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Authors:
Bekbulatova Alina,
Getmanova Sofia
Leaders:
Mogutova Tatyana Mikhailovna,
Deryushkina Oksana Valerievna
Introduction.
Theoretical part of the project:
- History of the development of fractal geometry.
- The concept of a fractal.
- Types of fractals:
a) geometric fractals, examples of geometric fractals;
b) algebraic fractals, examples of algebraic fractals;
c) stochastic fractals, examples.
- Natural fractals.
- Practical application of fractals:
- in literature;
- in telecommunications;
- in medicine;
- in architecture;
- in design;
- in economics;
- in games, cinema, music
- in natural sciences
- in physics;
- in biology
- fractals for housewives
- modern paintings – fractal graphics.
- Fractal graphics.
- The role of fractal geometry in life is a hymn to fractals!
The practical part of the project
- Creation of a scientific work “Journey to the world of fractals”
- Posting on the Internet.
- Participation in Olympiads and competitions.
- Create your own fractals.
- Creation of the brochure “The Amazing World of Fractals”
- Carrying out the festival “The Amazing World of Fractals.
Introduction
The geometry is often described as cold and dry. One reason is its inability to describe everything that surrounds us: the shape of a cloud, mountain, tree or seashore. Clouds are not spheres, mountains are not cones, coastlines are not circles, and the crust is not smooth, and lightning does not travel in a straight line. With great joy for us, we learned that in the modern world there is a new geometry - the geometry of fractals.
The discovery of fractals revolutionized not only geometry, but also physics, chemistry, biology, and all areas of our lives.
Relevance of the project:
- The role of fractals in the modern world is quite large
- Convincing arguments in favor of the relevance of studying fractals is the breadth of their application
Research hypothesis:
Fractal geometry is a modern, very interesting area of human knowledge. The emergence of fractal geometry is evidence of the ongoing evolution of man and the expansion of his ways of understanding the world.
Objective of the project:
Study the theory of fractals to create a scientific work “The Amazing World of Fractals” and develop and implement on a computer algorithms for drawing fractals on a plane.
Project objectives:
- Get acquainted with the history of the emergence and development of fractal geometry;
- Study the types of fractals and their application in the modern world.
- Execute fractal creation programs in Pascal and Logo programming languages
- Create a scientific work about fractals and publish it on the Internet.
- Create a brochure “The Amazing World of Fractals”
- Hold a festival “The Amazing World of Fractals” to familiarize school students with the results of our work.
We worked on the project for 4 months.
The main stages of our work:
- Gathering the necessary information: using the Internet, books, publications on this topic. (2 weeks)
- Sorting information by topic: systematizing and determining the order of writing the work. The work took 2 weeks.
- Preparation of text work: writing the text, partial preparation of systematized information. It took one month.
- Creation of the presentation: compression of systematized information, determination of the structure of the presentation, its creation and design, and took place over the course of a month.
- Learning a fractal creation program and creating your own fractals in the Pascal and Logo programming languages (until today)
Theoretical part of the project
We studied the history of the creation of fractal geometry.
Interest in fractal objects was revived in the mid-70s of the 20th century.
The birth of fractal geometry is usually associated with the publication of Mandelbrot’s book “The Fractal Geometry of Nature” in 1977. His works used the scientific results of other scientists who worked in the period 1875-1925 in the same field (Poincaré, Fatou, Julia, Cantor, Hausdorff But only in our time has it been possible to combine their work into a single system.
So what is a fractal?
Fractal - a geometric figure composed of several parts, each of which is similar to the entire figure.
A small part of a fractal contains information about the entire fractal. Today, the word “fractal” most often means a graphic representation of a structure that is similar to itself on a larger scale.
Fractals are divided into geometric, geometric and stochastic.
Geometric fractals are also called classical. They are the most visual, since they have so-called rigid self-similarity, which does not change when the scale changes. This means that no matter how close you zoom in on the fractal, you still see the same pattern.
Let us give the most famous examples of geometric fractals.
Snowflake Koch.
Invented in 1904 by German mathematician Helge von Koch.
To construct it, a single segment is taken, divided into three equal parts, and the middle link is replaced by an equilateral triangle without this link. In the next step, we repeat the operation for each of the four resulting segments. As a result of endless repetition of this procedure, a fractal curve is obtained.
Durer's pentagon.
A fractal looks like a bunch of pentagons squeezed together. In fact, it is formed by using a pentagon as an initiator and isosceles triangles, the ratio of the larger side to the smaller side is exactly equal to the so-called golden ratio. These triangles are cut out from the middle of each pentagon, resulting in a figure similar to 5 small pentagons glued to one big.
Sierpinski's napkin.
In 1915, Polish mathematician Waclaw Sierpinski came up with an interesting object.
