What is the golden ratio? Fibonacci Golden Ratio

Golden ratio - harmonic proportion

Golden ratio(golden proportion, division in extreme and average ratio) - division of a continuous value into two parts in such a ratio in which the smaller part is related to the larger one as the larger one is to the entire value.

Golden ratio- this is a section of a segment into two parts so that the length of the larger part is related to the length of the smaller part in the same way as the length of the entire segment is to the length of the larger part.

The ratio of the larger part to the smaller part in this proportion is expressed by quadratic irrationality

Golden ratio has many wonderful properties, but even more fictitious properties. Many people " strive to find» golden ratio in everything between one and a half and two.

Golden Triangle

To find segments golden ratio ascending and descending rows you can use the pentagram.

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 Durer (1471...1528). Let O be the center of the circle, A the point on the circle, and E the midpoint of the segment OA. The perpendicular to the radius OA, restored at point O, intersects the circle at point D. Using a compass, plot the segment CE = ED on the diameter. The side length of a regular pentagon inscribed in a circle is equal to DC. We plot the segments DC on the circle and get five points to draw a regular pentagon. We connect the corners of the pentagon through one another with diagonals 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 represents Golden Triangle. Its sides form an angle of 36° at the apex, and the base, laid on the side, divides it in the proportion of the golden ratio.

We draw straight AB. From point A we plot on it three times a segment O of arbitrary size, through the resulting point P we draw a perpendicular to line AB, on the perpendicular to the right and left of point P we plot segments O. The resulting points d and d 1 connect with straight lines to point A. Segment dd 1 put on line Ad 1 , getting point C. It divided the line Ad 1 in proportion to the golden ratio. LinesAd 1 and dd 1 used to construct a “golden” rectangle.

Great astronomer of the 16th century. Johannes Kepler called golden ratio one of the treasures of geometry. He is the first to pay attention to the meaning golden ratio for botany (plant growth and structure).

Kepler called golden ratio continuing itself

“It is structured in such a way,” he wrote, “that the two junior terms of this infinite proportion add up to the third term, and any two last terms, if added, give the next term, and the same proportion is maintained ad infinitum.”

Construction of a series of segments golden ratio can be done both in the direction of increase (ascending series) and in the direction of decrease (descending series).

Fibonacci series

With history golden ratio The name of the Italian mathematician monk Leonardo of Pisa, better known as Fibonacci (son of Bonacci), is indirectly connected. He traveled a lot in the East, introduced Europe to Indian (Arabic) numerals. In 1202, his mathematical work “The Book of the Abacus” (counting board) was published, which collected all the problems known at that time. One of the problems read “How many pairs of rabbits will be born from one pair in one year.” Reflecting on this topic, Fibonacci built the following series of numbers:

A series of numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc. known as the Fibonacci series. The peculiarity of the sequence of numbers is that each of its members, starting from the third, is equal to the sum of the previous two 2 + 3 = 5; 3 + 5 = 8; 5 + 8 = 13, 8 + 13 = 21; 13 + 21 = 34, etc., and the ratio of adjacent numbers in the series approaches the ratio of the golden division. So, 21: 34 = 0.617, and 34: 55 = 0.618. This relationship is denoted by the symbol F. Only this ratio - 0.618: 0.382 - gives a continuous division of a straight line segment into golden ratio, increasing it or decreasing it to infinity, when the smaller segment is related to the larger one as the larger one is to the whole.

Fibonacci also dealt with the practical needs of trade: what is the smallest number of weights that can be used to weigh a product? Fibonacci proves that the optimal system of weights is: 1, 2, 4, 8, 16...

The Fibonacci series could have remained only a mathematical incident, if not for the fact that all researchersgolden divisionin the plant and animal world, not to mentionart , invariably came to this series as an arithmetic expression of the lawgolden division.

Scientists continued to actively develop the theory of Fibonacci numbers and golden ratio. Yu. Matiyasevich solves Hilbert's 10th problem using Fibonacci numbers. Elegant methods are emerging for solving a number of cybernetic problems (search theory, games, programming) using Fibonacci numbers and the golden ratio. In the USA, even the Mathematical Fibonacci Association is being created, which has been publishing a special journal since 1963.

One of the achievements in this area is the discovery of generalized Fibonacci numbers and generalized golden ratios.

The Fibonacci series (1, 1, 2, 3, 5, 8) and the “binary” series of weights discovered by him 1, 2, 4, 8, 16... at first glance are completely different. But the algorithms for their construction are very similar to each other: in the first case, each number is the sum of the previous number with itself 2 = 1 + 1; 4 = 2 + 2..., in the second - this is the sum of the two previous numbers 2 = 1 + 1, 3 = 2 + 1, 5 = 3 + 2.... Is it possible to find a general mathematical formula from which we obtain “ binary series and Fibonacci series? Or maybe this formula will give us new numerical sets that have some new unique properties?

Indeed, let us set the numerical parameter S, which can take any values: 0, 1, 2, 3, 4, 5... Consider a number series, S+ 1 of the first terms of which are units, and each of the subsequent ones is equal to the sum of two terms of the previous one and separated from the previous one by S steps. If n We denote the th term of this series by φ S ( n), then we obtain the general formula φ S ( n) = φ S ( n– 1) + φ S ( n – S – 1).

