Tech UPTechnologyThe golden number: the divine formula

The golden number: the divine formula

But what is special about that number? Why is it not like the others? In the same way that the number Pi (3.141592 …) represents the most perfect geometric body, the sphere, 1.618033 … is the number of beauty. The 15th-century monk Luca Pacioli, perhaps influenced by the idea that new knowledge should be adapted to the beliefs of the Church, called it La Divina Pro Portion and stated: there is the same substance between three persons -Father, Son and Holy Spirit-, in the same way the same proportion will always be found between three terms, and never more or less “. What is hidden behind this esoteric phrase, more typical of alchemists and occultists than mathematicians, is that number, which is believed to have been baptized by Leonardo da Vinci with the name of the golden number. Centuries later the American mathematician Mark Barr assigned the Greek letter fi, in honor of the sculptor Phidias, who used it in his works.

The surprising beauty of an irrational number

The golden number belongs to the set of irrational numbers, that is, those that cannot be expressed as a quotient of two whole numbers. For example, the square root of two is irrational – a discovery that so upset the Pythagoreans that they hid it from the world. In our case, we can compute the golden number with a calculator if we follow these simple instructions: first, we calculate the square root of 5; Then we add 1 to the result and divide the total by 2. If we know how to program a computer, we can try to beat the record for the highest number of decimal places calculated: in 2000 and with less than 3 hours of computing, the first 1,500 million were found decimal places.

Mathematically speaking, we can define the golden number as the one that if we add one to it, the same result comes out as if we square it. Thus, if 4 were the golden number, to calculate its square it would not be necessary to do the operation of 4 by 4, which comes out 16, but it would simply be enough to add 1. And there are actually two golden numbers, one positive ( 1.618033 …) and another negative (-1.618033 …), but it is the first one that has taken all the glory.


The humanist who named 1.618033 …

So far all this may seem like pure numerology. It is as if someone, with very little work and a lot of free time, has taken the trouble to start looking for curious relationships with numbers. However, what is truly mysterious is that this strange number is found in the growth of plants, in cones, in the distribution of leaves on a stem or in the formation of shells. Also in the identity card, credit cards, a large part of business cards and in almost all packs of tobacco. Or in the Parthenon . Or in the classic example of what a harmonious body is: the Vitruvian Man by Leonardo da Vinci.

Following in the footsteps of those who most influenced him, the humanist Leon Battista Alberti and the sculptor Antonio Filarete, Leonardo believed that anatomy and architecture were related. It was in the 1480s, while trying to win over the Duke of Milan and the architects of the court, that he deepened this relationship that he expressed in his famous 1487 drawing, clearly based on the description of the architect Marcus Vitruvius Pollio.

Anatomical perfection is golden perfection

In it, Pollio affirms: ” In the human body, the central part is the navel. For if a man lies on his back, with his arms and legs extended, and a couple of measures are centered on the navel, the fingers of the hands and feet will touch the described circumference from that center. And it can also be inscribed in a square figure “. If we divide the side of the square (the height of the human being) by the radius of the circumference (the distance from the navel to the tips of the fingers) we will have the golden number. Thus, if the reader wants to know if it is beautifully perfect, he only has to pick up one ruler.

Little by little Leonardo became obsessed with the search for guidelines that related not only anatomy with architecture, but also with the harmonic structure of music and with nature itself. His search for proportions in the world around him, like his attempt to relate the circumference of the treetops to the length of their branches, was intense but futile. However, it was not a wrong idea, because looking at nature we can find the golden number in different contexts. But first we must look back and pay attention to a 13th century Italian mathematician who had a somewhat dark passion for rabbits and their reproductive rate.

The most beautiful number in a rabbit hole

In 1202, Leonardo Fibonacci wondered about how fast rabbits would spread across the Earth under ideal conditions. Suppose, it was said, that we have a single partner, that both members are ready to procreate within a month of existence, and that they give birth to a new partner after a month of gestation. How many couples will there be after a year? At the end of the first month the original couple is willing to procreate, but there is still only one couple. At the end of the second month we will have the original and her first couple-daughter. At the end of the third there will be in the field the original, the first couple, who is about to procreate, and a second couple-daughter. At the end of the fourth month we will have the original and its third couple-daughter, the first couple and its first couple-daughter, and the second couple-daughter, who is ready to procreate. In short, the sequence of pairs of rabbits is: 1, 1, 2, 3, 5. Can the reader guess the pattern behind this sequence? If we lengthen it a little, it is easier:1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 … Indeed, the so-called Fibonacci sequence, also called Fibonacci numbers, is obtained by adding the previous two to obtain the next one. Now, what does this sequence of numbers have to do with the golden number? Do the following experiment: take a calculator and divide any one by its immediate previous one. As you progress through the sequence, the quotient will get closer and closer to the golden number.