To construct it, take a solid equilateral triangle. In the first step, an inverted equilateral triangle is removed from the center. The second step removes three inverted triangles from the remaining three triangles, and so on.
Dragon Curve.
Invented by Italian mathematician Giuseppe Peano.
Sierpinski carpet.
A square is taken, divided into nine equal squares, the middle one is thrown away, and the same operation is repeated with the rest ad infinitum.
The second type of fractals is algebraic fractals.
They got their name because they are built on the basis of algebraic formulas. As a result of mathematical processing of this formula, a point of a certain color is displayed on the screen. The result is a strange figure in which straight lines turn into curves and self-similarity effects appear at various scale levels. Almost every point on a computer screen is like a separate fractal.
Examples of the most famous algebraic fractals.
Mandelbrot set.
Mandelbrot sets are the most common among algebraic fractals. It can be found in many scientific journals, book covers, postcards, and computer screen savers. This fractal resembles a carding machine with flaming tree-like and circular areas attached to it.
Lots of Julia.
The Julia set was invented by the French mathematician Gaston Julia. An equally famous algebraic fractal.
Newton Pools.
Stochastic fractals.
Fractals, during the construction of which in an iterative system some parameters change randomly, are called stochastic. The term "stochasticity" comes from the Greek word meaning "assumption".
In this case, the resulting objects are very similar to natural ones - asymmetrical trees, rugged coastlines, etc. Two-dimensional stochastic fractals are used in modeling terrain and sea surfaces.
These fractals are used in modeling terrain and sea surfaces, and the electrolysis process. This group of fractals has become widespread thanks to the work of Michael Barnsley from the Georgia Institute of Technology.
A typical representative of this class of fractals is "Plasma".
The most understandable for us are the so-called natural fractals.
“The Great Book of Nature is written in the language of geometry” (Galileo Galilei).
Natural fractals.
- In wildlife:
- Starfish and urchins
- Flowers and plants (broccoli, cabbage)
- Tree crowns and plant leaves
- Fruit (pineapple)
- Circulatory system and bronchi of humans and animals
- In inanimate nature:
- Borders of geographical objects (countries, regions, cities)
- Frosty patterns on window glass
- Stalactites, stalagmites, helictites.
Almost all natural formations: tree crowns, clouds, mountains, coastlines have a fractal structure.
What does it mean?
If you look at a fractal object as a whole, then at a part of it on an enlarged scale, then at a part of this part, it is not difficult to see that they look the same.
Marine fractals.
An octopus is a bottom-dwelling sea animal from the order of cephalopods.
Its bodies and suckers on all eight tentacles of this animal have a fractal structure.
Another typical representative of the fractal underwater world is coral.
There are over 3,500 species of corals known in nature.
Green fractal – fern leaves.
Fern leaves have the shape of a fractal figure - they are self-similar.
Onion is a fractal that makes you cry. Of course, it is a simple fractal: ordinary circles of different diameters, one might even say a primitive fractal.
A striking example of a fractal in nature is “Romanescu", also known as "Romanesque broccoli" or "coral cauliflower".
Cauliflower- typical fractal.
Let's look at the structure of cauliflower.
If you cut one of the flowers, it is obvious that the same cauliflower remains in your hands, only smaller in size. We can keep cutting again and again, even under a microscope - but all we get are tiny copies of the cauliflower.
Matryoshka - souvenir toy- a typical fractal. The principle of fractality is obvious when all the figures of a wooden toy are lined up in a row and not nested inside each other.
Man is a fractal.
A child is born and grows, and this process is accompanied by the principle of “self-similarity”, fractality.
The scope of fractals is wide.
Fractals in literature
Among literary works there are those that have a textual, structural or fractal nature. In literary fractals, elements of the text are endlessly repeated:
The priest had a dog
he loved her.
She ate a piece of meat
he killed her.
Buried in the ground
Caption wrote:
The priest had a dog...
“Here is the house.
Which Jack built.
And here is the wheat.
In the house,
Which Jack Built
And here is a cheerful tit bird,
Which cleverly steals wheat,
Which is stored in a dark closet
In the house,
Which Jack built..." .
Fractals in telecommunications.
To transmit data over distances, antennas with fractal shapes are used, which greatly reduces their size and weight.
Fractals in medicine.
Currently, fractals are widely used in medicine. The human body itself consists of many fractal structures: the circulatory system, muscles, bronchi, bronchial tracts in the lungs, arteries.
The theory of fractals is used to analyze electrocardiograms.
Assessing the magnitude and rhythms of the fractal dimension allows one to judge at an earlier stage and with greater accuracy and information about disturbances of homeostasis and the development of specific heart diseases.
X-ray images processed using fractal algorithms provide a higher-quality picture, and, accordingly, better diagnostics!!
Another area of active use of fractals is gastroenterology.