It is obvious that when S= 0 from this formula we get a “binary” series, with S= 1 – Fibonacci series, with S= 2, 3, 4. new series of numbers, which are called S-Fibonacci numbers.

Overall golden S-proportion is the positive root of the golden equation S-sections x S+1 – x S – 1 = 0.

It is easy to show that when S= 0, the segment is divided in half, and when S= 1 – familiar classic golden ratio.

Relations between neighbors S-Fibonacci numbers coincide with absolute mathematical accuracy in the limit with gold S-proportions! In such cases, mathematicians say that gold S-sections are numerical invariants S-Fibonacci numbers.

Facts confirming the existence gold S-sections in nature, cites the Belarusian scientist E.M. Soroko in the book “Structural Harmony of Systems” (Minsk, “Science and Technology”, 1984). It turns out, for example, that well-studied binary alloys have special, pronounced functional properties (thermal stable, hard, wear-resistant, resistant to oxidation, etc.) only if the specific gravities of the original components are related to each other by one of gold S-proportions. This allowed the author to put forward the hypothesis that gold S-sections there are numerical invariants of self-organizing systems. Once confirmed experimentally, this hypothesis may be of fundamental importance for the development of synergetics, a new field of science that studies processes in self-organizing systems.

Using codes gold S-proportions any real number can be expressed as a sum of powers of gold S-proportions with integer coefficients.

The fundamental difference between this method of encoding numbers is that the bases of the new codes, which are gold S-proportions, at S> 0 turn out to be irrational numbers. Thus, new number systems with irrational bases seem to put “ from head to toe» the historically established hierarchy of relations between rational and irrational numbers. The fact is that at first there were “ open» natural numbers; then their ratios are rational numbers. And only later - after the Pythagoreans discovered incommensurable segments - irrational numbers were born. For example, in decimal, quinary, binary and other classical positional number systems, natural numbers were chosen as a kind of fundamental principle - 10, 5, 2 - from which, according to certain rules, all other natural numbers, as well as rational and irrational numbers, were constructed.

A kind of alternative to existing methods of notation is a new, irrational system, as a fundamental principle, the beginning of which is an irrational number (which, recall, is the root of the equation golden ratio); other real numbers are already expressed through it.

In such a number system, any natural number can always be represented as finite - and not infinite, as previously thought! – sum of powers of any of gold S-proportions. This is one of the reasons why “irrational” arithmetic, having amazing mathematical simplicity and elegance, seems to have absorbed the best qualities of classical binary and “Fibonacci” arithmetic.

Golden ratio and symmetry

Golden ratiocannot be considered on its own, separately, without connection withsymmetry . The great Russian crystallographer G.V. Woolf (1863...1925) believedgolden ratioone of the manifestationssymmetry .

Golden divisionis not a manifestationasymmetry , something oppositesymmetry . According to modern ideasgolden division- This asymmetrical symmetry. Into the science of symmetry included concepts such asstatic And dynamic symmetry . Static symmetry characterizes peace and balance, while dynamic characterizes movement and growth. So, in nature it is static symmetry characterized by equal segments and equal values. Dynamicsymmetry is characterized by an increase in segments or their decrease, and it is expressed in quantitiesgolden ratioascending or descending series.

Pierre Curie formulated a number of profound ideas at the beginning of our centurysymmetry . He argued that it cannot be consideredsymmetry any body, without taking into account the symmetry of the environment.

Patterns "gold"symmetry manifest themselves in the energy transitions of elementary particles, in the structure of some chemical compounds, in planetary and cosmic systems, in the gene structures of living organisms. These patterns, as indicated above, exist in the structure of individual human organs and the body as a whole, and also manifest themselves in the biorhythms and functioning of the brain and visual perception.

Golden Wurf- this is a sequential series of segments when adjacent segments are in relation to the golden ratio.

Let's consider the harmonic process of string vibrations. Standing waves of fundamental and higher harmonics (overtones) can be created on the string. The lengths of the half-waves of the harmonic series correspond to the function 1/N, where N is a natural number. The lengths of half-waves can be expressed as a percentage of the half-wave length of the fundamental harmonic: 100% 50% 33% 25% 20%... One or another harmonic can be excited by influencing the corresponding section of the string. In the case of an impact on an arbitrary section of the string, all harmonics will be excited with different amplitude coefficients, which depend on the coordinates of the section, on the width of the section, and on the time-frequency characteristics of the impact.

Let us introduce the function of the string’s susceptibility to impulse action. Taking into account the different signs of the phases of even and odd harmonics, we obtain an alternating function, which, to a first approximation, corresponds to the Bessel function, and, by and large, to the Schrödinger psi function. It looks approximately like this:

If the anchoring point is taken as the reference point, and the middle of the string as 100%, then the maximum susceptibility for the 1st harmonic will correspond to 100%, for the 2nd - 50%, for the 3rd - 33%, etc. Let's see where our function will intersect the x-axis.

62% 38% 23.6% 14.6% 9% 5,6% 3.44% 2.13% 1.31% 0.81% 0.5% 0.31% 0.19% 0.12% ...

