Philosophical geometry. Part 4, Final. The golden ratio and the root of five
Oh god, fourth part! It is beyond my power! Calm, I am running out of pills, so this is the last article, and there will be revelations in it. Under the cut description of the process of fitting, concealing, entanglement and manipulation.
In the previous ( 1 , 2 , 3 ) parts we saw how different proportions were used in geometry, ancient art and modern industrial design. We have left undisclosed the theme of the golden section and another root - √5. Let's start the same.
One day, people came across the idea of proportions. The same patterns were constantly encountered in different figures. It was impressive. Then someone thought of measuring a couple of plants, animals and some parts of the body that are usually hidden from strangers. The patterns were there. It was even more impressive.
')
To tolerate there is no more urine left, the most common relationship has been declared sacred. Some saw in them a manifestation of divine intervention. Some are god himself. And since the sacred proportions are so often found, then you can fit everything you want under them, make a symbol out of it and scare the flock.
Hoaxes and registry of the most good intentions are found in history constantly. For example, the census takers of the classic work “The Church History of the People of Angles”, the Trouble of the Honorable, attached pieces to the text so that certain church questions would look more favorable. A 25-28 chapter VI of the book "Notes on the Gallic War," Caesar does not seem to be Caesar.
Also in symbolism. It is necessary that people feel its deep meaning, and the form itself is not so important. Take any picture, in it necessarily something will be found. The more ancient, the better. The oldest is Egypt, we practice it.
Here is a bas-relief map from the tomb of Petosiris, found in 1919.
After sitting for a sufficient time with a ruler and compass, you can find in it a golden section and a bunch of different relationships (in addition to text, wits, you do not need a compass for this).
It looks pretty cool, so there is no reason not to say that the Egyptians knew about the golden section and specifically everyone did it.
To mystify geometry is easy and simple. Now I will show you a couple of tricks. Look under the cat.
So, tricks. Well, firstly, all these proportions are so ancient that all the coolest constructions are well described. Having opened the reference book, we see how the most golden proportions can be said in a very simple way. We take a square, draw right next to it and draw an AC diagonal.
With the center O and radius OA build a circle. We extend the line FE to the point of intersection with circle K. Then we can repeat the focus with the circle for side AB, but it is easier to draw a perpendicular KH from K, extend AB and mark the point of intersection H.
Voila, we have a “golden rectangle”. The ratio of FE to EK is the same as FK to FE. Even more awesome, KF / FB is equal to that too. For brevity, the Greeks called this relationship φ (Phi). It is approximately equal to 1.618
We show a creative beginning and make a couple more passes. Connect the corners of the original square and the resulting rectangle. Draw a circle on the formed points.
Now, to open the veil over the great mystery, we need to find something to fit into this rectangle. Remembering how lucky we were with the epple, let's go the easiest way and make the obvious.
Hooray! We managed to declare iPod 4G a sacred device. It is obvious that now it will be necessary to buy.
Here it is - the benefits of studying textbooks. Geometry exists for so long that it costs nothing to take reference books, yes classical sources, to write books and earn a lot of profit from it.
You look, the book is being filmed with some kind of actor of the type who has gone nuts from the script, and the profit will be an order of magnitude even greater. You can compost the reader's brain, and you can compost the reader's brain with how you can compost the reader's brain - also, by the way, profitable (well, you know what I, my well-read commentators,). On a pinch, you can hit the contemporaries and earn some advantages by unraveling the secrets of the Arabic mosaic, linking it ... well, let's say with the bee.
In fact, in art, of course, it’s all about ordinary modular grids and guide lines, which artists and designers use to compose compositions. The idea is so old that even the most ancient of the ancient Egyptians used it.
Compositions are taught in art schools, and those who teach, too, someone once taught. So no mystery. Is that only the very first teacher something there invented. The rest is enough to take some kind of grid and arrange elements on it.
But the one-and-a-half side ratio and square modules are not interesting. There is no secret. It is better to take something more "sacred". Let's say another known relation is √5. The Zune player is just that.
Let's look in the directory and see what patterns can emerge.
Put them on the player. Hm Nothing like yet. Let's try to sort through all the options.
Ummm ... Somehow rotten. What is interesting is that the touch near the central button. Then we will play with 4itami. Drag the ruler, measure the size of the elements and try to find dividers.
Explains sarcoidosis and aggressive behavior ... damn it seems to be writing in the wrong window. In general, here again the dull squares that you will confuse very few people. Ruined such a good start.
We'll have to go back to the iPod. iPod Nano 2G also has an aspect ratio of √5 / 1. Let's look again at the diagram from the reference book and think about how to stick it to something. Yeah. There is a side √5 that can be crammed into the previous build.
From points A, O and C we draw arcs with radius AB (since this is 1x for us, and AC is √5x). From the same points we draw perpendiculars to AC before intersecting with arcs. Connect new intersections and now our iPod is almost ready.
Akhaylay, mahalay, abra-kadabra!
So, what have we learned?
1. If you want to confuse people with your head, you should always use the mystic numbers √2, √3, √5 and φ. With them there will always be the most "mysterious" coincidences. There are many more interesting things like Fibonacci sequences, spirals, gnome magnifications and all sorts of tricky divisions. But the simpler - the easier it is to confuse everyone, right?
2. Apple absolutely brainwashed people. After all, they have a whole set of young occultists:
3. From a simple square, you can construct a whole brain explosion, religion, algebra, the dichotomy of good and evil.
Everyone will see what he wants here, including boobs and Mickey Mouse (you can still see the star of David, if you attend to it).