A new research method in medicine, electrogastroenterography is a research method that allows you to evaluate the bioelectrical activity of the stomach, duodenum and other parts of the gastrointestinal tract.
Fractals in architecture.
The fractal principle of the development of natural and geometric objects penetrates deep into architecture both as an image of the external solution of the object, and as an internal principle of architectural form formation.
Designers from all over the world started use in your work remarkable fractal structures, only recently described by prominent mathematicians.
The use of fractals has brought almost all areas of modern design to a new level.
The introduction of fractal structures has increased both the visual and functional components of the design in many cases.
Designer Takeshi Miyakawa dreamed of becoming a mathematician as a child.
How else can we explain this piece of furniture: the Fractal 23 bedside table contains 23 drawers of various sizes and proportions, which somehow manage to coexist with each other inside the cubic body, filling almost all the space available to them.
Fractals in economics.
Recently, fractals have become popular among economists for analyzing stock exchange rates, currency and trading markets.
Fractals appear on the market quite often.
Fractals in games.
Today, many games (perhaps the most striking example of Minecraft), where various kinds of natural landscapes are present, use fractal algorithms in one way or another. A large number of programs have been created for generating landscapes and landscapes based on fractal algorithms.
Fractals in cinema.
In cinema, a fractal algorithm is used to create various fantastic landscapes. Fractal geometry allows special effects artists to easily create objects such as clouds, smoke, flames, starry skies, etc. What then can we say about fractal animation, it is truly an amazing sight.
Electonic music.
The spectacle of fractal animation is successfully used by VJs. Such video installations are especially often used at concerts of electronic music performers.
Natural Sciences.
Fractals are often used in geology and geophysics. It is no secret that the coasts of islands and continents have a certain fractal dimension, knowing which one can very accurately calculate the lengths of the coasts.
The study of fault tectonics and seismicity is sometimes also studied using fractal algorithms.
Geophysics uses fractals and fractal analysis to study magnetic field anomalies, to study the propagation of waves and oscillations in elastic media, to study climate and many other things.
Fractals in physics.
In physics, fractals are used very widely. In solid state physics, fractal algorithms make it possible to accurately describe and predict the properties of solid, porous, spongy bodies, and aerogels. This helps in the creation of new materials with unusual and useful properties.
An example of a solid is crystals.
The study of turbulence in flows adapts very well to fractals.
The transition to a fractal representation makes the work of engineers and physicists easier, allowing them to better understand the dynamics of complex systems.
Using fractals you can also simulate flames.
Fractals in biology.
In biology, they are used to model populations and to describe internal organ systems (the blood vessel system). After the creation of the Koch curve, it was proposed to use it when calculating the length of the coastline.
Fractals for housewives.
It is easy to transfer the theory of fractals to the home, including the kitchen.
The result of application can be anything: fractal earrings, fractal tasty liver and much more. You only need to use knowledge and ingenuity!
Fractal graphics are widely used in the modern world. The paintings are popular - the result of fractal graphics.
And this is no coincidence. Admire the beauty of fractal graphics!
Practical part of the project
- Created a scientific work “Journey to the World of Fractals”
- We studied programs for creating fractals in the Pascal and Logo programming languages.
- Created your own fractals.
- We made our own “Sierpinski Napkin” and “Sierpinski Carpet”
- Made “Fractal earrings”
- Created a series of paintings “Miracles of Fractal Graphics”
- Published the work “Journey to the World of Fractals” on the Internet.
- We took part with the work “Journey to the World of Fractals” in the VII All-Russian Olympiad for schoolchildren and students “Science 2.0” in the academic subject “Mathematics”. We took first place.
- We took part in the All-Russian competition “Great Discoveries and Inventions” with the work “Journey to the World of Fractals”. We took first place.
- We took part with the work “Journey to the World of Fractals” in the VIII All-Russian Olympiad for schoolchildren and students “I am a researcher” in the academic subject of Mathematics. We took first place.
- Created a presentation “The Amazing World of Fractals”
- Created brochures “Using Fractals” and “Fractals Around Us”
- We held the festival “The Amazing World of Fractals” for students in grades 8-11.”
So, we can say with complete confidence about the enormous practical application of fractals and fractal algorithms today.
The range of areas where fractals are used is very extensive and diverse.
And for sure, in the near future, fractals, fractal geometry, will become close and understandable to each of us. We cannot live without them in our lives!
Let's hope that the emergence of fractal geometry is evidence of the ongoing evolution of man and the expansion of his ways of knowing and understanding the world. Perhaps our children will also easily and meaningfully operate with the concepts of fractals and nonlinear dynamics, as we operate with the concepts of classical physics and Euclidean geometry.
Results of the project
- We got acquainted with the history of the emergence and development of fractal geometry;
- We studied the types of fractals and their application in the modern world.