This proportion golden wurf. Each next number is 0.618 times different from the previous one. The result is the following: Excitation of the string at a point dividing it in relation to the golden section at a frequency close to the fundamental harmonic will not cause vibrations of the string, i.e. The golden ratio point is the point of compensation, damping. For damping at higher frequencies, for example at the 4th harmonic, the compensation point must be selected at the 4th intersection of the function with the x-axis. If we create a rectangular flat resonator of electromagnetic oscillations, the sides of which are in proportion to the golden section, then the oscillations in such a resonator will be divided into two degrees of freedom, because vibrations along the larger side will not be able to excite vibrations along the smaller side, because for the smaller side, the length of the larger side corresponds to the compensation point. Now it becomes clear the reason that prompted the creation of rectangular cells with the proportion of the golden section on aircraft with electromagnetic energy sources. This made it possible to orient electromagnetic vibrations in the desired direction (vertically or horizontally). Further, these proportions have already been reflected inarchitecture religious buildings and became canonsart .

20.05.2017

The golden ratio is something every designer should know about. We'll explain what it is and how you can use it.

There is a general mathematical relationship found in nature that can be used in design to create pleasing, natural-looking compositions. It is called the Golden Ratio or the Greek letter “phi”. If you are an illustrator, art director or graphic designer, you should definitely use the Golden Ratio in every project.

In this article, we'll explain how to use it and also share some great tools for further inspiration and learning.

Closely related to the Fibonacci Sequence, which you may remember from math class or Dan Brown's Da Vinci Code, the Golden Ratio describes a perfectly symmetrical relationship between two proportions.

Approximately equal to the ratio 1:1.61, the Golden Ratio can be illustrated as the Golden Rectangle: a large rectangle containing a square (in which the sides are equal to the length of the shortest side of the rectangle) and a smaller rectangle.

If you remove the square from the rectangle, you will be left with another, small Golden Rectangle. This process can continue indefinitely, just like Fibonacci numbers, which work in reverse. (Adding a square with sides equal to the length of the longest side of the rectangle gets you closer to the Golden Rectangle and the Golden Ratio.)

Golden Ratio in action

The Golden Ratio is believed to have been used for around 4,000 years in art and design. However, many people agree that the construction of the Egyptian Pyramids also used this principle.

In more modern times, this rule can be seen in the music, art and design around us. By using a similar working methodology, you can bring the same design features into your work. Let's take a look at some inspiring examples.

Greek architecture

In ancient Greek architecture, the Golden Ratio was used to determine the pleasing spatial relationship between the width of a building and its height, the size of the portico, and even the position of the columns supporting the structure.

The result is a perfectly proportional structure. The Neoclassical architecture movement also used these principles.

last supper

Leonardo Da Vinci, like many other artists of yesteryear, often used the Golden Ratio to create pleasing compositions.

In the Last Supper, the figures are located in the lower two-thirds (the larger of the two parts of the Golden Ratio), and Jesus is perfectly sketched between the golden rectangles.

Golden ratio in nature

There are many examples of the Golden Ratio in nature - you can find them around you. Flowers, seashells, pineapples and even honeycombs show the same ratio.

How to calculate the Golden Ratio

The calculation of the Golden Ratio is quite simple, and starts with a simple square:

01. Draw a square

It forms the length of the short side of the rectangle.

02. Divide the square

Divide the square in half using a vertical line, creating two rectangles.

03. Draw a diagonal

In one of the rectangles, draw a line from one corner to the opposite.

04. Turn

Rotate this line so that it lies horizontal to the first rectangle.

05. Create a new rectangle

Create a rectangle using a new horizontal line and the first rectangle.

How to use the Golden Ratio

Using this principle is easier than you think. There are a couple of quick tricks you can use in your layouts, or take a little more time and fully flesh out the concept.

Fast way

If you've ever encountered the Rule of Thirds, you'll be familiar with the idea of ​​dividing a space into equal thirds vertically and horizontally, with where lines intersect to create natural points for objects.

The photographer places the key subject on one of these intersecting lines to create a pleasing composition. This principle can also be used in your page layout and poster designs.

The rule of thirds can be applied to any shape, but if you apply it to a rectangle with approximately 1:1.6 proportions, you will end up very close to the golden rectangle, which will make the composition more pleasing to the eye.

Full implementation

If you want to fully implement the Golden Ratio in your design, then simply arrange the main content and sidebar (in web design) in a ratio of 1: 1.61.

You can round the values ​​down or up: if the content area is 640px and the sidebar is 400px, then this markup is quite suitable for the Golden Ratio.

Of course, you can also divide the content and sidebar areas into the same relationship, and the relationship between the web page's header, content area, footer, and navigation can also be designed using the same principle.

Useful tools

Here are some tools to help you use the Golden Ratio in design and create proportional designs.

GoldenRATIO is an application for creating website designs, interfaces and templates suitable for the Golden Ratio. Available on the Mac App Store for $2.99. Includes a visual Golden Ratio calculator.

The application also has a “Favorites” function, which saves settings for recurring tasks and a “Click-thru” mod that allows you to minimize the application in Photoshop.

This Golden Ratio calculator from Pearsonified helps you create the perfect typography for your website. Enter the font size, container width in the field, and click the button Set my type! If you need to optimize the number of letters per line, you can additionally enter a CPL value.

This simple, useful, and free app is available for Mac and PC. Enter any number and the application will calculate the second digit according to the Golden Ratio principle.

