Fractals in nature
Fractal(lat. fractus- crushed) is a term meaning a geometric figure that has the property of self-similarity, that is, composed of several parts, each of which is similar to the entire figure.Nature often creates amazing and beautiful fractals, with ideal geometry and such harmony that you simply freeze with admiration.
From giant mountains to what we eat for lunch, perfect harmony can be seen everywhere.
Sea shells
Nautilus is one of the most famous examples of a fractal in nature.
Snowflakes
Lightning
Lightning terrifies and frightens and at the same time delights with its beauty. Fractals created by lightning are not arbitrary or regular.
Romanessa
This special type of broccoli, a cruciferous and tasty cousin of cabbage, is a particularly symmetrical fractal. You can prepare it for your favorite math teacher.
Fern
Fern is a good example of a fractal among flora
Queen Anne's Lace
Queen Anne's Lace wild carrot is a perfect example of a floral fractal. Each constellation is copied exactly the same, only smaller. The photo was taken from below to see it in all its glory.
Broccoli
Although broccoli is not as famously geometric as romanessa, it is also fractal.
Peacock
Peacocks are known to everyone for their colorful plumage, in which solid fractals are hidden. Have you ever seen an albino peacock? Look
A pineapple
Pineapple is an unusual fruit; it is, in fact, a fractal. Although it is often associated with Hawaii, the fruit is native to southern Brazil.
Clouds
Look out the window now. Almost at any moment you can see fractals in the sky.
Crystals
Ice, frosty patterns on windows are also fractals
Mountains
Mountain crevices and coastlines, although arbitrary in their lines, are also fractal
Trees and leaves
From an enlarged image of a leaf to the branches of a tree - fractals can be found in everything
Coastline
Individual fragments of the coast create fractality. And this is Florida
Rivers and fjords
From the western United States of America to the icy fjords of Norway, airline passengers can see it all. And we thank some for having the courage to photograph such beauty.
Sea urchins and starfish
Sea urchins are so small and compact, as if they came from the hand of a skilled jeweler. But who will surpass nature? And starfish are like a reflection of the heavenly ones
Stalagmites and stalactites
While stalagmites rise from the ground, stalactites reach towards it
How the fractal was discovered
The mathematical shapes known as fractals originate from the genius of the eminent scientist Benoit Mandelbrot. For most of his life he taught mathematics at Yale University in the USA. In 1977 - 1982, Mandelbrot published scientific works devoted to the study of “fractal geometry” or “geometry of nature”, in which he broke down seemingly random mathematical forms into component elements that, on closer examination, turned out to be repeating - which proved the presence of a certain model for copying . Mandelbrot's discovery had significant consequences in the development of physics, astronomy and biology.
Fractals in nature
In nature, many objects have fractal properties, for example: tree crowns, cauliflower, clouds, the circulatory and alveolar systems of humans and animals, crystals, snowflakes, the elements of which are arranged into one complex structure, coastlines (the fractal concept allowed scientists to measure the coastline of the British Isles and other previously unmeasurable objects).
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. In this simplest case, even a small part of the fractal contains information about the entire final structure.
Fractals in digital technology
Fractal geometry has made an invaluable contribution to the development of new technologies in the field of digital music, and also made it possible to compress digital images. Existing fractal image compression algorithms are based on the principle of storing a compressed image instead of the digital image itself. For a compressed image, the main image remains a fixed point. Microsoft used one of the variants of this algorithm when publishing its encyclopedia, but for one reason or another this idea was not widely used.
The mathematical basis of fractal graphics is fractal geometry, where the principle of inheritance from the original “parent objects” is the basis for the methods for constructing “heir images”. The very concepts of fractal geometry and fractal graphics appeared only about 30 years ago, but have already become firmly established in the everyday life of computer designers and mathematicians.
The basic concepts of fractal computer graphics are:
- Fractal triangle - fractal figure - fractal object (hierarchy in descending order)
- Fractal line
- Fractal composition
- “Parent object” and “Successor object”
Just like in vector and three-dimensional graphics, the creation of fractal images is mathematically calculated. The main difference from the first two types of graphics is that a fractal image is built according to an equation or system of equations - you don’t need to store anything other than the formula in the computer’s memory to perform all the calculations - and this compactness of the mathematical apparatus allowed the use of this idea in computer graphics. Simply by changing the coefficients of the equation, you can easily get a completely different fractal image - using several mathematical coefficients, surfaces and lines of very complex shapes are specified, which allows you to implement composition techniques such as horizontals and verticals, symmetry and asymmetry, diagonal directions and much more.
