Frost patterns on the window, the intricate and unique form of snowflakes, sparkling lightning in the night sky fascinate and captivate with their extraordinary beauty. However, few people know that all this is a complex fractal structure.
Infinitely self-similar figures, each fragment of which is repeated when zoomed out, are called fractals.. The human vascular system, the animal alveolar system, the meanders of the sea shores, clouds in the sky, the contours of trees, antennas on the roofs of houses, the cell membrane and star galaxies - all this amazing product of the chaotic movement of the world is fractals.
The first examples of self-similar sets with unusual properties appeared in the 19th century. The term "fractals", which comes from Latin word"fractus" - fractional, broken, was introduced by Benoit Mandelbrot in 1975. Thus, a fractal is a structure consisting of parts similar to the whole. It is the property of self-similarity that sharply distinguishes fractals from objects of classical geometry.
Simultaneously with the publication of the book "The Fractal Geometry of Nature" (1977), fractals gained worldwide fame and popularity.
T The term "fractal" is not a mathematical concept and therefore does not have a strict generally accepted mathematical definition. Moreover, the term fractal is used in relation to any figures that have any of the following properties:
Nontrivial structure on all scales. This property distinguishes fractals of such regular figures as a circle, an ellipse, a graph of a smooth function, etc.
Increase The scale of a fractal does not lead to a simplification of its structure, that is, on all scales we see an equally complex picture, while when considering a regular figure on a large scale, it becomes similar to a fragment of a straight line.
Self-similarity or approximate self-similarity.
Metric or fractional metric dimension, significantly superior to the topological one.
Construction is possible only with the help of a recursive procedure, that is, the definition of an object or action through itself.
Thus, fractals can be divided into regular and irregular. The former are mathematical abstractions, that is, a figment of the imagination. For example, the Koch snowflake or the Sierpinski triangle. The second kind of fractals is the result of natural forces or human activity. H irregular fractals, unlike regular fractals, retain the ability to self-similarity within limited limits.
Every day fractals find more and more applications in science and technology - they are the best description of the real world. It is possible to give examples of fractal objects indefinitely, they surround us everywhere. Fractal as a natural object is a vivid example of eternal continuous movement, formation and development.
Fractals are widely used in computer graphics to build an image of natural objects, for example, trees, bushes, mountain ranges, sea surfaces, etc. The use of fractals in decentralized networks has become effective and successful. For example, the Netsukuku IP assignment system uses the principle of fractal information compression to compactly store information about network nodes. Due to this, each node of the Netsukuku network stores only 4 Kb of information about the status of neighboring nodes, moreover, any new node connects to the common network without the need for central regulation of the distribution of IP addresses, which, for example, is actively used on the Internet. Thus, the principle of fractal information compression ensures the most stable operation of the entire network.
Very promising is the use of fractal geometry in the design of "fractal antennas".
Currently, fractals are actively used in nanotechnology. Fractals have become especially popular with traders. With their help, economists analyze the course of stock exchanges, gross and trading markets.In petrochemistry, fractals are used to create porous materials. In biology, fractals are used to model the development of populations, as well as to describe systems of internal organs.Even in literature, fractals have found their niche. Among works of art works with textual, structural and semantic fractal nature were found.
/BDE mathematics/
Julia set (in honor of the French mathematician Gaston Julia (1893-1978), who, together with Pierre Fatou, was the first to study fractals.In the 1970s, his work was popularized by Benoit Mandelbrot.)
geometric fractals
The history of fractals in the 19th century began precisely with the study of geometric fractals. Fractals clearly reflect the property of self-similarity. The most obvious examples of geometric fractals are:
Koch curve is a non-self-intersecting continuous curve of infinite length. This curve has no tangent at any point.
Cantor setis a nondense uncountable perfect set.
Sponge Menger - this is an analogue of the Cantor set with the only difference that this fractal is built in three-dimensional space.
Triangle or Sierpinski carpetis also an analogue of the Cantor set on the plane.
Weierstrass and van der Waerden fractalsare a non-differentiable continuous function.
Brownian particle trajectoryalso not differentiable.
Peano Curve is a continuous curve that passes through all points of the square.
Tree of Pythagoras.
Consider the triadic Koch curve.
To construct a curve, there is a simple recursive procedure for generating a fract of curves on a plane. First of all, it is necessary to define an arbitrary broken line with a finite number of links, the so-called generator. Further, each link is replaced by a generating element, more precisely a broken line similar to a generator. As a result of such a replacement, a new generation of the Koch curve is formed. In the first generation, the curve consists of four straight links, the length of each of which is 1/3. To get the third generation of the curve, the same algorithm is performed - each link is replaced by a reduced generatrix. Thus, to obtain each subsequent generation, all links of the previous one are replaced by a reduced generating element. Then, the nth generation curve for any finite n is called a prefractal. When n tends to infinity, the Koch curve becomes a fractal object.
Let's turn to another way of constructing a fractal object. To create it, you need to change the construction rules: let the generatrix be two equal segments connected at a right angle. In the zero generation, we replace the unit segment with a generating element so that the angle is on top. That is, with such a replacement, the middle of the link is shifted. Subsequent generations are built according to the rule: the first link on the left is replaced by a generating element in such a way that the middle of the link is shifted to the left of the direction of movement. Further, the replacement of links alternates. The limiting fractal curve constructed according to this rule is called the Harter-Hateway dragon.
AT computer graphics geometric fractals are used to model images of trees, bushes, mountain ranges, coastlines. 2D geometric fractals are widely used to create 3D textures.
After graduating from university, Mandelbrot moved to the United States, where he graduated from the California Institute of Technology. On his return to France, he received his doctorate from the University of Paris in 1952. In 1958, Mandelbrot finally settled in the United States, where he began work at the IBM Research Center in Yorktown..
