When we look at a beautiful landscape, we are covered all around. Then we pay attention to details. A babbling river or a majestic tree. We see a green field. We notice how the wind hugs him gently and the juror sways the grass from side to side. We can feel the aroma of nature and hear the birds singing... Everything is harmonious, everything is interconnected and gives a sense of peace, a sense of beauty. Perception goes in stages in slightly smaller shares. Where will you sit on the bench: on the edge, in the middle, or anywhere? Most will answer that a little further from the middle. An approximate number in bench proportion from your body to the edge would be 1.62. So it is in the cinema, in the library - everywhere. We instinctively create harmony and beauty, which I call the “Golden Section” all over the world.
The Golden Ratio in Mathematics
Have you ever wondered if it is possible to define the measure of beauty? It turns out that mathematically it is possible. Simple arithmetic gives the concept of absolute harmony, which is displayed in impeccable beauty, thanks to the principle of the Golden Section. The architectural structures of other Egypt and Babylon were the first to conform to this principle. But Pythagoras was the first to formulate the principle. In mathematics, this division of the segment is slightly more than half, or rather 1.628. This ratio is represented as φ =0.618= 5/8. A small segment \u003d 0.382 \u003d 3/8, and the entire segment is taken as one.
A:B=B:C and C:B=B:A 
Great writers, architects, sculptors, musicians, people of art, and Christians who draw pictograms (five-pointed stars, etc.) with its elements in temples, escaping evil spirits, and people studying exact sciences, problem solving cybernetics.
Golden section in nature and phenomena.
Everything on earth taking shape grows up, sideways or in a spiral. Archimedes paid close attention to the latter, having drawn up an equation. A cone, a shell, a pineapple, a sunflower, a hurricane, a web, a DNA molecule, an egg, a dragonfly, a lizard are arranged along the Fibonacci series ... 
Ticirius proved that our entire Universe, space, galactic space, everything is planned based on the Golden Principle. Absolutely in everything living and not living you can read the highest beauty. 
The golden ratio in man.
The bones are thought out by nature, also according to the proportion 5 / 8. This excludes people's reservations about “big bones”. Most body parts in ratios apply to the equation. If all parts of the body obey the Golden formula, then the external data will be very attractive and ideally folded. 
Segment from the shoulders to the top of the head and its size = 1:1.618
Segment from the navel to the top of the head and from the shoulders to the top of the head = 1:1.618
Segment from the navel to the knees and from them to the feet = 1: 1.618
The segment from the chin to the extreme point of the upper lip and from it to the nose \u003d 1: 1.618 
All facial distances give a general idea of the ideal proportions that attract the eye.
Fingers , palm , also obey the law . It should also be noted that the segment of the spread arms with the torso is equal to the height of a person. Why , all organs , blood , molecules correspond to the Golden formula . True harmony inside and outside of our space.
Parameters from the physical side of the surrounding factors.
Sound volume. Highest point sound that causes discomfort and pain in the auricle = 130 decibels. This number can be divided by the proportion 1.618, then it turns out that the sound of a human scream will be = 80 decibels.
Using the same method, moving on, we get 50 decibels, which is typical for the normal volume of human speech. And the last sound that we get thanks to the formula is the pleasant sound of a whisper = 2.618.
According to this principle, it is possible to determine the optimal-comfortable, minimum and maximum number of temperature, pressure, humidity. The simple arithmetic of harmony is embedded in our entire environment.
The golden ratio in art.
In architecture, the most famous buildings and structures: the Egyptian pyramids, the Mayan pyramids in Mexico, Notre Dame de Paris, the Greek Parthenon, the Petrovsky Palace, and others. 
In music: Arensky, Beethoven, Havan, Mozart, Chopin, Schubert, and others.
In painting: almost all the paintings of famous artists are painted according to the section: the versatile Leonardo da Vinci and the inimitable Michelangelo, Shishkin and Surikov are so close in writing, the ideal of the purest art is the Spaniard Raphael, and the Italian Botticelli, who gave the ideal of female beauty, and many, many others.
In poetry: the ordered speech of Alexander Sergeevich Pushkin, especially “Eugene Onegin” and the poem “Shoemaker”, the poetry of the wonderful Shota Rustaveli and Lermontov, and many other great masters of the word.