How to build a fractal?
The creator of fractals plays the role of an artist, photographer, sculptor, and scientist-inventor at the same time. What are the upcoming stages of creating a drawing from scratch?
- set the shape of the drawing using a mathematical formula
- investigate the convergence of the process and vary its parameters
- select image type
- choose a color palette
Among fractal graphic editors and other graphic programs we can highlight:
- "Art Dabbler"
- “Painter” (without a computer, no artist will ever achieve the capabilities laid down by programmers only through a pencil and a brush pen)
- “Adobe Photoshop” (but here the image is not created “from scratch”, but, as a rule, only processed)
Let us consider the structure of an arbitrary fractal geometric figure. In its center there is the simplest element - an equilateral triangle, which received the same name: “fractal”. On the middle segment of the sides, we will construct equilateral triangles with a side equal to one third of the side of the original fractal triangle. Using the same principle, even smaller successor triangles of the second generation are built - and so on ad infinitum. The resulting object is called a “fractal figure”, from the sequences of which we obtain a “fractal composition”.
Source: http://www.iknowit.ru/
Fractals and ancient mandalas
In principle, a mandala is a geometric symbol of a complex structure, which is interpreted as a model of the Universe, a “map of the cosmos.” This is the first sign of fractality!
They are embroidered on fabric, painted on sand, made with colored powders and made of metal, stone, wood. Its bright and mesmerizing appearance makes it a beautiful decoration for the floors, walls and ceilings of temples in India. In the ancient Indian language, “mandala” means the mystical circle of the relationship between the spiritual and material energies of the Universe, or in other words, the flower of life.
I wanted to write a very short review of fractal mandalas, with a minimum of paragraphs, showing that the relationship clearly exists. However, trying to understand and connect information about fractals and mandalas into a single whole, I had the feeling of a quantum leap into a space unknown to me.
I demonstrate the immensity of this topic with a quote: “Such fractal compositions or mandalas can be used in the form of paintings, design elements for living and working spaces, wearable amulets, in the form of videotapes, computer programs...” In general, the topic for the study of fractals is simply enormous.
One thing I can say for sure is that the world is much more diverse and richer than the poor ideas our minds have about it.
Fractal sea animals
My guesses about fractal sea animals were not groundless. Here are the first representatives. An octopus is a bottom-dwelling sea animal from the order of cephalopods.
Looking at this photo, the fractal structure of its body and suckers on all eight tentacles of this animal became obvious to me. The number of suckers on the tentacles of an adult octopus reaches up to 2000.
An interesting fact is that the octopus has three hearts: one (the main one) drives blue blood throughout the body, and the other two - gills - push the blood through the gills. Some types of these deep-sea fractals are poisonous.
By adapting and camouflaging itself to its environment, the octopus has the very useful ability to change color.
Octopuses are considered the most “smart” of all invertebrates. They get to know people and get used to those who feed them. It would be interesting to look at octopuses that are easy to train, have a good memory and even recognize geometric shapes. But the lifespan of these fractal animals is short - a maximum of 4 years.
Man uses the ink of this living fractal and other cephalopods. They are sought after by artists for their durability and beautiful brown tone. In Mediterranean cuisine, octopus is a source of vitamins B3, B12, potassium, phosphorus and selenium. But I think that you need to know how to cook these sea fractals in order to enjoy eating them as food.
By the way, it should be noted that octopuses are predators. With their fractal tentacles they hold prey in the form of mollusks, crustaceans and fish. It’s a pity if such a beautiful mollusk becomes the food of these sea fractals. In my opinion, he is also a typical representative of the fractals of the sea kingdom.
This is a relative of snails, the gastropod nudibranch Glaucus, also known as Glaucus, also known as Glaucus atlanticus, also known as Glaucilla marginata. This fractal is also unusual in that it lives and moves under the surface of the water, being held in place by surface tension. Because the mollusk is a hermaphrodite, then after mating both “partners” lay eggs. This fractal is found in all oceans of the tropical zone.
Fractals of the sea kingdom
Each of us, at least once in our lives, held a sea shell in our hands and examined it with genuine childish interest.
