𝕯𝖎𝖛𝖎𝖓𝖊 𝖕𝖗𝖔𝖕𝖔𝖗𝖙𝖎𝖔𝖓.
Divine proportion.
“Philosophy is written in that enormous book that is continually open before our eyes (I am talking about the universe), but which cannot be understood if we do not first learn to understand the language and to know the characters with which it has been written. It is written in mathematical language, and the characters are triangles, circles and other geometric figures without which it is humanly impossible to understand a word; without them it is vainly wandered through a dark labyrinth.”
Isaac Newton, that because of those coincidences in history. (as referred to by the author in other publications), was born the same year that Galileo died; not only did he come to agree with Galileo, but he also demonstrated, based exclusively on mathematics, the laws that governed the movement of all the stars of the Solar System. From then until today, countless natural phenomena have been explained by mathematics.
It is no coincidence that a crew member of Apollo 11, when the ship was leaving Earth's orbit, told the world that "now it is Newton who is leading us".
Since then, not only in physics, but in all sciences, in biology, chemistry, medicine and economics; mathematics has proved to be an essential tool for understanding and explaining the universe around us.
But not only to understand and explain it, but also to make it more comfortable. From simply turning on a light, calling a friend on the phone, heating a meal on the Microwave, to taking pictures of the surface of Saturn thousands of miles away, all of this would be unthinkable without the support that mathematics has given to technicians, engineers, and physicists in the most varied fields.
See the world with mathematical eyes.
In previous publications in this series, the author spoke of the gold, his power and influence. To the surprise of many, there is also some of this power in mathematics; the golden power. Let's see.
This bill, it's rectangular in shape. The rectangle is one of the geometric shapes that most commonly appear in our environment. Many objects we see and touch every day have this rectangle shape.
Most sports are played on fields that are rectangular, many of their areas are themselves rectangles.
Of all the rectangles, there are some very particular ones, which are especially harmonious to such an extent that the first credit cards had the proportion of those rectangles. Here we have two blue rectangles, those rectangles are harmonious, because they have the proportions between their determined sides and a strange property; let's observe:
For this demonstration, the author built a pair of blue rectangles, previously calculated, and cut out one of them, the largest possible square.
The new rectangle was compared with the original one, and we noticed that both have the same shape; in the mathematical sciences they are said to be similar; that is, they have the proportional sides.
This property does not occur with all rectangles. To demonstrate this, the author took a rectangle with other measurements, such as the following green rectangles:
Proceed as with the blue rectangles, cut the largest possible square from one of them:
Indeed, these two rectangles are not at all similar, their dimensions are not proportional.
Rectangles - like the blue one - which if they comply with this property are called "golden rectangles". If we divide the length of the long side (a) of one of these special rectangles by the length of the short side (b), we get a very special number, so special that it is called the "gold number".
Why is this rectangle so special? In fact, the Greeks already knew him, he is present in many of his artistic manifestations, especially in his temples and sculptures.
How did they find out? I'm sure it was trying to divide a segment in a very special way. They wanted the ratio between the largest part of the segment and the smallest part to be the same as the ratio between the total length of the segment and the length of the largest part. Let's watch:
For those who like equations, the problem comes down to a simple equation: let's assign the value of 1 to the length of the smallest segment, and call the length of the largest segment X. The relationship established by the Greeks would be:
If we transform this equation, we would get this simpler one:
That is to say,
And finally we get a simple second-degree equation:
whose solution is:
This expression shows the close link between the number of gold and the square root of five.
The fact that the Greeks and later artists of all ages have adopted this proportion as a model of harmony and beauty would be enough reason to treat this strange number with respect. Artists and mathematicians such as Lucas Pacioli, Leonardo Da Vinci o Alberto Durero, have designated this number with such striking names as golden section, golden reason or divine proportion.
Since the Renaissance, great painters have used dimensions related to golden reason in their masterpieces.
But without a doubt, the most surprising case is the Pyramid of Keops. Many speculated, and sometimes with very little foundation, about the concise numerical relationships found in the proportions of the pyramids of Egypt. In the case of the pyramid of Keops, built around 2600 B.C., using more than two million stones of about twenty tons each; these relationships do not seem to be the result of chance, and constitute the first and perhaps the most spectacular appearance in architecture of the gold number.
Heródoto, the famous Greek historian of the 5th century B.C. He tells that the Egyptian priests had shown him the fact that the dimensions of the pyramid were such that the square of the total height was exactly equal to the area of one of the faces. This fact, attributed to the meticulousness of the Egyptian architect, is striking; as well as the geometric characteristics that can be deduced from this, we can admire with astonishment that the Egyptians, 3000 years ago, already knew and applied the golden reason.
In fact, the gold number appears, not once, but up to three times in numerical relationships between different elements of the pyramid. Thus, the ratio between the height of one face and half of the side of the base is 1.618..., that is, the number of gold. But the surprises do not end here, the quotient between the total area and the lateral area of the pyramid is also the number of gold. And to top it off, the quotient between the lateral area and the base area is still the golden number.