In sculpture: a statue of Apollo Belvedere, Olympian Zeus, beautiful Athena and graceful Nefertiti, and other sculptures and statues.
Photography uses the “rule of thirds”. The principle is this: the composition is divided into 3 equal parts vertically and horizontally, the key points are located either on the intersection lines (horizon) or at the intersection points (object). Thus the proportions are 3/8 and 5/8. 
There are many tricks in according to the Golden Ratio that should be analyzed in detail. I will describe them in detail in the next one. 
golden ratio is a simple principle that will help make design visually pleasing. In this article, we will explain in detail how and why to use it.
A common mathematical proportion in nature called the Golden Ratio, or the Golden Mean, is based on the Fibonacci Sequence (which you most likely heard about in school, or read in Dan Brown's The Da Vinci Code), and implies an aspect ratio of 1 :1.61.
Such a ratio is often found in our lives (shells, pineapples, flowers, etc.) and therefore is perceived by a person as something natural, pleasing to the eye.
→ The golden ratio is the relationship between two numbers in the Fibonacci sequence 
→ Plotting this sequence to scale gives spirals that can be seen in nature.
It is believed that the Golden Ratio has been used by mankind in art and design for more than 4,000 years, and possibly even more, according to scientists who claim that the ancient Egyptians used this principle in the construction of the pyramids.
Famous examples
As we have already said, the Golden Ratio can be seen throughout the history of art and architecture. Here are some examples that only confirm the validity of using this principle:
Architecture: Parthenon
In ancient Greek architecture, the Golden Ratio was used to calculate the ideal proportion between the height and width of a building, the size of a portico, and even the distance between columns. Later, this principle was inherited by neoclassical architecture.
Art: The Last Supper
For artists, composition is the foundation. Leonardo da Vinci, like many other artists, was guided by the principle of the Golden Ratio: in the Last Supper, for example, the figures of the disciples are located in the lower two thirds (the larger of the two parts of the Golden Ratio), and Jesus is placed strictly in the center between two rectangles.
Web design: Twitter redesign in 2010
Twitter creative director Doug Bowman posted a screenshot on his Flickr account explaining the use of the golden ratio for the 2010 redesign. “Anyone who is interested in #NewTwitter proportions - know that everything is done for a reason,” he said.
Apple iCloud
The iCloud service icon is also not a random sketch at all. As explained by Takamasa Matsumoto in his blog (original Japanese version) everything is based on the mathematics of the Golden Ratio, the anatomy of which can be seen in the figure on the right.
How to build the Golden Ratio?
The construction is quite simple, and begins with the main square:
Draw a square. This will form the length of the "short side" of the rectangle.
Divide the square in half with a vertical line so that you get two rectangles.
In one rectangle, draw a line by joining opposite corners.
Expand this line horizontally as shown in the figure.
Create another rectangle using the horizontal line you drew in the previous steps as a base. Ready!
"Golden" tools
If drawing and measuring is not your favorite pastime, leave all the “dirty work” to tools that are designed specifically for this. With the help of the 4 editors below, you can easily find the Golden Ratio!
The GoldenRATIO app helps you design websites, interfaces and layouts according to the Golden Ratio. It's available on Mac App Store for $2.99, and has a built-in calculator with visual feedback, and a handy Favorites feature that stores settings for recurring tasks. Compatible with Adobe Photoshop.
This calculator will help you create the perfect typography for your site in accordance with the principles of the Golden Ratio. Just enter the font size, content width in the field on the site, and click "Set my type"!
It's simple and free app for Mac and PC. Just enter a number and it will calculate the proportion for it according to the golden section rule.
A handy program that will save you from the need for calculations and drawing grids. Finding the perfect proportions is easy with her! Works with all graphic editors, including Photoshop. Despite the fact that the tool is paid - $ 49, it is possible to test the trial version for 30 days.
What do you have in common Egyptian pyramids, paintings of "Mona Lisa" by Leonardo da Vinci and the logos of Twitter and Pepsi?
