Often, brilliant discoveries made in science can radically change our lives. For example, the invention of a vaccine can save many people, but the creation of new weapons leads to murder. Literally yesterday (on the scale of history) man “tamed” electricity, and today he can no longer imagine his life without it. However, there are also discoveries that, as they say, remain in the shadows, despite the fact that they also have one or another impact on our lives. One of these discoveries was the fractal. Most people have never even heard of this concept and will not be able to explain its meaning. In this article we will try to understand the question of what a fractal is and consider the meaning of this term from the perspective of science and nature.
Order in chaos
In order to understand what a fractal is, we should begin the debriefing from the position of mathematics, but before delving into it, 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. Today, the concept of 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 Lewy 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 he is distinguished by his wealth and 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 large. 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 ones 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 developments 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. It claims to correspond 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.
Nature is a perfect creation, scientists are convinced, who discover the proportions of the golden section in the structure of the human body, and fractal figures in the head of a cauliflower.
“The study and observation of nature gave birth to science,” wrote Cicero in the first century BC. In later times, with the development of science and its distance from the study of nature, scientists are surprised to discover what was known to our ancestors, but was not confirmed by scientific methods.
It is interesting to find similar formations in the micro- and macrocosm; it can also be inspiring that science can describe the geometry of these formations. The circulatory system, a river, lightning, tree branches... all of these are similar systems, consisting of different particles and different in scale.
Proportions of the “golden ratio”
Even the ancient Greeks, and possibly the Egyptians, knew the proportion of the “golden section”. Luca Pacioli, a Renaissance mathematician, called this ratio the “divine proportion.” Later, scientists discovered that the golden ratio, which is so pleasing to the human eye and which is often found in classical architecture, art and even poetry, can be found everywhere in nature.
The golden ratio is a division of a segment into two unequal parts, in which the short part is related to the long one as the long part is to the entire segment. The ratio of the long part to the entire segment is an infinite number, an irrational fraction 0.618..., the ratio of the short part is 0.382...
If you build a rectangle with sides whose ratio is equal to the proportion of the “golden ratio”, and inscribe another “golden rectangle” into it, another one into that one, and so on ad infinitum inwards and outwards, then a spiral can be drawn along the corner points of the rectangles. It is interesting that such a spiral will coincide with a cut of a nautilus shell, as well as other spirals found in nature.
Illustration: Homk/wikipedia.org
Nautilus fossil.
Photo: Studio-Annika/Photos.com
Nautilus shell.
Photo: Chris 73/en.wikipedia.org
The proportion of the golden ratio is perceived by the human eye as beautiful and harmonious. And the proportion 0.618... is equal to the ratio of the previous to the next number in the Fibonacci series. Fibonacci numbers appear everywhere in nature: this is the spiral along which the branches of a plant adjoin the stem, the spiral along which the scales on a pine cone grow or the grains on a sunflower. Interestingly, the number of rows spinning counterclockwise and clockwise are adjacent numbers in the Fibonacci series.
The head of a broccoli cabbage and a ram's horn twists in a spiral... And in the human body itself, of course, healthy and of normal proportions, the golden ratio ratios are found.
Vitruvian Man. Drawing by Leonardo da Vinci.
1, 1, 2, 3, 5, 8, 13, 21, ... are numbers in the Fibonacci series, in which each subsequent term is obtained from the sum of the previous two. Distant spiral galaxies photographed by satellites also spin in Fibonacci spirals.
Spiral galaxy.
Photo: NASA
Three tropical cyclones.
Photo: NASA
The DNA molecule is twisted in a double helix.
Twisted human DNA.
Illustration: Zephyris/en.wikipedia.org
The hurricane twists in a spiral, the spider weaves its web in a spiral.
Web of a cross spider.
Photo: Vincent de Groot/videgro.net
The “golden proportion” can also be seen in the body structure of the butterfly, in relation to the thoracic and abdominal parts of its body, as well as in the dragonfly. And most eggs fit, if not into the rectangle of the golden ratio, then into a derivative of it.