The golden number will bring us many more surprises. Before we discover some of them, let's look at a simple mechanism to build golden rectangles.
From any square, a golden rectangle can be obtained by marking the midpoint of one of the sides of the square and drawing an arc of circumference whose radius is the distance from this midpoint to the upper vertex until it finds or intercepts the extension of the lower vertex; this is the first vertex of our golden rectangle, the second is obtained by drawing parallels to the sides of the square.
From now on, new golden rectangles are easier to obtain. Just draw a square on the longest side of the previous rectangle, and you get your second golden rectangle. If on this scheme we draw a new square along the lake side of the obtained rectangle, we will have drawn our third golden rectangle and thus infinitely...
These rectangles have an interesting property, if we join by means of arcs of circumference, the consecutive vertices of the squares, we will obtain a very special curve that is denominated "Durero's Spiral", for many, this is a very familiar curve, a spiral that reminds us very much of the shells of some snails and the horns of some ruminants. This spiral was discovered by the Renaissance painter Alberto Durero, and since then, many scientists and mathematicians have associated this spiral with the growth of mollusk shells.
In nature they surprise us from time to time, phenomena of the most unsuspected in which this curve appears, from a galaxy to a hurricane it seems that they feel attracted by the beauty of this curve.
In the plant world, the examples also puzzle us. This explains why many scientists have argued with great enthusiasm that many animals and plants whose growth occurs by maintaining their shape and retaining the proportions between their parts are directly related to the number of gold.
Nature presents with suspicious frequency, forms related to the pentagons. These phenomena will be related to our friend the golden number?
To answer this question, we have to go back to a symbol used by the Pythagorean School. Pythagoras and his school, constituted a secret society of religious-philosophical character and their followers identified themselves using this symbol. This symbol is called a pentagram or starry pentagon. In this starry pentagon we detail segments of different lengths.
All these segments are related to each other with the gold number. Thus, the quotient between the first and second segments is the same as the quotient between the second and third segments and the third and fourth segments. Is this the reason why plants and animals prefer pentagonal forms?
There is also a golden angle. We can divide the circumference into two angles in such a way that the ratio between the largest and the smallest angle is exactly the number of gold. The smaller of these angles measures approximately 137°30'.
Expression they can do at home.
Try to measure the length and width of an egg from any bird by using a plastic tape measure or an appropriate square; if you have a vernier, much better and more accurate. Take both measurements and divide the largest length by the smallest length of each egg and you will notice that possibly neither of the quotients matches the gold number, but what you can see is that, with any egg, we divide the maximum height by its maximum width, we will get a number that will always be between the root of the gold number that measures 1.27..., and the number of gold that is 1.61.....
REFERENCES
Biographical information.
D´Arcy Thompson. Sobre el crecimiento y la forma. Editorial Hermann Blume. 2017.
CASTELNUOVO Emma. Matemática nella Realitá. Editorial Boringhiere. 2016.
LOUELMO María Jesús. Geometría en la naturaleza. Editorial M.E.C. 2016.
MATILA G., Ghyka. El número de oro y estética de las proporciones en la naturaleza y el arte. Editorial Poseidon, 2017.
Electronics
Alberto Durero. Biographies and Lives.
Galileo Galilei. Biographies and Lives.
Heródoto, el historiador viajero. National Geographic.
Isaac Newton. Biographies and Lives.
La espiral de Durero. Nicolas Project.
Número áureo: belleza matemática. ABC.es
Audiovisual material consulted
The Divine Proportion or Golden Proportion. YouTube.
The Divine Proportion | Gold Number (Phi) | FIBONACCI Sequence. YouTube.
The great pyramid of Keops. ABC Ciencia.
Luca Pacioli. The divine proportion. YouTube.
All the material is original and can be reviewed in my profile of Google+
Tools used in image editing, animation and editing:
- Paint Windows 10.
- PhotoScape X.
- Gif Windows 10 (Tienda Microsoft).
- Photoshop.
- PowerPonit Office 2016.
- Adobe XD.
- Lightroom Classic.
- After Effects.
- Crazy Talk Animator.
- Lunapic.
- MessLetters.
But this does not end here, in future publications I will provide you with interesting and entertaining material, which without being specialists in chemistry, biology, paleontology, physics or mathematics, will entertain you and teach you the mysteries and curiosities of our universe, our history...
But this does not end here, in future publications I will provide you with interesting and entertaining material, which without being specialists in chemistry, biology, paleontology, physics or mathematics, will entertain you and teach you the mysteries and curiosities of our universe, our history...
Hey, your posts in amazing. I really enjoyed it.
Could you tell me how you set the format of the title? I love it 😊
Thank you for your appreciation. For different fonts type what you want, then copy and paste the font of your choice into the Messletters application. A few months ago I published a post showing you how to do it... Thank you for your visit and support.