Usually shells are a beautiful souvenir reminiscent of a trip to the sea. When you look at this spiral formation of invertebrate molluscs, there is no doubt about its fractal nature.
We humans are somewhat like these soft-bodied mollusks, living in well-appointed concrete fractal houses, placing and moving our bodies in fast cars.
Another typical representative of the fractal underwater world is coral.
There are over 3,500 varieties of corals known in nature, with a palette of up to 350 color shades.
Coral is the skeletal material of a colony of coral polyps, also from the invertebrate family. Their huge accumulations form entire coral reefs, the fractal method of formation of which is obvious.
Coral can with full confidence be called a fractal from the sea kingdom.
It is also used by humans as a souvenir or raw material for jewelry and ornaments. But it is very difficult to replicate the beauty and perfection of fractal nature.
Once again, performing the ritual in the kitchen with a knife and cutting board, and then, dipping the knife into cold water, I was in tears and once again figured out how to deal with the tear fractal that appears before my eyes almost every day.
The principle of fractality is the same as that of the famous nesting doll - nesting. This is why fractality is not immediately noticed. In addition, the light, uniform color and its natural ability to cause unpleasant sensations do not contribute to close observation of the universe and the identification of fractal mathematical patterns.
But the lilac-colored salad onion, due to its color and the absence of tear-producing phytoncides, made me think about the natural fractality of this vegetable. Of course, it is a simple fractal, ordinary circles of different diameters, one might even say the most primitive fractal. But it would not hurt to remember that the ball is considered an ideal geometric figure within our Universe.
Many articles have been published on the Internet about the beneficial properties of onions, but somehow no one has tried to study this natural specimen from the point of view of fractality. I can only state the usefulness of using a fractal in the form of an onion in my kitchen.
P.S. I have already purchased a vegetable cutter for chopping fractals. Now we have to think about how fractal such a healthy vegetable as ordinary white cabbage is. The same principle of nesting.
Fractals in folk art
The story of the world famous Matryoshka toy caught my attention. Taking a closer look, we can say with confidence that this souvenir toy is 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.
My small research into the history of the appearance of this toy fractal on the world market showed that the roots of this beauty are Japanese. The matryoshka doll has always been considered an original Russian souvenir. But it turned out that she was the prototype of the Japanese figurine of the old sage Fukuruma, once brought to Moscow from Japan.
But it was the Russian toy industry that brought this Japanese figurine world fame. Where the idea of fractal nesting of a toy came from remains a mystery to me personally. Most likely, the author of this toy used the principle of nesting figures inside each other. And the easiest way to invest is similar figures of different sizes, and this is already a fractal.
An equally interesting object of study is the painting of a fractal toy. This is a decorative painting - Khokhloma. Traditional elements of Khokhloma are herbal patterns of flowers, berries and branches.
Again all signs of fractality. After all, the same element can be repeated several times in different versions and proportions. The result is a folk fractal painting.
And if you won’t surprise anyone with the newfangled painting of computer mice, laptop covers and phones, then fractal tuning of a car in a folk style is something new in auto design. One can only be amazed at the manifestation of the world of fractals in our lives in such an unusual way in such ordinary things for us.
Fractals in the kitchen
A typical representative of a fractal from the plant world was on my kitchen table.
With all my love for cauliflower, I always came across specimens with a uniform surface without visible signs of fractality, and even a large number of inflorescences nested within each other did not give me a reason to see a fractal in this useful vegetable.
But the surface of this particular specimen with its clearly defined fractal geometry did not leave the slightest doubt about the fractal origin of this type of cabbage.
Another trip to the hypermarket only confirmed the fractal status of cabbage. Among the huge number of exotic vegetables was a whole box of fractals. It was Romanescu, or Romanesque broccoli, cauliflower.
It turns out that designers and 3D artists admire its exotic fractal-like shapes.
Cabbage buds grow in a logarithmic spiral. The first mention of Romanescu cabbage came from Italy in the 16th century.
And brocolli cabbage is not a frequent guest in my diet, although it is many times superior to cauliflower in terms of the content of nutrients and microelements. But its surface and shape are so uniform that it never occurred to me to see a vegetable fractal in it.
Fractals in quilling
Having seen openwork crafts using the quilling technique, I never lost the feeling that they reminded me of something. The repetition of the same elements in different sizes is, of course, the principle of fractality.
