difference between golden rectangle and golden ratio ? (answer must be in points)
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Answer:
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)).
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.
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.
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
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
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.