This application allows you to design with golden proportions, saving a lot of time on calculations.

You can change shapes and sizes to focus on your project. A permanent license costs $49, but you can download a free version for a month.

Golden Section Training

Here are some useful tutorials on the Golden Ratio (English):

In this Digital Arts tutorial, Roberto Marras shows how to use the Golden Ratio in your artistic work.

Tutorial from Tuts+ showing how to use the golden principles in web design projects.

A tutorial from Smashing Magazine about proportions and the rule of thirds.

They say that “divine proportion” is inherent in nature, and in many things around us. You can find it in flowers, beehives, sea shells, and even our bodies.

This divine ratio, also known as the golden ratio, divine ratio, or golden ratio can be applied to various forms of art and learning. Scientists say that the closer an object is to the golden ratio, the better the human brain perceives it.

Since this relationship was discovered, many artists and architects have used it in their works. You can find the golden ratio in several Renaissance masterpieces, architecture, painting, and more. The result is a beautiful and aesthetically pleasing masterpiece.

Few people know what the secret of the golden ratio is, which is so pleasing to our eyes. Many believe that the fact that it appears everywhere and is a “universal” proportion forces us to accept it as something logical, harmonious and organic. In other words, it simply “feels” what we need.

So what is the golden ratio?

The golden ratio, also known as “phi” in Greek, is a mathematical constant. It can be expressed by the equation a/b=a+b/a=1.618033987, where a is greater than b. This can also be explained by the Fibonacci sequence, another divine proportion. The Fibonacci sequence starts with 1 (some say 0) and adds the previous number to it to get the next one (i.e. 1, 1, 2, 3, 5, 8, 13, 21...)

If you try to find the quotient of two subsequent Fibonacci numbers (i.e. 8/5 or 5/3), the result is very close to the golden ratio of 1.6 or phi.

The golden spiral is created using a golden rectangle. If you have a rectangle of squares 1, 1, 2, 3, 5 and 8 respectively as shown in the picture above, you can start building the golden rectangle. By using the side of the square as the radius, you create an arc that touches the points of the square diagonally. Repeat this procedure with each square in the golden triangle and you will end up with a golden spiral.

Where can we see it in nature

The Golden Ratio and Fibonacci Sequence can be found in flower petals. For most flowers, the number of petals is reduced to two, three, five or more, which is similar to the golden ratio. For example, lilies have 3 petals, buttercups have 5, chicory flowers have 21, and daisies have 34. Flower seeds probably also follow the golden ratio. For example, sunflower seeds germinate from the center and grow toward the outside, filling the seed head. They are usually spiral-shaped and resemble a golden spiral. Moreover, the number of seeds is usually reduced to Fibonacci numbers.

Hands and fingers are also an example of the golden ratio. Look closer! The base of the palm and the tip of the finger are divided into parts (bones). The ratio of one part in comparison to another is always 1.618! Even the forearms and hands are in the same ratio. And fingers, and face, and the list goes on...

Application in art and architecture

The Parthenon in Greece is said to have been built using golden proportions. It is believed that the dimensional ratios of height, width, columns, distance between pillars, and even the size of the portico are close to the golden ratio. This is possible because the building looks proportionally perfect and has been like this since ancient times.

Leonardo Da Vinci was also a fan of the golden ratio (and many other curiosities, in fact!). The wondrous beauty of the Mona Lisa may be due to the fact that her face and body represent the golden ratio, just like real human faces in life. In addition, the numbers in the painting “The Last Supper” by Leonardo Da Vinci are arranged in the order that is used in the golden ratio. If you draw golden rectangles on the canvas, Jesus will be right in the central lobe.

Application in logo design

It's no surprise that you can also find the use of the golden ratio in many modern projects, particularly design. For now, let's focus on how this can be used in logo design. First, let's look at some of the world's most famous brands that have used the golden ratio to perfect their logos.

Apparently Apple used circles from Fibonacci numbers, joining and cutting the shapes to create the Apple logo. It is unknown whether this was done intentionally or not. However, the result is a perfect and visually aesthetic logo design.

The Toyota logo uses the ratio of a and b, forming a grid in which three rings are formed. Notice how this logo uses rectangles instead of circles to create the golden ratio.

The Pepsi logo is created by two intersecting circles, one larger than the other. As shown in the picture above, the larger circle is proportional to the smaller circle - you guessed it! Their latest non-emboss logo is simple, effective and beautiful!

Besides Toyota and Apple, the logos of several other companies such as BP, iCloud, Twitter, and Grupo Boticario are also believed to have used the golden ratio. And we all know how famous these logos are - all because the image immediately springs to mind!

Here's how you can apply it in your projects

Sketch a golden rectangle as shown above in yellow. This can be achieved by constructing squares with height and width from numbers belonging to the golden ratio. Start with one block and place another next to it. And place another square, whose area is equal to those two, above them. You will automatically receive a side of 3 blocks. After building this 3 block structure, you will end up with a side of 5 quads from which you can make another (5 block area) box. This can go on as long as you like until you find the size you need!

The rectangle can move in any direction. Select small rectangles and use each one to assemble a layout that will serve as a logo design grid.

