportion")- it does no even rise to the level of being transcendental. It is an algebraic integer being the root of x^2 - x -1 = 0. This leads to the functional equation of phi= (1 + 1/phi) which follows easily from its minimal polynomial. However, somehow the golden ratio has managed to gain mysterious value because objects having aspect ratios similar to this number seem pleasing to the eye. It is also related to the Fibonacci numbers - again following from the properties of its minimal polynomial and the properties of the field extension Q(sqrt(5)).

But its reputation has grown - but only a few pyramids actually have ratios close to the golden ratio, the Great Pyramid of Giza, being an example of one.

So for mathematics - it is not any more or less important than any algebraic number or any element in the algebraic field Q(sqrt(5)) where it is a fundamental unit. However, the golden ratio must have a vey good PR agent and for numerology - the golden ratio seems to have reached the pinnacle of importance. However, it does have importance in mathematics education as it is a wonderful example of a concept that can be used to grab the attention of children at an early age to stimulate their interest in mathematics. That may be its biggest contribution to mathematics.

Anton Vrdoljak can you also give some recommendation to this question and some of my articles

In the world of art, architecture, and design, the golden ratio has earned a tremendous reputation. Greats like Le Corbusier and Salvador Dalí have used the number in their work. The Parthenon, the Pyramids at Giza, the paintings of Michelangelo, the Mona Lisa, even the Apple logo are all said to incorporate it.

The Golden Ratio has rather broad applications in almost every field. Here are the some of them:

1-Architecture - Many works of art are claimed to have been designed using the golden ratio. Eg: The Parthenon, according to some studies, has many proportions that approximate the golden ratio.

2-Logos and trademarks - For construction of any kind of design, one can use the beauty of the golden ratio to make great composition decisions quickly and easily that have an inherent visual harmony that cannot be produced with any other ratio.

3-Cosmetic and corrective dentistry - Dentists uses the golden ratio during corrective surgeries to have a relative proportion between the teeths.

4-Reconstructive and corrective facial surgery - It is used in Plastic

5-Surgery for better facial proportion.

6-Design, layout and composition of marketing and advertising materials

7-Photo editing and composition

8-Stock market analysis - It’s used in the analysis of financial markets to predict price and timing inflection points in stock price movements.

9-To Develop an Automotive Designs

Beyond that, the knowledge and understanding of the golden ratio can simply help you to appreciate the beauty and aesthetics, in nature as well as in the arts. As with any study or tool, the more you know about it, the more you will be able to do with it and appreciate the results.