After watching another master class on quilling, there was no longer any doubt about the fractal nature of quilling. After all, to make various elements for quilling crafts, a special ruler with circles of different diameters is used. Despite all the beauty and uniqueness of the products, this is an incredibly simple technique.
Almost all the main elements for quilling crafts are made from paper. To stock up on free quilling paper, take a look at your bookshelves at home. Surely, you will find a couple of bright glossy magazines there.
Quilling tools are simple and inexpensive. Everything you need to perform amateur quilling work can be found among your home stationery supplies.
And the history of quilling begins in the 18th century in Europe. During the Renaissance, monks from French and Italian monasteries used quilling to decorate book covers and were not even aware of the fractal nature of the paper-rolling technique they had invented. Girls from high society even took quilling courses in special schools. This is how this technique began to spread across countries and continents.
This video quilling master class on making luxurious plumage can even be called “do-it-yourself fractals.” With the help of paper fractals, wonderful exclusive Valentine cards and many other interesting things are obtained. After all, fantasy, like nature, is inexhaustible.
It's no secret that the Japanese are very limited in space in life, and therefore they have to try their best to use it effectively. Takeshi Miyakawa shows how this can be done both effectively and aesthetically. His fractal cabinet confirms that the use of fractals in design is not only a tribute to fashion, but also a harmonious design solution in conditions of limited space.
This example of using fractals in real life, in relation to furniture design, showed me that fractals are real not only on paper in mathematical formulas and computer programs.
And it seems that nature uses the principle of fractality everywhere. You just need to take a closer look at it, and it will manifest itself in all its magnificent abundance and infinity of being.
Completed by 7th grade student Polina Karpyuk
Prioda is created from self-similar figures, we just don’t notice it. In this gallery we have collected images in which fractality is clearly visible.
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Fractals in nature Completed by: 7th “B” class student Polina Karpyuk Supervisor: Molchanova Irina Pavlovna Rubtsovsk-2015
Mathematics, when looked at correctly, reflects not only truth, but also incomparable beauty. Bertrand Russell
What do a tree, a seashore, a cloud, or the blood vessels in our hand have in common? There is one property of structure that is inherent in all of the listed objects: they are self-similar. From a branch, as from a tree trunk, smaller shoots extend, from them even smaller ones, etc., that is, a branch is similar to the whole tree. The circulatory system is structured in a similar way: arterioles depart from the arteries, and from them the smallest capillaries through which oxygen enters the organs and tissues. The American mathematician Benoit Mandelbrot called this property of objects fractality, and such objects themselves - fractals. The word “fractal” itself is translated from Latin as “partial”, “divided”, “fragmented”, and as for the content of this term, there is no formulation as such. It is usually interpreted as a self-similar set, a part of the whole, which repeats its structure at the micro level. .
Space photographs of Earth's landscapes often provide excellent examples of fractals.
Coastlines usually have a fractal shape, but vary in the degree to which they are rugged. This example shows two characteristic properties of natural fractals: Individual channels are not a copy of each other, but have similar curvilinear outlines, as if they were drawn by the same pattern. Large ducts are similar in outline to small and very small ducts. If we enlarge, for example, the lower left corner of the picture, we will get something similar to the entire picture
The interaction of water and land gives rise to fractal structures in landscapes - be they mountains, rivers or coastlines.
Probably everyone knows the painting by the Japanese artist Hokusai “The Great Wave”, where a tsunami wave is depicted against the backdrop of Fuji. If you look closely at this picture, you will notice that when drawing the crest of a wave, the artist used a fractal, as if consisting of numerous predatory water paws. Therefore, this picture is often used as an illustration for books on chaos theory and fractals.
When a sand dune is eroded by water, it replicates on a tiny scale what gives fractal shape to larger Earth landscapes.
Lightning discharge is one example of natural fractals.
This picture illustrates not only the fractal nature of tree crowns, it suggests another interesting consideration: the forest as a biological community is also a fractal. Individual trees - large and small - then act as branches of the fractal. They are similar, but do not repeat each other.
Leaf veins are a flat natural fractal. For each plant, the characteristic pattern is unique, just as the papillary pattern on a person’s hand is unique. Goethe (poet and scientist) believed that the leaf is the most expressive part of the plant, which reflects its entire morphology.