If the logo is more rounded, then you will need a circular version of the golden rectangle. You can achieve this by drawing circles proportional to the Fibonacci numbers. Create a golden rectangle using only circles (this means the largest circle will have a diameter of 8, and the smaller circle will have a diameter of 5, and so on). Now separate these circles and place them so that you can form the basic outline for your logo. Here's an example of the Twitter logo:

Note: You don't have to draw all the golden ratio circles or rectangles. You can also use the same size more than once.

How to use it in text design

It's easier than designing a logo. A simple rule for applying the golden ratio in text is that subsequent larger or smaller text must conform to Phi. Let's look at this example:

If my font size is 11, then the subtitle should be written in a larger font. I multiply the text font by the golden ratio number to get a larger number (11*1.6=17). This means the subtitle should be written in font size 17. And now the title or title. I’ll multiply the subtitle by the proportion and get 27 (1*1.6=27). Like this! Your text is now proportional to the golden ratio.

How to apply it in web design

But here it’s a little more complicated. You can stay true to the golden ratio even in web design. If you are an experienced web designer, you have already guessed where and how it can be applied. Yes, we can effectively use the golden ratio and apply it to our web page grids and UI layouts.

Take the total number of grid pixels as the width or height and use it to construct the golden rectangle. Divide the largest width or length to get smaller numbers. This can be the width or height of your main content. What's left could be the sidebar (or bottombar if you applied it to height). Now continue using the golden rectangle to further apply it to windows, buttons, panels, images and text. You can also build a full mesh based on small versions of the golden rectangle placed both horizontally and vertically to create smaller interface objects that are proportional to the golden rectangle. To get the proportions you can use this calculator.

Spiral

You can also use the golden spiral to determine where to place content on your site. If your home page loads with graphic content, such as an online store website or photography blog, you can use the golden spiral method that many artists use in their work. The idea is to place the most valuable content in the center of the spiral.

Content with grouped material can also be placed using a golden rectangle. This means that the closer the spiral moves to the central squares (to one square block), the “dense” the contents there.

You can use this technique to indicate the placement of your header, images, menus, toolbar, search box, and other elements. Twitter is famous not only for its use of the golden rectangle in its logo design, but also for its use in web design. How? Through the use of the golden rectangle, or in other words the golden spiral concept, in the users' profile page.

But this won't be easy to do on CMS platforms, where the content author determines the layout instead of the web designer. The Golden Ratio is suitable for WordPress and other blog designs. This is probably because a sidebar is almost always present in a blog design, which fits nicely into the golden rectangle.

An easier way

Very often, designers skip complex mathematics and apply the so-called “rule of thirds”. It can be achieved by dividing the area into three equal parts horizontally and vertically. The result is nine equal parts. The intersection line can be used as the focal point of the form and design. You can place a key theme or main elements on one or all of the focal points. Photographers also use this concept for posters.

The closer the rectangles are to the ratio 1:1.6, the more pleasant the picture is perceived by the human brain (since it is closer to the golden ratio).

Introduction……………………………………………………….………3

1. Dynamic symmetry in nature and architecture………………3

2. Golden ratio – harmonic proportion…………………..6

3. Second golden ratio…………………………………………..7

4. History of the golden ratio………………………………………..7

5. Fibonacci series……………………………………………………11

6. Nature………………………………………………………12

Conclusion……………………………………………………………13

References……………………………………………………...15

Introduction.

The idea that the physical world is dominated by harmony and order, which can be expressed mathematically, goes back to ancient Greece. In Europe during the Renaissance, Galileo said that the book of the universe was written in the language of mathematics. Scientists who lived after him also expressed amazement at the fact that all the laws of the universe could be translated into mathematical language.

Realizing this “universal applicability” of mathematics, unknown to chemical and biological sciences, the great physicist James Jones said: “The architect of the universe must have been a mathematician.” It is known that Einstein's theory of relativity is not just the result of reflection; it was put forward after certain mathematical developments.

Bearing in mind the intelligibility that physical laws acquire when translated into the language of mathematics, Einstein said: “The only incomprehensible quality of the universe is its comprehensibility.”

And how not to be amazed even at the simplest example - the expression of the force of mutual attraction of bodies in the form of a mathematical formula:

F = Y-mi-1712/ r

In this formula, the constant value of the constant “Y” in all cases - from the force of attraction between electrons and protons in an atom to the mutual attraction of stars, from our planet to worlds billions of light years distant from us, demonstrates amazing simplicity, that is, the phenomenality of the formula and its enduring value, as a kind of universal currency.

The extremely effective and unexpected results of the application of mathematics to other branches of science still seem to us a mystery. Some scientists associate this with the orientation of other sciences towards the development of mathematical knowledge.

1. Dynamic symmetry in nature and architecture

Term "dynamic symmetry" was first used by the American architectural researcher D. Hambidge, denoting a certain principle of proportionation in architecture. Later, this term independently appeared in physics, where it was introduced to describe physical processes characterized by invariants. Finally, the term dynamic symmetry a pattern of natural morphogenesis is named, which in terms of origin also turns out to be unrelated to Hambidge’s idea and, even more so, to the appearance of this term in physics. However, all three options are deeply interconnected in content.