Such a property of man is to pervert simple ideas. But that's where it all began:
In the previous ( 1 , 2 , 3 ) parts we saw how different proportions were used in geometry, ancient art and modern industrial design. We have left undisclosed the theme of the golden section and another root - √5. Let's start the same.
One day, people came across the idea of proportions. The same patterns were constantly encountered in different figures. It was impressive. Then someone thought of measuring a couple of plants, animals and some parts of the body that are usually hidden from strangers. The patterns were there. It was even more impressive.
')
To tolerate there is no more urine left, the most common relationship has been declared sacred. Some saw in them a manifestation of divine intervention. Some are god himself. And since the sacred proportions are so often found, then you can fit everything you want under them, make a symbol out of it and scare the flock.
Hoaxes and registry of the most good intentions are found in history constantly. For example, the census takers of the classic work “The Church History of the People of Angles”, the Trouble of the Honorable, attached pieces to the text so that certain church questions would look more favorable. A 25-28 chapter VI of the book "Notes on the Gallic War," Caesar does not seem to be Caesar.
Also in symbolism. It is necessary that people feel its deep meaning, and the form itself is not so important. Take any picture, in it necessarily something will be found. The more ancient, the better. The oldest is Egypt, we practice it.
Here is a bas-relief map from the tomb of Petosiris, found in 1919.
After sitting for a sufficient time with a ruler and compass, you can find in it a golden section and a bunch of different relationships (in addition to text, wits, you do not need a compass for this).
It looks pretty cool, so there is no reason not to say that the Egyptians knew about the golden section and specifically everyone did it.
To mystify geometry is easy and simple. Now I will show you a couple of tricks. Look under the cat.
So, tricks. Well, firstly, all these proportions are so ancient that all the coolest constructions are well described. Having opened the reference book, we see how the most golden proportions can be said in a very simple way. We take a square, draw right next to it and draw an AC diagonal.
With the center O and radius OA build a circle. We extend the line FE to the point of intersection with circle K. Then we can repeat the focus with the circle for side AB, but it is easier to draw a perpendicular KH from K, extend AB and mark the point of intersection H.
Voila, we have a “golden rectangle”. The ratio of FE to EK is the same as FK to FE. Even more awesome, KF / FB is equal to that too. For brevity, the Greeks called this relationship φ (Phi). It is approximately equal to 1.618
We show a creative beginning and make a couple more passes. Connect the corners of the original square and the resulting rectangle. Draw a circle on the formed points.
Now, to open the veil over the great mystery, we need to find something to fit into this rectangle. Remembering how lucky we were with the epple, let's go the easiest way and make the obvious.
Hooray! We managed to declare iPod 4G a sacred device. It is obvious that now it will be necessary to buy.
Here it is - the benefits of studying textbooks. Geometry exists for so long that it costs nothing to take reference books, yes classical sources, to write books and earn a lot of profit from it.
You look, the book is being filmed with some kind of actor of the type who has gone nuts from the script, and the profit will be an order of magnitude even greater. You can compost the reader's brain, and you can compost the reader's brain with how you can compost the reader's brain - also, by the way, profitable (well, you know what I, my well-read commentators,). On a pinch, you can hit the contemporaries and earn some advantages by unraveling the secrets of the Arabic mosaic, linking it ... well, let's say with the bee.
In fact, in art, of course, it’s all about ordinary modular grids and guide lines, which artists and designers use to compose compositions. The idea is so old that even the most ancient of the ancient Egyptians used it.
Compositions are taught in art schools, and those who teach, too, someone once taught. So no mystery. Is that only the very first teacher something there invented. The rest is enough to take some kind of grid and arrange elements on it.
But the one-and-a-half side ratio and square modules are not interesting. There is no secret. It is better to take something more "sacred". Let's say another known relation is √5. The Zune player is just that.
Let's look in the directory and see what patterns can emerge.
Put them on the player. Hm Nothing like yet. Let's try to sort through all the options.
Ummm ... Somehow rotten. What is interesting is that the touch near the central button. Then we will play with 4itami. Drag the ruler, measure the size of the elements and try to find dividers.
Explains sarcoidosis and aggressive behavior ... damn it seems to be writing in the wrong window. In general, here again the dull squares that you will confuse very few people. Ruined such a good start.
We'll have to go back to the iPod. iPod Nano 2G also has an aspect ratio of √5 / 1. Let's look again at the diagram from the reference book and think about how to stick it to something. Yeah. There is a side √5 that can be crammed into the previous build.
From points A, O and C we draw arcs with radius AB (since this is 1x for us, and AC is √5x). From the same points we draw perpendiculars to AC before intersecting with arcs. Connect new intersections and now our iPod is almost ready.
Akhaylay, mahalay, abra-kadabra!
So, what have we learned?
1. If you want to confuse people with your head, you should always use the mystic numbers √2, √3, √5 and φ. With them there will always be the most "mysterious" coincidences. There are many more interesting things like Fibonacci sequences, spirals, gnome magnifications and all sorts of tricky divisions. But the simpler - the easier it is to confuse everyone, right?
2. Apple absolutely brainwashed people. After all, they have a whole set of young occultists:
3. From a simple square, you can construct a whole brain explosion, religion, algebra, the dichotomy of good and evil.
Everyone will see what he wants here, including boobs and Mickey Mouse (you can still see the star of David, if you attend to it).
Such a property of man is to pervert simple ideas. But that's where it all began:
Source: https://habr.com/ru/post/57806/
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