Ferns are an example of natural fractals that are very similar to computer fractals. Moreover, they are also interesting because ferns are one of the most evolutionarily ancient plants, along with various mosses and other lower plants
This is another famous and very impressive example of a natural fractal that has mathematically clear forms. There are at least three levels of self-similar ingenious pyramids Romanesco cabbage
A magically beautiful fractal that could well inspire some artist. Meanwhile, take a closer look: this is just a tight bunch of cabbage leaves.
These are interesting examples of fractal structure in the mineral world. Carbonate Apatite Gold Nugget is an exquisite treasure crafted by nature itself.
Have you ever thought that we literally think in fractals? There is something to think about here - who would argue that the brain is one of the most amazing and unique creations of nature. And it turns out that outwardly it has the same fractal features as atmospheric clouds or the root system of nettles.
Here everything is even more complicated: two separate fractal trees are intertwined - venous blood is supplied to one, and oxygen-enriched arterial blood is discharged to the other. And in totality, the lung is an amazingly complex system of three fractals - one respiratory and two circulatory.
The retina contains light-sensitive cells that allow us to see. In this photo they are yellowish-green. They do form a network (the retina), but this network is chaotic and fractal.
This is the belly of a pig. Its color patterns also seem to follow fractal rules. This is an interesting topic and, most importantly, has many applications, including military significance. By what rules should a camouflage pattern be drawn up so that its wearer blends with natural forms - landscape and vegetation?
Thank you for your attention!!!
In order to understand what a fractal is, we should start the debriefing from the position of mathematics, but before delving into the exact sciences, we will philosophize a little. Every person has a natural curiosity, thanks to which he learns about the world around him. Often, in his quest for knowledge, he tries to use logic in his judgments. Thus, by analyzing the processes that occur around him, he tries to calculate relationships and derive certain patterns. The greatest minds on the planet are busy solving these problems. Roughly speaking, our scientists are looking for patterns where there are none, and there should not be any. And yet, even in chaos there is a connection between certain events. This connection is what the fractal is. As an example, consider a broken branch lying on the road. If we look closely at it, we will see that with all its branches and twigs it itself looks like a tree. This similarity of a separate part with a single whole indicates the so-called principle of recursive self-similarity. Fractals can be found all over the place in nature, because many inorganic and organic forms are formed in a similar way. These are clouds, sea shells, snail shells, tree crowns, and even the circulatory system. This list can be continued indefinitely. All these random shapes are easily described by a fractal algorithm. Now we have come to consider what a fractal is from the perspective of exact sciences. 
Some dry facts
The word “fractal” itself is translated from Latin as “partial”, “divided”, “fragmented”, and as for the content of this term, there is no formulation as such. It is usually interpreted as a self-similar set, a part of the whole, which repeats its structure at the micro level. This term was coined in the seventies of the twentieth century by Benoit Mandelbrot, who is recognized as the father of fractal geometry. Today, the concept of a fractal means a graphic image of a certain structure, which, when scaled up, will be similar to itself. However, the mathematical basis for the creation of this theory was laid even before the birth of Mandelbrot himself, but it could not develop until electronic computers appeared.
Historical background, or How it all began
At the turn of the 19th and 20th centuries, the study of the nature of fractals was sporadic. This is explained by the fact that mathematicians preferred to study objects that could be researched on the basis of general theories and methods. In 1872, the German mathematician K. Weierstrass constructed an example of a continuous function that is not differentiable anywhere. However, this construction turned out to be entirely abstract and difficult to perceive. Next came the Swede Helge von Koch, who in 1904 constructed a continuous curve that had no tangent anywhere. It's fairly easy to draw and turns out to have fractal properties. One of the variants of this curve was named after its author - “Koch snowflake”. Further, the idea of self-similarity of figures was developed by the future mentor of B. Mandelbrot, the Frenchman Paul Levy. In 1938, he published the article "Plane and spatial curves and surfaces consisting of parts similar to the whole." In it, he described a new type - the Levy C-curve. All of the above figures are conventionally classified as geometric fractals. 
Dynamic or algebraic fractals
The Mandelbrot set belongs to this class. The first researchers in this direction were the French mathematicians Pierre Fatou and Gaston Julia. In 1918, Julia published a paper based on the study of iterations of rational complex functions. Here he described a family of fractals that are closely related to the Mandelbrot set. Despite the fact that this work glorified the author among mathematicians, it was quickly forgotten. And only half a century later, thanks to computers, Julia’s work received a second life. Computers made it possible to make visible to every person the beauty and richness of the world of fractals that mathematicians could “see” by displaying them through functions. Mandelbrot was the first to use a computer to carry out calculations (such a volume cannot be done manually) that made it possible to construct an image of these figures.