In mathematical terms, this means that the sequence of numbers created by dividing a Fibonacci number by its immediate preceding tends, or has as its limit, the golden number. That is, this infinite succession of numbers ends, at infinity, in the golden number.


The drone’s pedigree follows a mathematical model

The problem with Fibonacci rabbits is that they are ideal. Is there a more realistic example of this golden succession being found in nature? Yes, for example in the family tree of any drone from a honeycomb. It hatches from the queen’s unfertilized egg, then has a mother, but no father. On the contrary, both the queen (the only one that can lay eggs) and the workers hatch from the egg fertilized by a male. They therefore have a father and a mother. With this in mind, a drone’s family tree looks as follows: it has 1 mother, 2 grandparents (male and female), 3 great-grandparents (two from the grandmother’s family and one from the grandfather’s), 5 great-great-grandparents, 8 great-great-great-grandparents … The drone family tree is a Fibonacci sequence! And not only that. In 1966, Doug Yanega of the University of California Museum of Entomological Research discovered that the relationship between female and male bees in a community is close to the golden number.

The logarithmic spiral of the nautilus shell

Now let’s convert the numbers into squares. Let’s put two of the same, one next to the other, of any size, whose sides we will take as a unit. On top of them, let’s draw another whose side is double the previous ones. On the right, let’s add another one, triple on one side. Below, the one corresponding to 5, and so on, so that each new square has on its side the sum of the two previous squares. If we now draw a quarter of a circumference inside each square (starting with the first one), as in the photograph of the conch at the beginning of the report, we will have a logarithmic spiral that is precisely the one presented by the nautilus shell.

Now take a pencil and draw a line that goes from the center to the outside. Notice two points where this line cuts the shell, with the only condition that the spiral has made a complete turn between them. You will see that the outermost one is 1,618 times further from the center than the innermost one. This means that the shell’s growth factor is the golden number.

The Fibonacci numbers are also found in the number of spirals to the left and to the right that we can count in the seeds of sunflowers and in the cones of pines; in the number of flower petals (3 in the iris; 5 or 8 in some buttercups; daisies and sunflowers usually have 13, 21, 34, 55 or 85 …) and in the number of flowers in the spirals of cauliflower and broccoli. In fact, each of them is a tiny cauliflower itself. If you count the spirals in both directions coming out of these mini-flowers, what number do you get? You can also look for Fibonacci numbers in the banana and in the apple. Even the leaves around the stem follow this order.


The best possible ordering system

Why this taste of nature for the Fibonacci sequence? Leaves, petals and seeds are arranged in the plants following a fixed angle because this is the best packing system even if the plant grows. If we place the golden number of leaves per turn on the stem, we obtain the best packaging so that all of them receive the maximum light without hiding one from the other and, in the case of flowers, the best exposure to attract pollinating insects. . Fibonacci numbers are the best approximation that exists to the golden number. Given all this, it is not surprising that the Parthenon can be framed in a golden rectangle – one in which the ratio of its length to its height comes out as the golden number. The same happens with credit cards. Is there anything more beautiful than a Visa with no spending limit?

The architecture of the Divine Proportion

The Parthenon of Athens – below these lines – is a good example of Greek architectural beauty and, as such, it can be framed within a golden rectangle. Some mathematicians have claimed to see the golden number in the Great Pyramid of Cheops -to the right-. Thus, if the distance from the base of one of the faces of the pyramid to the top vertex is divided by the height of the pyramid, we obtain 1.6. Is it perhaps something intentional? Some think so, but the truth is that there is no basis for thinking about it. The Rhind Papyrus (1650 BC), one of the oldest surviving mathematical works, does not mention the golden number, although it does solve some problems related to the construction of pyramids.

Fibonacci, the rabbit man

Leonardo of Pisa is best known by his nickname, Fibonacci (from Filius Bonacci, son of Bonacci). Born in northern Italy, but educated in North Africa, he spent all his youth traveling the Mediterranean, as his father was the representative of the merchants of the Republic of Pisa.

He was one of the first to introduce the decimal numbering system in Europe. In 1202 he published his Liber abacci (Calculation Book), where he explained how to add, subtract, divide and multiply with this new system. It was here that what is now his famous succession appears. The funny thing is that it was presented as a problem posed for readers to learn to use the decimal system, and not as a consequence of their reflections on arithmetic.

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