The Golden Ratio OR The Divine Proportion
Mathematics represents the universal language of humanity, with the help of which we can understand God. (Dana-Georgiana FLOREA)
I know this equation, it is characteristic of segments divided in the ratio of the golden ratio. And now, let’s draw the conclusion: in the meridian triangle of Cheops’ pyramid, the hypotenuse is in the golden ratio with the smallest of the legs. This is the triangle that bears the name of the Egyptian triangle. But the golden section is also included in the dimensions of the king’s chamber, located inside the pyramid, which is reached through a corridor that communicates with the outside through an opening in one of the outer walls, located about 14 meters above the ground. The dimensions of this parallelepipedal room are the base – being a double square (with sides 1 and 2), and the height equal to half of the diagonal of this rectangle.
Who, and when, it was discovered that if two segments of a figure are in the ratio of the golden ratio, that figure makes an entirely different impression from others in which this division is not considered, will never be known, for it is a good handed down from prehistory. Assumptions exist and we will discuss them later. It is certain that the pyramid of Cheops, and others in which this ratio was respected, imposes through its grandeur, through the harmony and solidity of the form. Many wondered if she would make the same impression if she were taller or flatter. The sympathizers of the golden ratio, however, claim that its perfect proportionality is due only to this ratio, which got the name of the golden ratio or golden number from Leonardo da Vinci – and has kept it ever since.
Luca Pacioli wrote a whole book about this subject and entitled it “Divine Proportion”, and his friend Leonardo da Vinci put aside the dozens of plans and tasks he had, just to make the necessary drawings for this book. That, in the 15th century, and in the 20th century, Matila Ghyka published several books on this subject, one even titled The Golden Number. Then, the golden ratio appears as a mathematical element that intervenes in the expression of beauty in art. And, there is a third reason: we will have so many other issues to discuss that I don’t think we will have time to return to this one, if we don’t decide to say everything we have to say now !
Is the content rendered by the image of your creation perceived as a scientific obligation for the development of a link between man and the Universe?
- You specified well, sir: we will only say what we know about the golden ratio, because there will be a lot more that we don’t know !
- And among those we know, we will have to mention those who do not recognize any merit in the golden ratio ! Among them is Benedetto Croce, I think you have read about this too?
- Could he not have done it? Don’t I have the memories of the past so that I don’t know what would await me if I skipped some important line from a book taken from you? And if we are going to discuss the gold standard, then we have to start with the Greeks, because they said it directly, most likely after importing it from the Egyptians. It preoccupied the Pythagoreans a lot, then Plato and, of course, Euclid, with whom I prefer to start, to present it first from a strictly geometric point of view. In Elements, it is mentioned for the first time in problem 11 of the second Book, in this form: “Let a given line be cut, in such a way that the rectangle contained by the whole line and one of the segments is equal to the square of the remaining segment”. First, let me specify that the Greek mathematicians used the term “line” with the meaning of “line segment”, then that this problem is an example of solving, geometrically, a quadratic equation. The method used, known as “application of areas”, leads to the determination of point C on the segment AB, so as to realize the required relationship (fig. 6). Denoting AB a and AC x, then a (ax) = xz.
It seems that in Book Two this problem serves as an introduction to problem 30 from Book VI, which is formulated as follows: “Let a given finite straight line be cut according to an extreme and average ratio”. Here, in Book VI, there is also the definition that was not introduced in Book II: “A straight line is said to be cut according to an extreme and mean ratio when, as the whole to the larger segment, so the greatest to the least”. Here is the golden section which for Euclid and the other Greek geometers had the name of dividing a segment into mean and extreme ratio.
- In truth, the construction of regular convex or stellate polygons with 5, 10 and any even multiple of 5 sides is based on the golden section, and this was the work of the Pythagoreans. These constructions are found in Book IV of Euclid’s Elements, starting with the 11th sentence, or rather even with the problem in the 10th sentence: “Let an isosceles triangle be constructed having each of the angles from double base of the remainder”. This problem, which is solved by applying the division of a segment in the golden section, is then used in the following constructions. In V. Marian’s Commentary it is clearly stated that these sentences, like the entire Book IV, “are a discovery of the Pythagoreans”. And further: “The Pythagorean origin of the construction of the side of the pentagon also emerges from the fact that the recognition sign of the Pythagoreans was the star pentagon (pentagram)… It even seems that the choice of the pentagram was made precisely to remind this important discovery”.
Can you create another vision of the way in which the compositional experimentation of the idea of “primordiality of creation” is desired through an intensification of the manifestations of life?
- Admiration is an understatement, I think they adored it, especially if we think about the fact that the golden section can be built with a ruler and compass.
- You are right, for the Greek mathematicians this argument weighs heavily, or the golden ratio is, first of all, a proportional average, therefore, it can be constructed with a ruler and a compass. And, effectively, how would the operation be carried out?
- It is not difficult to understand that this property of the segments found in the golden section, to transmit their properties to each other endlessly, delighted the Greek mathematicians, and stunned Luca Pacioli to such an extent that it made him attribute a divine origin to this proportion ! It seems to be a harmony that is transmitted from one segment to another, like a permanence of life !
- But that is how it is in reality, because the golden number also expresses one of the manifestations of life: the law of organic growth, which Leonardo Fibonacci established at the beginning of the 13th century. The problem, which has since become famous, entitled the house rabbits, is in the twelfth chapter of the Book of the Abacus. Given its charm, I would read it right after the original:
“How many pairs of house rabbits are born in a year, from a single pair of rabbits? To find out how many pairs of rabbits are born in a year, someone placed a pair of rabbits in a walled area, knowing that after a month, a pair of rabbits gives birth to another pair, and rabbits begin to give birth to offspring at the age of one month. Since the first pair gives offspring in the first month, the pair doubles, and in this month two pairs are obtained, of which one pair, namely the first, will have offspring in the following month, so that in the second month there will be three pairs; of these in the following month two pairs will have offspring so that, in the third month, two more pairs of rabbits are born, and the number of pairs of rabbits in this month is five. Among them, three pairs will have offspring in the same month, and the number of rabbits in the fourth month will be eight. Among them, five pairs will give rise to five other pairs, which added to the eight other pairs form in the fifth month thirteen pairs.
Thus, Leonardo Fibonacci found that there is a numerical law by which a property of living matter can be expressed, namely, in the form of a string of whole numbers: 1, 1, 2, 3, 5, 8, 13,... which has the property that each term of the string, starting from the third, is the sum of the two previous terms: (1) 2=1+1; Or 3=2+1;...13=8+5. This string, double additive, bears the name of the discoverer, the “Fibonacci string” or the “law of organic growth”.
The primordiality of Creation refers to the act of establishing a point of support, such as a mathematical calculation, with the help of which you can determine a similarity between “that which is above” and “that which is below”, between the manifestations of life and the behavior of the universe.
The Golden Ratio OR The Divine Proportion highlights the mathematical means by which a solution can be determined that intervenes in the expression of beauty in art, and not only that. Also, as Pythagoras claimed, the whole universe is mathematically ordered and everything can be expressed by numbers. And in 1989, the legendary physicist John Archibald Wheeler suggested that the Universe is fundamentally mathematical and can be seen as made up of information.
* Note: Câmpan, Florica - Povestiri cu proporţii şi simetrii, Editura Albatros, 1985.