First, let us note the strategic similarity of our research direction with Hambidge. This is a well-known historically established direction, which in the field of architecture and art is motivated by the search for patterns of harmony, and therefore focused on the study of natural objects. Typically, architects are interested in the structural patterns of natural shape formation and especially in the golden ratio and Fibonacci numbers

Patterns that are notable for their intriguing role in architectural formation. It is no coincidence that architect-researchers so often pay attention to the botanical phenomenon phyllotaxis, which is characterized by these patterns.

Phyllotaxis turned out to be the object of attention of the author of the first version of the concept of dynamic symmetry, D. Hambidge. As a result of studying this phenomenon, D. Hambidge concludes law so-called monotonous growth, and offers its geometric interpretation - spiral of monotonous growth, or else

- golden spiral (Fig. 1).

Fig 1. Construction of the golden spiral according to Hambidge.

However, the main generalization made by D. Hambidge as a result of studying the laws of natural morphogenesis (phyllotaxis), as well as the proportions of classical architecture, comes down to the idea of ​​architectural proportioning, called dynamic symmetry. Hambidge illustrates it using a simple geometric diagram (Fig. 2).

Fig 2. Proportional system “Dynamic symmetry” by D. Hambidge.

This is a sequential system of rectangles, the first of which is a square, and each subsequent one is built on the side of the original square, equal to 7, and on the diagonal of the previous rectangle. The result is a series of rectangles, the ratio of the sides of which is expressed by the series. In this series, Hambidge distinguishes between two types of rectangles - static and dynamic. For static rectangles, the aspect ratios are expressed as integers, while for dynamic rectangles they are expressed as irrational ones. Dynamic rectangles, according to D. Hambidge, express the idea of ​​growth, movement and development. Of these, he first of all singles out three whose long sides are equal But it attaches special importance to the rectangle that is directly related to "golden rectangle" Hambidge conducts a thorough geometric study, discovering various manifestations of the golden section in the rectangle system. Studying the geometric properties of this rectangle, he shows the possibility of its use for analyzing the proportions of objects of classical architecture and art (Fig. 3, 4).

This, in a nutshell, is the essence of D. Hambidge's idea of ​​dynamic symmetry. As we see, it does not directly follow from the properties of phyllotaxis. Hambidge, generally speaking, does not delve into the mathematics of phyllotaxis. In his various diagrams illustrating the patterns of uniform growth, or some ideas of proportioning, he uses well-known numerical relationships characteristic of phyllotaxis, incl. golden ratio.

2. GOLDEN RATIO - harmonic proportion.

In mathematics, a proportion is the equality of two ratios: a: b = c: d.
A straight line segment AB can be divided by point C into two parts in the following ways:
into two equal parts AB: AC = AB: BC;

into two unequal parts in any respect (such parts do not form proportions);
thus, when AB: AC = AC: BC.

The latter is the golden division or division of a segment in extreme and average ratio.

Golden ratio- this is such a proportional division of a segment into unequal parts, in which the entire segment relates to the larger part as the larger part itself relates to the smaller; or in other words, the smaller segment is to the larger as the larger is to the whole

Segments of the golden proportion are expressed as an infinite irrational fraction 0.618..., if c is taken as one, a = 0.382. The numbers 0.618 and 0.382 are the Fibonacci sequence ratios. The basic geometric figures are based on this proportion.
A rectangle with this aspect ratio became known as the golden rectangle. It also has interesting properties. If you cut a square from it, you will again be left with a golden rectangle. This process can be continued indefinitely. And if you draw a diagonal of the first and second rectangles, then the point of their intersection will belong to all the resulting golden rectangles.
Of course there is also a golden triangle. This is an isosceles triangle whose side length to base length ratio is 1.618.
There is also a golden cuboid - this is a rectangular parallelepiped with edges having lengths of 1.618, 1 and 0.618.

In a star pentagon, each of the five lines that make up the figure divides another in relation to the golden ratio, and the ends of the star are golden triangles.

3. Second GOLDEN RATIO

The figure shows the position of the line of the second golden ratio. It is located midway between the golden ratio line and the middle line of the rectangle.

Dividing a rectangle with the line of the second golden ratio

4. 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 Cheops pyramid, temples, bas-reliefs, household items and jewelry from the tomb of Tutankhamun indicate that 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 a tomb named after him, holds in his hands measuring instruments in which the proportions of the golden division are recorded.

The Greeks were skilled geometers. They even taught arithmetic to their children using geometric figures. The Pythagorean square and the diagonal of this square were the basis for the construction of 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 Pythagorean school and, in particular, to the issues of the golden division. The façade of the ancient Greek temple of the Parthenon features golden proportions. During its excavations, compasses were discovered that were used by architects and sculptors of the ancient world. The Pompeian compass (museum in Naples) also contains the proportions of the golden division.