Let's not delay with the answer - they are all created using the golden section rule. The golden ratio is the ratio of two quantities a and b, which are not equal to each other. This proportion is often found in nature, and the golden ratio is also actively used in fine arts and design - compositions created using the "divine proportion" are well balanced and, as they say, pleasing to the eye. But what exactly is the golden ratio and can it be used in modern disciplines, for example, in web design? Let's figure it out.
A LITTLE MATH
Suppose we have a certain segment AB, divided in two by point C. The ratio of the lengths of the segments: AC / BC = BC / AB. That is, the segment is divided into unequal parts in such a way that the larger part of the segment is the same share in the whole, undivided segment, which the smaller segment is in the larger one.
This unequal division is called the golden ratio. The golden ratio is denoted by the symbol φ. The value of φ is 1.618 or 1.62. In general, speaking quite simply, this is a division of a segment or any other value in relation to 62% and 38%.
The "divine proportion" has been known to people since ancient times, this rule was used in the construction of the Egyptian pyramids and the Parthenon, the golden ratio can be found in the paintings of the Sistine Chapel and in the paintings of Van Gogh. The golden ratio is widely used today - examples that are constantly before our eyes are the Twitter and Pepsi logos.
The human brain is designed in such a way that it considers beautiful images or objects in which an unequal ratio of parts can be found. When we say about someone that "he is proportionately complex," we, without knowing it, are referring to the golden ratio.
The golden ratio can be applied to various geometric shapes. If we take a square and multiply one of its sides by 1.618, we get a rectangle.
Now, if we superimpose a square on this rectangle, we can see the golden ratio line:
If we continue to use this proportion and break the rectangle into smaller parts, we get this picture:
It is not yet clear where this fragmentation will lead us. geometric shapes. A little more and everything will become clear. If in each of the squares of the scheme we draw a smooth line equal to a quarter of a circle, then we will get the Golden Spiral.
This is an unusual spiral. It is also sometimes called the Fibonacci spiral, after the scientist who studied the sequence in which each number is earlier than the sum of the previous two. The bottom line is that this mathematical relationship, visually perceived by us as a spiral, is found literally everywhere - sunflowers, sea shells, spiral galaxies and typhoons - everywhere there is a golden spiral.
HOW CAN YOU USE THE GOLDEN RATIO IN DESIGN?
So, the theoretical part is over, let's move on to practice. Can the golden ratio be used in design? Yes, you can. For example, in web design. Given this rule, you can get the correct ratio of the compositional elements of the layout. As a result, all parts of the design, down to the smallest ones, will be harmoniously combined with each other.
If we take a typical layout with a width of 960 pixels and apply the golden section rule to it, then we get this picture. The ratio between the parts is already known 1:1.618. As a result, we have a two-column layout, with a harmonious combination of two elements.
Sites with two columns are very common and this is far from accidental. Take, for example, the National Geographic website. Two columns, golden section rule. Good design, orderly, balanced and respectful of visual hierarchy.
One more example. Design studio Moodley developed the brand identity for the Bregenz Performing Arts Festival. When the designers were working on the poster of the event, they clearly used the golden ratio rule in order to correctly determine the size and location of all elements and, as a result, get the perfect composition.
Lemon Graphic, which created the visual identity for Terkaya Wealth Management, also used a 1:1.618 ratio and a golden spiral. Three design elements business card fit perfectly into the scheme, as a result of which all parts are very well combined with each other
And here is another interesting use of the golden spiral. Before us is the National Geographic website again. Taking a closer look at the design, you can see that there is another NG logo on the page, only smaller, which is located closer to the center of the spiral.
Of course, this is not accidental - the designers knew perfectly well what they were doing. This is a great place to duplicate the logo as our eye naturally moves towards the center of the composition when looking at the site. This is how the subconscious works and this must be taken into account when working on design.
GOLDEN CIRCLE
"Divine proportion" can be applied to any geometric shapes, including circles. If you inscribe a circle in squares, the ratio between which is 1: 1.618, then we get golden circles.
Here is the Pepsi logo. Everything is clear without words. And the ratio, and how the smooth arc of the white logo element was obtained.
With the Twitter logo, things are a little more complicated, but here you can see that its design is based on the use of golden circles. It does not follow the rule of "divine proportion" a little, but for the most part all its elements fit into the scheme.