- Created our own fractals in the Pascal and Logo programming languages
- Created a scientific paper on fractals.
- Created brochures “Fractals around us” and “Use of fractals”
- We held the festival “The Amazing World of Fractals” for students in grades 8-11.
Fractal(eng. fractal) is a geometric figure with a fractional dimension, which has the property of recursiveness (each part of the fractal is a reduced copy of the whole structure). Price fluctuations on charts of different scales can also be considered as fractals.
According to Bill Williams, a fractal is a graphical pattern consisting of 5 candles (bars), the central one of which is characterized by the highest maximum (fractal up) or the lowest minimum (fractal down). It should be noted that fractals are used in almost all areas of science, for example, in computer modeling of physical structures that do not have simple geometric patterns (mountain landscapes, clouds, coastlines, etc.).
Basic properties of fractals:
- Fractional dimension;
- Self-similarity form with approximate form assumption;
- Irregularity, which allows them to be described in traditional geometric language;
- Fine structure (content of arbitrarily small scales).
Theory and construction of fractals in the Forex market
In classical technical analysis, a fractal is a figure that consists of five bars. On a price chart, fractals are indicated as icons above exchange rate bars. A lower fractal (marks under the price bar) is a structure (bar) in which two subsequent and previous Low values are equal or higher (see Fig. 1). The up fractal is the structure with the highest price and is critical for entering and exiting the market (see Fig. 2).
(Fig. 1 – Fractal down)
(Fig. 2 – Fractal up)
B. Williams Fractal
It should be noted that in fact the Bill Williams fractal is not one, since it is capable of identifying only extreme points (see Fig. 3). A fractal has a remarkable property of completeness, causing a dynamic transition from one structure to another. However, it is impossible to describe the entire variety of wave oscillations with the B. Williams fractal.
Rice. 3 – Bill Williams Fractal
Trading using fractals
Like most trading indicators, fractals are recommended to be used in conjunction with other analysis methods and indicators. The most common confirmation of a fractal is the Alligator indicator.
Features of using fractals:
- The higher the timeframe where the fractal search occurs, the more reliable the signal will be. However, it should be taken into account that the longer the time period, the smaller the number of trading signals;
- Fractal is a lagging indicator, it is better to use it as confirmation of other indicators and indicators;
- Fractals are recommended to be used in combination with each other and on several timeframes at once;
- Fractals should be used in combination with other system indicators, as they are more effective as decision support rather than used alone.
Provides its services in most small towns and villages of the Kharkov region. In addition to Internet access, on their website you can order video surveillance services, satellite TV, intercoms, etc.
But I’ll tell you specifically about the Internet. I have been using the services for a year and a half now and am completely satisfied with the quality.
At the moment they have 3 tariffs:
8 Mbit/s for 100 UAH/month (private houses) and 50 UAH (apartment buildings)
25 Mbit/s for 150 UAH/month (private) and 70 UAH (apartments)
110 Mbit/s for 200 UAH/month (private) and 90 UAH (apartments).
As you can immediately notice, the prices for private houses are much more expensive than those of other providers in big cities (!), but if you compare with the same Ukrtelecom, the price is much cheaper in terms of quality and affordable speed for the same money in small cities.
I would also like to draw your attention to the fact that in our city, Fractal is the fastest Internet.
I use a 110 Mbit tariff. The speed is fully consistent with the declared speed, never sags.
Stability and speed
Below are Speedtest speed measurements and torrent downloads. I would like to warn you that measurements are taken via Wi-Fi connection and the speed is slightly reduced. If you connect your computer via twisted pair cable, the torrent download speed increases to approximately 12.5Mb/s. (the maximum speed I achieved was 13.2Mb/sec).
There are almost no access interruptions, and if they do happen, it is only for some reason. For example, when support networks are damaged. Everything is being restored promptly, but there was one case when, after heavy snowfall and icing, the entire city was left without electricity and, of course, without the Internet - then everything was restored in about 15 days, although many other clients were luckier, they were connected earlier (extensive damage + strong frosts).
Personal Area
It is possible to block the balance if, for example, you are leaving somewhere. In your personal account, all expenses and top-ups are listed from the very beginning of connection.
You can top up both at self-service terminals and by non-cash methods using a bank card or Webmoney.
Support
About those. support the first time using the impressions were not very good, because... it was almost impossible to get through. But over time, communication showed that they seemed to have improved.
Connection
In most cases, clients are connected in batches, i.e. everyone at once, as soon as a certain number of people are recruited in the area. However, if there is already a switch near you, then you won’t have to wait.
The cost of connection was 2500 UAH, now the price may have changed.
In general, if in your locality it is possible to connect to this provider, you can safely do so. I'm glad I ran away from the old provider called ***telecom, where daily connection dropouts are a common thing that they don't care about. As far as I know, the Fractal network is currently being actively expanded.