In the ancient literature that has come down to us, the golden division was first mentioned in Euclid’s Elements. In the 2nd book of the Elements, a geometric construction of the golden division is given. After Euclid, the study of the golden division was carried out by Hypsicles (2nd century BC), Pappus (3rd century AD), and others. In medieval Europe, they became acquainted with the golden division through Arabic translations of Euclid’s Elements. The translator J. Campano from Navarre (III century) made comments on the translation. The secrets of the golden division were jealously guarded and kept in strict secrecy. They were known only to initiates.
IN Renaissance Interest in the golden division is increasing among scientists and artists due to its application both in geometry and in art, especially in architecture. Leonardo da Vinci, an artist and scientist, saw that Italian artists have a lot of 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 of Italy in the period between Fibonacci and Galileo. Luca Pacioli was a student of the artist Piero Della Franceschi, who wrote two books, one of which was called “On Perspective in Painting.” He is considered the creator of descriptive geometry. Luca Pacioli perfectly understood the importance of science for art. In 1496, at the invitation of the Duke of Moreau, he came to Milan, where he lectured on mathematics. Leonardo da Vinci also worked in Milan at the Moro court at that time. In 1509, Luca Pacioli's book "The Divine Proportion" was published in Venice with brilliantly executed illustrations, which is why it is believed that they were made by Leonardo da Vinci. The book was an enthusiastic hymn to the golden ratio. Among the many advantages of the golden proportion, the monk Luca Pacioli did not fail to name its “divine essence” as an expression of the divine trinity: God the Son, God the Father and God the Holy Spirit (it was implied that the small segment is the personification of God the Son, the larger segment is the God of the Father, and the entire segment - God of the Holy Spirit).
Leonardo da Vinci He also paid a lot of attention to the study of the golden division. He made sections of a stereometric body formed by regular pentagons, and each time he obtained rectangles with aspect ratios in the golden division. Therefore, he gave this division the name golden ratio. So it still remains as the most popular.
At the same time, in the north of Europe, in Germany, Albrecht Dürer was working on the same problems. He sketches the introduction to the first version of the treatise on proportions. Dürer writes: “It is necessary that someone who knows how to do something should teach it to others who need it. This is what I set out to do.” Judging by one of Dürer's letters, he met with Luca Pacioli while in Italy. Albrecht Durer develops in detail the theory of proportions of the human body. Dürer assigned an important place in his system of relationships to the golden section. A person's height is divided in golden proportions by the line of the belt, as well as by a line drawn through the tips of the middle fingers of the lowered hands, the lower part of the face by the mouth, etc. Dürer's proportional compass is well known.

Great astronomer of the 16th century. Johann Kepler called the golden ratio one of the treasures of geometry. He was the first to draw attention to the importance of the golden proportion for botany (plant growth and their structure). Kepler called the golden proportion self-continuing. “It is structured in such a way,” he wrote, “that the two lowest terms of this endless proportion add up to the third term, and any two last terms, if added together, give the next term, and the same proportion remains until infinity."
In subsequent centuries, the rule of the golden proportion turned into an academic canon, and when, over time, the struggle against academic routine began in art, in the heat of the struggle “they threw out the baby with the bathwater.” The golden ratio was “discovered” again in the middle of the 19th century.
In 1855, the German researcher of the golden ratio, Professor Zeising, published his work “Aesthetic Studies”. He absolutized the proportion of the golden section, declaring it universal for all phenomena of nature and art. Zeising had numerous followers, but there were also opponents who declared his doctrine of proportions “mathematical aesthetics.”

Zeising did a tremendous job. He measured about two thousand human bodies and came to the conclusion that the golden ratio expresses the average statistical law. The division of the body by the navel point is the most important indicator of the golden ratio. The proportions of the male body fluctuate within the average ratio of 13: 8 = 1.625 and are somewhat closer to the golden ratio than the proportions of the female body, in relation to which the average value of the proportion is expressed in the ratio 8: 5 = 1.6. In a newborn the proportion is 1:1, by the age of 13 it is 1.6, and by the age of 21 it is equal to that of a man. The proportions of the golden ratio also appear in relation to other parts of the body - the length of the shoulder, forearm and hand, hand and fingers, etc. Zeising tested the validity of his theory on Greek statues. He developed the proportions of Apollo Belvedere in the most detail. Greek vases, architectural structures of various eras, plants, animals, bird eggs, musical tones, and poetic meters were studied. Zeising gave a definition to the golden ratio and showed how it is expressed in straight line segments and in numbers. When the numbers expressing the lengths of the segments were

Golden proportions in the human figure

obtained, Zeising saw that they constituted a Fibonacci series, which could be continued indefinitely in one direction or the other. His next book was titled “The Golden Division as the Basic Morphological Law in Nature and Art.” In 1876, a small book, almost a brochure, was published in Russia outlining this work of Zeising. The author took refuge under the initials Yu.F.V. This edition does not mention a single work of painting. At the end of the 19th – beginning of the 20th centuries. Many purely formalistic theories appeared about the use of the golden ratio in works of art and architecture. With the development of design and technical aesthetics, the law of the golden ratio extended to the design of cars, furniture, etc.

5. Fibonacci series

The name of the Italian mathematician monk Leonardo of Pisa, better known as Fibonacci (son of Bonacci), is indirectly connected with the history of the golden ratio. He traveled a lot in the East, introduced Europe to Indian (Arabic) numerals. In 1202, his mathematical work “The Book of the Abacus” (counting board) was published, which collected all the problems known at that time. One of the problems read “How many pairs of rabbits will be born from one pair in one year.” Reflecting on this topic, Fibonacci built the following series of numbers:

Fibonacci proves that the optimal system of weights is: 1, 2, 4, 8, 16...