CONCLUSION
As you can see, despite the fact that the rule of the golden ratio has been known since time immemorial, it has not become outdated at all. Hence, it can be used in design. You don't have to go out of your way to fit into a schema—the design discipline is imprecise. But if you need to achieve a harmonious combination of elements, then trying to apply the principles of the golden ratio will not hurt.
A person distinguishes objects around him by shape. Interest in the form of an object may be dictated by vital necessity, or it may be caused by the beauty of the form. The form, which is based on a combination of symmetry and the golden ratio, contributes to the best visual perception and the emergence of a sense of beauty and harmony. The whole always consists of parts, parts of different sizes are in a certain relationship to each other and to the whole. The principle of the golden section is the highest manifestation of the structural and functional perfection of the whole and its parts in art, science, technology and nature.
Golden Ratio - Harmonic Proportion
In mathematics proportion(lat. proportio) call the equality of two relations:
a : b = c : d.
Line segment AB can be divided into two parts in the following ways:
- into two equal parts AB : AC = AB : BC;
- into two unequal parts in any ratio (such parts do not form proportions);
- so when AB : AC = AC : BC.
The latter is the golden division or division of the segment in the extreme and average ratio.
The golden section is such a proportional division of a segment into unequal parts, in which the entire segment relates to the larger part in the same way as the larger part itself relates to the smaller one; or in other words, the smaller segment is related to the larger one as the larger one is to everything:
a : b = b : c
or
c : b = b : a.
Rice. 1. geometric image golden ratio
Practical acquaintance with the golden ratio begins with dividing a straight line segment in the golden ratio using a compass and ruler.
Rice. 2.BC = 1/2 AB; CD = BC
From a point B a perpendicular is restored equal to half AB. Received point C connected by a line to a dot A. A segment is drawn on the resulting line BC, ending with a dot D. Line segment AD transferred to a straight line AB. The resulting point E divides the segment AB in the golden ratio.
Segments of the golden ratio are expressed by an infinite irrational fraction AE= 0.618... if AB take as a unit BE\u003d 0.382 ... For practical purposes, approximate values \u200b\u200bof 0.62 and 0.38 are often used. If the segment AB taken as 100 parts, then the largest part of the segment is 62, and the smaller is 38 parts.
The properties of the golden section are described by the equation:
x 2 – x – 1 = 0.
Solution to this equation:
The properties of the golden section created a romantic aura of mystery and almost mystical worship around this number.
The second golden ratio
The Bulgarian magazine "Fatherland" (No. 10, 1983) published an article by Tsvetan Tsekov-Karandash "On the second golden section", which follows from the main section and gives a different ratio of 44: 56.
Such a proportion is found in architecture, and also takes place in the construction of compositions of images of an elongated horizontal format.
Rice. 3.
The division is carried out as follows. Line segment AB is divided according to the golden ratio. From a point C the perpendicular is restored CD. Radius AB there is a point D, which is connected by a line to a point A. Right angle ACD is divided in half. From a point C a line is drawn until it intersects with a line AD. Dot E divides the segment AD in relation to 56:44.
Rice. 4.
The figure shows the position of the line of the second golden section. It is located in the middle between the line of the golden section and middle line rectangle.
Golden Triangle
To find segments of the golden ratio of the ascending and descending series, you can use pentagram.
Rice. 5. Building regular pentagon and pentagrams
To build a pentagram, you need to build a regular pentagon. The method of its construction was developed by the German painter and graphic artist Albrecht Dürer (1471...1528). Let O- the center of the circle A is a point on the circle and E- the middle of the segment OA. Perpendicular to Radius OA, restored at the point O, intersects the circle at a point D. Using a compass, set aside a segment on the diameter CE = ED. The length of a side of a regular pentagon inscribed in a circle is DC. Putting segments on the circle DC and get five points to draw a regular pentagon. We connect the corners of the pentagon through one diagonal and get a pentagram. All diagonals of the pentagon divide each other into segments connected by the golden ratio.
Each end of the pentagonal star is a golden triangle. Its sides form an angle of 36 ° at the apex, and the base laid on the side divides it in proportion to the golden section.