Illustration: Adolphe Millot
Fractals
Other interesting shapes that we can see everywhere in nature are fractals. Fractals are figures made up of parts, each of which is similar to the whole figure - doesn't this remind you of the principle of the golden ratio?
Trees, lightning, bronchi and the human circulatory system have a fractal shape; ferns and broccoli are also called ideal natural illustrations of fractals. “Everything is so complicated, everything is so simple” is how nature works, people notice, listening to it with respect.
“Nature has endowed man with the desire to discover the truth,” wrote Cicero, with whose words I would like to end the first part of the article on geometry in nature.
Broccoli is a perfect natural illustration of a fractal.
Photo: pdphoto.org
Fern leaves have the shape of a fractal figure - they are self-similar.
Photo: Stockbyte/Photos.com
Green fractals: fern leaves.
Photo: John Foxx/Photos.com
Veins on a yellowed leaf, shaped like a fractal.
Photo: Diego Barucco/Photos.com
Cracks on a stone: fractal in macro.
Photo: Bob Beale/Photos.com
Branches of the circulatory system on the ears of a rabbit.
Photo: Lusoimages/Photos.com
Lightning strike - fractal branch.
Photo: John R. Southern/flickr.com
Branch of arteries in the human body.
Winding river and its branches.
Photo: Jupiterimages/Photos.com
Ice frozen on glass has a self-similar pattern.
Photo: Schnobby/en.wikipedia.org
An ivy leaf with branching veins - fractal in shape.
Photo: Wojciech Plonka/Photos.com
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 mollusk 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.
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).
The mathematical shapes known as fractals originate from the genius of the eminent scientist Benoit Mandelbrot. He spent most of his life in the United States, where he taught mathematics at Yale University. In 1977 and 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 constituent elements that, on closer inspection, turned out to be repeating - which proves the existence of some kind of pattern for copying . Mandelbrot's discovery had significant positive consequences in the development of physics, astronomy and biology.
How does a fractal work?
A fractal (from the Latin “fractus” - broken, crushed, broken) is a complex geometric figure that is made up of several infinite sequences of parts, each of which is similar to the entire figure, and is repeated as the scale decreases.
The structure of the fractal on all scales is non-trivial. Here we need to clarify what is meant. So, regular figures, such as a circle, an ellipse or the graph of a smooth function, are arranged in such a way that when considering a small fragment of a regular figure on a sufficiently large scale, it will be similar to a fragment of a straight line. For fractals, an increase in scale does not lead to a simplification of the structure of the figure, and on all scales we see a uniformly complex picture.
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.
A striking example of a fractal in nature is “Romanescu”, also known as “Romanescu broccoli” or “coral cauliflower”. The first mention of this exotic vegetable dates back to Italy in the 16th century. The buds of this cabbage grow in a logarithmic spiral. 3D artists, designers and chefs never cease to admire her. The latter, moreover, especially value the vegetable for the most refined taste (sweet and nutty, not sulfurous) that cabbage can have, and for the fact that it is less crumbly than ordinary cauliflower. In addition, romaine broccoli is rich in vitamin C, antioxidants and carotenoids.
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 principle of fractal compression of information for compact storage of information about network nodes “Netsukuku” is used by the system for assigning IP addresses. Each node stores 4 kilobytes of information about the state of neighboring nodes. Any new node connects to the general Internet without requiring central regulation of the distribution of IP addresses. We can conclude that the principle of fractal compression of information ensures decentralized operation of the entire network, and therefore work in it proceeds as stably as possible.
Fractals are widely used in computer graphics - when constructing images of trees, bushes, sea surfaces, mountain landscapes, and other natural objects. Thanks to fractal graphics, an effective way was invented to implement complex non-Euclidean objects whose images are similar to natural ones: these are algorithms for synthesizing fractal coefficients, which make it possible to reproduce a copy of any picture as close as possible to the original. Interestingly, in addition to fractal “painting”, there are also fractal music and fractal animation. In the fine arts, there is a direction that deals with obtaining an image of a random fractal - “fractal monotype” or “stochatypy”.
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”
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)