6. Nature.

Now let's move on to Nature, which gives a huge number of manifestations of the Golden Section and Fibonacci numbers. Let us give several illustrative examples of the manifestation of the Golden Section in Nature.

"Golden" spirals in sea shells

These illustrative examples could be continued ad infinitum. One thing is clear: The Golden Ratio and Fibonacci numbers reflect some fundamental patterns of living nature.

Now let’s talk about another modern scientific discovery that establishes connection of the genetic code with Fibonacci numbers and the Golden Ratio. In 1990, French researcher Jean-Claude Perez, who was working at that time as a researcher at IBM, made a very unexpected discovery in the field of genetic coding. He discovered a mathematical law governing the self-organization of bases T, C, A, G inside DNA. He discovered that successive sets of DNA nucleotides are organized into long-range order structures called RESONANCES . Resonance represents a special proportion that ensures the division of DNA in accordance with the Fibonacci numbers (1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...).

The key idea behind Jean-Claude Perez's discovery, called DNA SUPRA code , is as follows. Let us consider any segment of the genetic code consisting of bases of the type T, C, A, G, and let the length of this segment be equal to the Fibonacci number, for example, 144. If the number of bases is like T in the DNA segment under consideration is 55 (Fibonacci number) and the total number of bases of the type A, C And G is equal to 89 (Fibonacci number), then the segment of the genetic code in question forms resonance, that is, resonance there is a proportion between three adjacent Fibonacci numbers (55-89-144). The discovery is that each DNA forms many resonances of the considered type, that is, as a rule, segments of the genetic code with a length equal to the Fibonacci number Fn, are divided by the golden ratio into many bases like T(the number of which in the considered segment of the genetic code is equal to Fn- 2) and the total set of remaining bases (the number of which is equal to Fn- 1). If we carry out a systematic study of all possible “Fibonacci” segments of the genetic code, then we will obtain a certain set resonances, called SUPRA DNA code .

Since 1990, this pattern has been repeatedly tested and confirmed by many outstanding biologists, in particular Professors Montagniere and Sherman, who studied the DNA of the AIDS virus.

There is no doubt that the discovery in question belongs to the category of outstanding discoveries in the field of DNA that determine the development of genetic engineering. According to the author of the discovery, Jean-Claude Perez, the DNA SUPRA code is a universal bio-mathematical law, which indicates the highest level of self-organization of nucleotides in DNA according to the principle of the “Golden Section”.

Conclusion.

So, when the Lord created the universe, he was not content with merely caring for the perfection of his laws, which were to be established, but also gave them a beauty that elevates the human spirit. He wove a beautiful and graceful pattern into this grandiose lace, woven by the power of science. And as the son of the human race revealed the secrets of the pattern on this lace, mathematical science was born. Each was initiated into the secret of one thread, different from the others, and a grandiose picture appeared to us in its present form. Having gained this knowledge, we will either concentrate it at a single point and lock it in the human brain, or scatter it across the tablets of the books of the universe. The fact that we become familiar with existing truths only at a certain level of development indicates that mathematics belongs to the primordial.

Bibliography.

1. D. Pidou. Geometry and art. – M.: Mir, 1999

2. Stakhov A. Codes of the golden proportion.

3. Kepler I. About hexagonal snowflakes. – M., 1982.

4. Magazine “Mathematics at School”, 1994, No. 2; No. 3.

5. Tsekov-Pencil Ts. About the second golden section. – Sofia, 1983.

6. www.trinitas.ru/rus/doc/0232/004a/02321053.htm

7. http://www.noviyegrani.com/archives_show.php?ID=13&ISSUE=3

The Golden Ratio is a simple principle that can help make a 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 Golden Mean, is based on the Fibonacci Sequence (which you most likely heard about in school, or read about in Dan Brown's book "The Da Vinci Code"), and implies an aspect ratio of 1 :1.61.

This ratio is often found in our lives (shells, pineapples, flowers, etc.) and therefore is perceived by a person as something natural and pleasing to the eye.

→ The golden ratio is the relationship between two numbers in the Fibonacci sequence
→ Plotting this sequence to scale produces the 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 thousand years, and perhaps even more, according to scientists who claim that the ancient Egyptians used this principle when building 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 dimensions of a portico, and even the distance between columns. Subsequently, this principle was inherited by the architecture of neoclassicism.

Art: 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 principle for the 2010 redesign. “Anyone interested in #NewTwitter proportions, know that everything was done for a reason,” he said.

Apple iCloud

The iCloud service icon is also not a random sketch. As Takamasa Matsumoto explained in his blog (original Japanese version), everything is built on the mathematics of the Golden Ratio, the anatomy of which can be seen in the picture on the right.

How to construct the Golden Ratio?

The construction is quite simple, and starts 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 guide. Ready!

"Golden" instruments

If drawing and measuring is not your favorite activity, leave all the “grunt 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 application helps you develop websites, interfaces and layouts in accordance with the Golden Ratio. It's available on the 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 website according to the principles of the Golden Ratio. Just enter the font size, content width in the field on the site, and click “Set my type”!

This is a simple and free application for Mac and PC. Just enter a number and it will calculate the proportion for it according to the Golden Ratio rule.

A convenient program that will relieve you of the need for calculations and drawing grids. It makes finding ideal proportions easier than ever! 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.