2023-09-26 09:42:00
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At the heart of nature, in the patterns that adorn the petals of flowers, the spirals of seashells or even the branches of trees, there is an elegant and mysterious mathematical imprint: the golden ratio. It is a concept that has caught the attention of mathematicians, artists, philosophers and scientists throughout history, reaching notable influence on art and architectureas is the case of Leonardo da Vinci’s Mona Lisa, or the structure of the Parthenon in Athens.
THE GOLDEN NUMBER
The golden number, the number of God, the divine number, phi… The golden number responds to multiple names that make it seem like one of the most important and emblematic figures of mathematics, but what does it really refer to? Well, numerically, it is nothing more than the relationship between two segments of a line, that is, a simple geometric proportion (golden ratio).
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The geometric proportion that gives rise to the golden ratio. Segments AB and BC are perpendicular and equal to one. With center in O we draw the circumference radius 1/2. Finally, joining A with O and prolonging we obtain P. The relationship between AP and AB results in the golden ratio.
It is an irrational number, or what is the same, that cannot be expressed as a ratio of two integers and, therefore, It’s infinite: You can always calculate some more figure and it does not have a final value. The numerical value it responds to is 1.618033988…
THE GOLDEN RATIO
One way to better understand how the golden ratio works is through Fibonacci sequence, a series of figures in which the sum of two consecutive numbers always results in the next one and, in addition, the relationship between each number always approaches the golden ratio. That is, the succession is defined as: 0,1,1,2,3,5,8,13,21,34… where the third number is the sum of the first and the second, the fourth is the sum of the second and the third, the fifth is the sum of the third and the fourth… And to find the golden ratio we only have to divide each number by the previous oneHowever, it is better that you start at number 5, because you must let the succession form well.
And does the golden spiral? Well, that element is the “practical application” of our proportion. To find the spiral you must imagine a rectangle and draw a square inside that divides the rectangle into two unequal parts. Then, you must draw another square in the small part, and so on. The sides of the squares will represent each one of the values of the sequence (the two smallest would have a side of magnitude 1, the next 2, then 3 and then 5…) The spiral will leave the smallest square and cross the middle of the square with a curve.
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The golden spiral formed from the construction of rectangles, as indicated by the Fibonacci sequence.
THE GOLDEN NUMBER IN NATURE
However, the most incredible thing about this pattern is perhaps not all that mathematical part, but the way in which appears constantly in nature, leaving its mark on multiple plant, animal and even human systems. The most recognized example is that of the shells of marine animals: there are various species of nautilustypical of the family Nautiliade where the proportion between the spirals inside the shell responds to the golden ratio, following the trajectory marked by the spiral.
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Nautilus where the golden spiral is perfectly visible.
Other striking examples are the distribution of the petals of certain flowers, the number of spirals that make up a pine cone, or the relationship between the thickness of the main branches of a tree and its trunk. Furthermore, in the sunflowers It is very evident to appreciate the famous golden spiral in its central part, right in the seeds placed between the flowers.
Humans are not saved. The most curious example is that, for each individual, the golden ratio appears as the relationship of the distance between the navel and the sole, with the total height. Furthermore, the beauty of the proportion influences the way in which we perceive whether a person is “more or less beautiful” because, the closer their face is to the golden distribution, the more prototypically beautiful their features will seem to us.
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Traces of the golden spiral in sunflower seeds
THE GOLDEN NUMBER IN ART AND ARCHITECTURE
The observation of this divine proportion in natural systems over the years represented a clear influence for various artists, who decided to include it in their works and creations. Many of them went down in history and are presented today as paintings, symphonies or the most emblematic architectural works. A clear example is the Pyramid of Giza: the height of the monument divided by half of the base is approximately equal to the golden ratio. In addition, the sections into which the pyramid is divided, such as the height to the midpoint and the top, would also respond according to the proportion.
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The relationship between the height and size of the base of the pyramid of Giza, in Egypt, responds to the golden ratio.
The Mona Lisa It is perhaps the most recognized work that meets the golden ratio, although the painting also appears on that list atomic leda by Dalí, which stands out for having been painted with the advice of the mathematician Matila Ghyka. In da Vinci’s work, what stands out above all is the way in which facial patterns resemble the golden ratio, or the comparison of the rectangular aspect of the portrait with the golden ratio. Some researchers have even identified curves within the painting that resemble golden spirals.
In architecture I would also highlight the Athens Parthenon due to the relationship that appears between the measurements of the ceiling and the columns, or the violins, since its design implies that the location of its “es” is directly related to the golden ratio. Even in music It is possible to find the divine number, as is the case of the formal structures that appear in the sonatas of Wolfgang Amadeus Mozart, or in the Fifth Symphony of Ludwig van Beethoven. However, experts opt for the idea that both were probably composed based on the balance of sound masses, with the authors not being really aware of the use of the proportion.
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