A person with spatial imagination
Mandelbrot began his scientific career at IBM Research Center. While studying the possibilities of transmitting data over long distances, scientists were faced with the fact of large losses that arose due to noise interference. Benoit was looking for ways to solve this problem. Looking through the measurement results, he noticed a strange pattern, namely: the noise graphs looked the same on different time scales. 
A similar picture was observed both for a period of one day and for seven days or for an hour. Benoit Mandelbrot himself often repeated that he does not work with formulas, but plays with pictures. This scientist was distinguished by imaginative thinking; he translated any algebraic problem into the geometric area, where the correct answer is obvious. So it is not surprising that such a person, distinguished by rich spatial thinking, became the father of fractal geometry. After all, awareness of this figure can only come when you study the drawings and think about the meaning of these strange swirls that form the pattern. Fractal patterns do not have identical elements, but they are similar at any scale.
Julia–Mandelbrot
One of the first drawings of this figure was a graphic interpretation of the set, which was born out of the work of Gaston Julia and was further developed by Mandelbrot. Gaston tried to imagine what a set would look like based on a simple formula that was iterated through a feedback loop. Let's try to explain what has been said in human language, so to speak, on the fingers. For a specific numerical value, we find a new value using a formula. We substitute it into the formula and find the following. The result is a large number sequence. To represent such a set, it is necessary to perform this operation a huge number of times: hundreds, thousands, millions. This is what Benoit did. He processed the sequence and transferred the results to graphical form. Subsequently, he colored the resulting figure (each color corresponds to a certain number of iterations). This graphic image was named “Mandelbrot fractal”. 
L. Carpenter: art created by nature
The theory of fractals quickly found practical application. Since it is very closely related to the visualization of self-similar images, artists were the first to adopt the principles and algorithms for constructing these unusual forms. The first of them was the future founder of Pixar, Lauren Carpenter. While working on a presentation of aircraft prototypes, he came up with the idea of using an image of mountains as a background. Today, almost every computer user can cope with such a task, but in the seventies of the last century, computers were not able to perform such processes, because there were no graphic editors or applications for three-dimensional graphics at that time. And then Loren came across Mandelbrot’s book “Fractals: Form, Randomness and Dimension.” In it, Benoit gave many examples, showing that fractals exist in nature (fyva), he described their varied shapes and proved that they are easily described by mathematical expressions. The mathematician cited this analogy as an argument for the usefulness of the theory he was developing in response to a barrage of criticism from his colleagues. They argued that a fractal is just a pretty picture, has no value, and is a by-product of the work of electronic machines. Carpenter decided to try this method in practice. After carefully studying the book, the future animator began to look for a way to implement fractal geometry in computer graphics. It took him only three days to render a completely realistic image of the mountain landscape on his computer. And today this principle is widely used. As it turns out, creating fractals does not take much time and effort.
Carpenter's solution
The principle Lauren used was simple. It consists of dividing larger geometric shapes into small elements, and those into similar smaller ones, and so on. Carpenter, using large triangles, split them into 4 small ones, and so on, until he had a realistic mountain landscape. Thus, he became the first artist to use a fractal algorithm in computer graphics to construct the required image. Today this principle is used to imitate various realistic natural forms.
The first 3D visualization using a fractal algorithm
A few years later, Lauren applied his work in a large-scale project - the animated video Vol Libre, shown on Siggraph in 1980. This video shocked many, and its creator was invited to work at Lucasfilm. Here the animator was able to realize his full potential; he created three-dimensional landscapes (an entire planet) for the feature film “Star Trek”. Any modern program (“Fractals”) or application for creating 3D graphics (Terragen, Vue, Bryce) uses the same algorithm for modeling textures and surfaces. 
Tom Beddard
Formerly a laser physicist and now a digital artist and artist, Beddard created a number of very intriguing geometric shapes, which he called Fabergé fractals. Outwardly, they resemble decorative eggs from a Russian jeweler; they have the same brilliant, intricate pattern. Beddard used a template method to create his digital renderings of the models. The resulting products amaze with their beauty. Although many refuse to compare a handmade product with a computer program, it must be admitted that the resulting forms are extremely beautiful. The highlight is that anyone can build such a fractal using the WebGL software library. It allows you to explore various fractal structures in real time.