Rice. 6. Construction of the golden triangle
We draw a straight line AB. from point A lay a segment on it three times O arbitrary value, through the resulting point P draw a perpendicular to the line AB, on the perpendicular to the right and left of the point P set aside segments O. Received points d And d 1 connect with straight lines to a point A. Line segment dd 1 set aside on the line Ad 1 , getting a point C. She split the line Ad 1 in proportion to the golden ratio. lines Ad 1 and dd 1 is used to build a "golden" rectangle.
History of the golden ratio
It is generally accepted that the concept of the golden division was introduced into scientific use by Pythagoras, an ancient Greek philosopher and mathematician (VI century BC). There is an assumption that Pythagoras borrowed his knowledge of the golden division from the Egyptians and Babylonians. Indeed, the proportions of the pyramid of Cheops, temples, bas-reliefs, household items and decorations from the tomb indicate that the Egyptian craftsmen used the ratios of the golden division when creating them. The French architect Le Corbusier found that in the relief from the temple of Pharaoh Seti I in Abydos and in the relief depicting Pharaoh Ramses, the proportions of the figures correspond to the values of the golden division. The architect Khesira, depicted on a relief of a wooden board from the tomb of his name, holds measuring instruments in his hands, in which the proportions of the golden division are fixed.
The Greeks were skilled geometers. Even arithmetic was taught to their children with the help of geometric figures. The square of Pythagoras and the diagonal of this square were the basis for constructing dynamic rectangles.
Rice. 7. Dynamic Rectangles
Plato (427...347 BC) also knew about the golden division. His dialogue "Timaeus" is devoted to the mathematical and aesthetic views of the school of Pythagoras and, in particular, to the issues of the golden division.
In the facade of the ancient Greek temple of the Parthenon there are golden proportions. During its excavations, compasses were found, which were used by architects and sculptors of the ancient world. The Pompeian compass (Museum in Naples) also contains the proportions of the golden division.
Rice. 8.
In the ancient literature that has come down to us, the golden division is first mentioned in Euclid's Elements. In the 2nd book of the "Beginnings" the geometric construction of the golden division is given. After Euclid, Hypsicles (2nd century BC), Pappus (3rd century AD) and others studied the golden division. medieval Europe they got acquainted with the golden division from the Arabic translations of the "Beginnings" of Euclid. The translator J. Campano from Navarre (3rd century) commented on the translation. The secrets of the golden division were jealously guarded, kept in strict secrecy. They were known only to the initiates.
During the Renaissance, interest in the golden division among scientists and artists increased in connection with its use both in geometry and in art, especially in architecture Leonardo da Vinci, an artist and scientist, saw that Italian artists had great empirical experience, but little knowledge . He conceived and began to write a book on geometry, but at that time a book by the monk Luca Pacioli appeared, and Leonardo abandoned his idea. According to contemporaries and historians of science, Luca Pacioli was a real luminary, the greatest mathematician in Italy between Fibonacci and Galileo. Luca Pacioli was a student of the painter Piero della Francesca, who wrote two books, one of which was called On Perspective in Painting. He is considered the creator of descriptive geometry.
Luca Pacioli was well aware of the importance of science for art. In 1496, at the invitation of Duke Moreau, he came to Milan, where he lectured on mathematics. Leonardo da Vinci also worked at the Moro court in Milan at that time. In 1509, Luca Pacioli's Divine Proportion was published in Venice, with brilliantly executed illustrations, which is why they are believed to have been made by Leonardo da Vinci. The book was an enthusiastic hymn to the golden ratio. Among the many advantages of the golden ratio, the monk Luca Pacioli did not fail to name its “divine essence” as an expression of the Divine Trinity - God the Father, God the Son and God the Holy Spirit (it was understood that the small segment is the personification of God the Son, the larger segment is God the Father, and the entire segment - God the Holy Spirit).
E-books:
- Mario Livio.
Modern web design includes 2 features that must be clearly observed: aesthetics and the right scope. If you follow these concepts, web design can be considered successful.
