Jorge de Luna
Beauty or aesthetics are subjects that, in some cases, have been classified as subjective, placing them in the realm of philosophy. Other voices claim that they are related to "gender," referring to social attributes and opportunities associated with being male or female; however, they have a mathematical basis.
"The exact application is called Phi (Φ), in Spanish fi, a Greek letter representing an irrational algebraic number known as the 'number of God': 1.618 033 988... infinity, and it conditions the 'golden ratio,' the geometric order of the universe, duality, good and evil, light and darkness; it is the divine proportion of beauty," explained Professor and Researcher at the Escuela Superior de Ingeniería Textil (ESIT) of the Instituto Politécnico Nacional (IPN), José Javier Ocampo Silva.
It might seem that the "golden ratio," as it is also known, leans more towards the mystical side, but its exact application was in ancient Greece, in architecture, sculpture, and nature, where it instinctively aroused a sense of beauty or perfection among the Greeks. During the Renaissance, in the 15th century, the golden ratio was expressed in the human body, its harmonic relationship, and proportion of its parts.
In an interview for Selección Gaceta Politécnica, the Master of Visual Arts emphasized that the mathematical foundation for the design and composition of garments taught to future textile engineers is based on the calculations of the golden ratio.
He argued that humans have three nourishments, and to demonstrate this, he drew an equilateral triangle on canvas, showing at each corner the foundations: mathematics, sports, and the arts.
The number of God conditions the golden ratio, which during the Renaissance was expressed in the human body.
He said that numbers are precise, but it is not only necessary to instill this knowledge in young people but also to encourage them to engage in sports, as it complements them. Additionally, "we need the nourishment of creativity, and this is the arts; once complemented, we have equal sides, equal angles, all supported by mathematics."
"There can be no knowledge without mathematics; knowledge cannot exist if it is not justified mathematically, and that's why his teaching techniques at ESIT on fashion design and artistic creations have this foundation," said Ocampo Silva.
"A true artist has to justify their creations mathematically, apart from having studied anatomy, perspective, color, and technical skills, but they also need knowledge of composition, and what better way to learn than the golden ratio," he added.
The golden ratio dates back to 300 B.C. in the classical Greek work "Elements" by Euclid, but in 500 B.C., Phidias, the world's most famous sculptor and mathematician of the classical world, studied Phi and applied it to the design of his statue housed in the Parthenon in ancient Greece.
Literature suggests that the golden ratio is found everywhere in nature: in tree leaves, sunflowers, and pinecone shells.
Another emblematic theme around the "divine proportion" is the Fibonacci sequence, one of the most famous number sequences in history. It is called the "secret code of nature" or the "divine sequence" because it appears again and again in natural structures.
Although this sequence was discovered in 1200 by the Italian mathematician Leonardo Pisano, also known as Leonardo Fibonacci, the golden ratio received its name later in 1509 from the work "De Divina Proportione" by Luca Pacioli, which included illustrations by Leonardo Da Vinci, who first called it the "divine proportion."
Professor José Javier Ocampo Silva mentioned that for over 20 years in his lectures, he has used the "divine proportion" as the basis and has its mathematical reasoning in a second-degree equation.
He explained that the "number of God" is the result of an equation used by the Greek sculptor Phidias to achieve the most symmetrical features in his works.
Although to arrive at this infinite and irrational number, the professor commented that you first have to place a point, from which a line is drawn, which is divided in half and extended to an extreme reason.
The golden ratio is the divine proportion of beauty
"It is a mathematical ratio. We have a measure X, in this case, the unit was taken, so from the half with extreme reason, this mathematical relationship emerges. As you can see, it is inverse, and you have to solve the second-degree equation."
"So, we have this mathematical ratio: X is 1, as 1 is X minus 1. To solve this, you multiply across."
"We arrive at this second-degree equation that has its solution with the quadratic formula. So, by extracting the solution, we have that X is equal to this formula, and it will give us 1.618; this is the number we call Phi, and it is the golden ratio," he detailed.
In this regard, to exemplify and bring numbers to life, he sketched a drawing where a right-angled triangle was divided into golden proportions to find this spiral (known as the golden or snail spiral) that arises from the Fibonacci numerical series, an infinite sequence of natural numbers; starting from 0 and 1, they are added in pairs, so each number is equal to the sum of its two previous ones, such as: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55...
"By dividing a number from the Fibonacci sequence by the previous one, a result increasingly close to 1.618 is obtained," said José Javier Ocampo.
"This is what I teach my students because it awakens their creativity, and they are already seeing and saying, 'Oh wow, I didn't imagine that as an engineer, I would have to design a garment with mathematics,'" he expressed.
The mathematical basis for garment design that Professor Javier teaches to future textile engineers is based on golden ratio calculations.
The peculiarity of the shapes of ceremonial centers like the pyramids of Teotihuacán in Mexico and those in Egypt is ordered in golden proportions, just as galaxies and the universe are created in cosmic spirals. Hence, the importance of the Fibonacci mathematical series to determine the golden spiral, noted the ESIT professor.
The professor stated that the famous spiral resembles the shape of an accent mark, and based on it, students have to sketch, resembling a snail emerging from the square-rectangle.
"This geometry is for drawing a human head from the front. We use a compass that is the golden ratio, which is not equal parts; it is not divided in half or in thirds, so it is divided between the half and the extreme reason of that distance," he said.
The distance between the nose and the chin is what measures the nose; the ears measure the same as the nose in all humans, and therefore everything that is designed has to be done in these proportions, "if we do it this way, we are projecting our geometry of a human to our space," Ocampo Silva explained.
The ESIT professor said that the future designer expresses themselves with their tools, with all those materials they use for expression and to capture new things in their sketches. Color, shapes, but "it requires the technique that gives it geometric order. Yes, that's how it is in relation to beauty, because if we give geometric order, it is beauty."
The importance of applying it to a garment, dress, or textile product projects the soul and forms of each race, "we are all different, but the proportion is the same," he emphasized.
Selección Gaceta Politécnica #167. (October 31st, 2023). IPN Imagen Institucional: Read the full magazine in Spanish here
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