Fractals in nature
Few people pay attention, but these amazing figures are present everywhere. Nature is created from self-similar figures, we just don’t notice it. It is enough to look through a magnifying glass at our skin or a leaf of a tree, and we will see fractals. Or take, for example, a pineapple or even a peacock’s tail - they consist of similar figures. And the Romanescu broccoli variety is generally striking in its appearance, because it can truly be called a miracle of nature. 
Musical pause
It turns out that fractals are not only geometric shapes, they can also be sounds. Thus, musician Jonathan Colton writes music using fractal algorithms. He claims that such a melody corresponds to natural harmony. The composer publishes all of his works under a CreativeCommons Attribution-Noncommercial license, which provides for free distribution, copying, and transfer of works to others.
Fractal indicator
This technique has found a very unexpected application. On its basis, a tool for analyzing the stock exchange market was created, and, as a result, it began to be used in the Forex market. Nowadays, the fractal indicator is found on all trading platforms and is used in a trading technique called price breakout. This technique was developed by Bill Williams. As the author comments on his invention, this algorithm is a combination of several “candles”, in which the central one reflects the maximum or, conversely, the minimum extreme point.
Finally
So we looked at what a fractal is. It turns out that in the chaos that surrounds us, there actually exist ideal forms. Nature is the best architect, ideal builder and engineer. It is arranged very logically, and if we cannot find a pattern, this does not mean that it does not exist. Maybe we need to look on a different scale. We can say with confidence that fractals still hold many secrets that we have yet to discover.
What do a tree, a seashore, a cloud, or the blood vessels in our hand have in common? There is one property of structure that is inherent in all of the listed objects: they are self-similar. From a branch, as from a tree trunk, smaller shoots extend, from them even smaller ones, etc., that is, a branch is similar to the whole tree. The circulatory system is structured in a similar way: arterioles depart from the arteries, and from them the smallest capillaries through which oxygen enters the organs and tissues. Let's look at satellite images of the sea coast: we will see bays and peninsulas; Let's look at it, but from a bird's eye view: we will see bays and capes; Now imagine that we are standing on the beach and looking at our feet: there will always be pebbles that protrude further into the water than the rest. That is, the coastline, when zoomed in, remains similar to itself. The American (although he grew up in France) mathematician Benoit Mandelbrot called this property of objects fractality, and such objects themselves - fractals (from the Latin fractus - broken).
There is an interesting story connected with the coastline, or more precisely, with the attempt to measure its length, which formed the basis of Mandelbrot’s scientific article, and is also described in his book “The Fractal Geometry of Nature.” We are talking about an experiment carried out by Lewis Fry Richardson, a very talented and eccentric mathematician, physicist and meteorologist. One of the directions of his research was an attempt to find a mathematical description of the causes and likelihood of an armed conflict between two countries. Among the parameters that he took into account was the length of the common border of the two warring countries. When he collected data for numerical experiments, he discovered that data on the common border of Spain and Portugal differed greatly from different sources. This led him to the following discovery: the length of a country's borders depends on the ruler with which we measure them. The smaller the scale, the longer the border. This is due to the fact that with greater magnification it becomes possible to take into account more and more new bends of the coast, which were previously ignored due to the coarseness of the measurements. And if, with each increase in scale, previously unaccounted for bends of lines are revealed, then it turns out that the length of the boundaries is infinite! True, this does not actually happen - the accuracy of our measurements has a finite limit. This paradox is called the Richardson effect.
Nowadays, the theory of fractals is widely used in various areas of human activity. In addition to fractal painting, fractals are used in information theory to compress graphic data (the property of self-similarity of fractals is mainly used here - after all, to remember a small fragment of a picture and the transformations with which you can obtain the remaining parts, much less memory is required than to store the entire file). By adding random disturbances to the formulas that define a fractal, you can obtain stochastic fractals that very plausibly convey some real objects - relief elements, the surface of reservoirs, some plants, which is successfully used in physics, geography and computer graphics to achieve greater similarity of simulated objects with real. In radio electronics, in the last decade, antennas with a fractal shape began to be produced. Taking up little space, they provide high-quality signal reception. And economists use fractals to describe currency rate fluctuation curves (this property was discovered by Mandelbrot more than 30 years ago).