As for aesthetics, here we mean that when drawing this or that image of an object, we use many different manipulations: creating a grid, layout, using typographic techniques in order to get a good structure of the object. It is important to maintain a sense of harmony, order and visual balance in any graphic processing. The Golden Ratio and the Rule of Three will help us with this.
You have probably heard of these terms before. And, perhaps, you have an idea in which specific projects they can be used. The "Golden Ratio" and the "Rule of Three" are used to change the image and present it in at its best than it actually is. Such technologies help to improve even the most primitive picture.
Let's take a closer look at these features and find out in which areas of web design they can be applied.
What is the "Golden Ratio" and how did it appear?
This term may not be clear at first glance. Why "Golden"? Why use this technology? To date, it still remains a mystery who invented the "Golden Section", where this name came from. However, it is known that the technology has been used for 2400 years. It is also worth noting that the golden ratio is used in various branches of science: in astronomy, mathematics, architecture, music, painting and many others.
The golden ratio is formed from a simple mathematical equation that shows a ratio. In its simplest mathematical form, this relationship looks like this:
As you can see, this is a unique equation that separates the relationship between two line sizes and proportions. In decimal, b divided by a equals 1.618033... if a>b. In the example below, let's say that b is 5. Then the equation will look like this:
You may have heard of the Fibonacci sequence before. How does it actually work? For example, there is a series of numbers in which any given number created by adding the previous two. Starting from 0, the sequence is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34… etc:
The written expression is represented as a formula: xn = xn-1 + xn-2 .
The sequence is closely related to the golden ratio, because if you take any two consecutive numbers and divide by the previous one, the fraction will be very close to the golden ratio. As the value of the number increases, the fraction gets even closer to the golden ratio. For example, 8/5 is 1.6, 34/21 is 1.619, and so on.
"Golden Spiral" Rectangle
So, you must have seen similar equations. But why do designers use geometry in their designs? Why is overlaying shapes necessary? The scheme is called the Fibonacci Spiral. It is actually quite simple and is the most optimal for many geometric shapes. The spiral is created using quarter circles that are drawn inside an array of squares based on the Fibonacci sequence.
The diagram below shows an example:
It turns out that each subsequent radius is greater than the previous one by a number close to the golden ratio. The resulting spiral is used in many areas, most often in drawing and architecture, but it can also be observed in natural phenomena.
"Rule of Three"
This rule is one of the variants of the golden spiral and is often used when cropping photos and videos. Used to trim frames and give them an aesthetic look. To apply the "Rule of Three", you need to divide the image by 9 equal parts. Draw 2 horizontal lines and 2 vertical lines. It is important to place them evenly. The point is to align focus with the leftmost vertical divider. The horizon or "vanishing point" must be level with the horizontal divider.
Application of the "Golden Spiral"
As already noted, the Fibonacci sequence is closely related to the golden ratio. The application of the golden section is performed using a traced spiral. The image shows an example of use this method. So, we see a rectangle, the base of which extends from the woman's right wrist to her left elbow.
The rectangle expands vertically until it reaches the crown. If we draw squares inside the golden rectangle, all the important parts of the woman are on the edges of the inner squares: her chin, eyes and lips. Leonardo da Vinci used the golden ratio many times in his work. Below are examples of the golden spiral in nature and space.
Application in web design
Many designers make the mistake of thinking that by simply dividing or multiplying by 1.61... you can get a harmonious proportion. This is far from true, it is just the basis of the process. You can't just use this or that number and get the magic proportion. However, there are certain ways that help to get the golden ratio. Some artists tend to think that the golden ratio theory is a myth. Here is another example of how the golden ratio works. Let's take a prototype site and consider the application of the golden ratio on it.
Looks pretty simple, right? Yes, in fact it is. The design is based on a 960 pixel grid. The design is represented using the golden ratio. If you use 1 golden spiral that spans 960px, you can see how the header, logo, etc. were positioned.
We move our spiral lower and rely on its dimensions
It turns out a kind of cascade of spirals in which the main design elements are inscribed in rectangles with a golden ratio
A grid based on the golden ratio has a number of proportional relationships within it that are clearly proportional within a rectangle. At the bottom of this article, I have attached a PSD file that contains my example, you can try using it in your project to experiment with the golden ratio.