He worked in the fields of linguistics, game theory, economics, aeronautics, geography, physiology, astronomy, and physics.
Fractal (lat. fractus - crushed) - a term introduced by Benoit Mandelbrot in 1975. There is still no rigorous mathematical definition of fractal sets. O he was able to generalize and systematize "unpleasant" sets and build a beautiful and intuitive theory. He opened wonderful world fractals, the beauty and depth of which sometimes amaze the imagination, delight scientists, artists, philosophers ... Mandelbrot's work was stimulated by advanced computer technology, which made it possible to generate, visualize and explore various sets.
Japanese physicist Yasunari Watanaba created computer program drawing beautiful fractal ornaments. A calendar of 12 months was presented at the international conference "Mathematics and Art" in Suzdal.
In the age of digital technology, computer graphics will not surprise anyone. However, not everyone has heard about such a direction as fractal graphics. What is fractal graphics? What is a fractal and how to draw it?
fractal principle
Before answering these questions, let's take a look at history. The term "fractal" appeared in 1975 thanks to the mathematician, the creator of fractal geometry, Benoit Mandelbrot. He made a huge contribution to the understanding of this phenomenon in nature and life. Lot interesting information on this topic can be found in his famous book "The Fractal Geometry of Nature".
Now let's consider what is a fractal? In short, a fractal is a repeating self-similarity. This word comes from the Latin fractus - which means crushed, broken. That is, a figure consisting of parts that are similar to it - and there is a fractal.
If we take examples from nature, then snowflakes, a winding coastline, tree crowns are fractals. The properties of a fractal are very well demonstrated by a snowflake. The smallest crystals of which it consists are repeated and form the same crystals, but already bigger size. The same can be seen in trees. From a large branch, the same branch grows, but already smaller, and an even smaller branch grows from this branch, etc. That is, branches of the same shape are repeated, decreasing in size. And this is a fractal - a repeating self-similarity.
By the way, if we want to enlarge a picture with a fractal structure, then this will be “running in a circle”, since the fractal will increase indefinitely. We will see the same picture despite the zoom. Infinity when increasing or decreasing is an amazing property of fractals.
How is a fractal built?
To draw a fractal, we will use the Sierpinski triangle. Proposed by the Polish mathematician Vaclav Sierpinski back in 1915, this fractal has become widely known and perfectly illustrates the principle of constructing fractals. Here is a diagram of its construction:
An equilateral triangle is used here as the main figure. We mark the middle on each of its sides. Then we connect these three points with lines. As a result, three more triangles are formed inside our triangle, but of a smaller size. Next, we repeat the crushing of each of these three triangles. We already get nine new figures, then twenty-seven ... And so on ad infinitum. And all this set is inside the original triangle. Therefore, when the picture is zoomed in in electronic format there is a feeling of infinity.
fractal graphics
So, what is fractal graphics? It is no coincidence that we have considered the essence of a fractal and the principle of its construction, because this is what fractal graphics are based on. To create such graphic image artists use special editors. The fractal image in them is formed from parent objects and child objects and is calculated by means of mathematical formulas. Therefore, graphic files in these programs weigh a little (unlike raster graphics). An example of a fractal graphics editor is ChaosPro. This is free generator fractals, working in real time. Here are some interesting images generated by ChaosPro:
Through fractal geometry, you can generate the surface of water, clouds, mountains. It is possible to calculate surfaces of complex shape using several coefficients. In this way, amazing abstract paintings are created that look like a fantastic alien world. The properties of fractals can also be used in technical computer graphics. But if we ignore the practical application and focus on the beauty of fractal graphics, then isn't this fantastic creativity, worthy of being an independent direction in fine arts and just pleasing to the eye?
The evolution of fractals
A simplified scientific definition of a fractal (from the Latin fractus - “crushed,
broken, broken”) is a set with the property of self-similarity.
This concept also denotes a self-similar geometric figure,
each fragment of which is repeated when its scale is reduced.
Untitled Wang Fu 14th century
Fractals have long and firmly settled in the visual arts, starting from the sunken
in the summer of the civilizations of the Aztecs, Incas and Maya, ancient Egyptian and ancient Roman.
Firstly, they are quite difficult to avoid when depicting wildlife, where
fractal-like forms are found all the time.
Farewell on the Shen Zhou River XV century
One of the earliest and most pronounced examples of fractal painting
- landscape traditions of ancient and medieval China.
Wang Meng, Untitled
Shen Zhou, Untitled
In the 20th century, fractal structures were most widely used in the directions
op-art (optical art) and imp-art (from the word impossible - impossible).
The first of them grew in the 1950s from abstractionism, more precisely, spun off
from geometric abstraction. Victor Vasarely was one of the pioneers of Op Art.
french artist with Hungarian roots.
Klonopin
Guiva
But in the field of imp-art, which is distinguished as an independent trend inside
optical art, the Dutch artist Maurits Cornelis Escher became famous.
He applied techniques based on mathematical principles in the creation of works.
butterflies
Less and less
Escher got his hand in the image of "impossible figures": the creation optical illusions,
misleading viewers and forcing the vestibular apparatus to strain.
Presentation on the topic: Fractals in art and architecture Prepared by a 10th grade student Varchenkov Vadim Valerievich, head - Stiplina Galina Nikolaevna Comprehensive school 9" Tel.: , Safonovo, Smolensk region 2014 Nomination: "Mathematical models of real processes in nature and society"
Fractal is a mathematical term, has complex exact calculations and is based on exact mathematical principles, is widely used in computer graphics and the construction of many computer processes. Now, the use of the fractal is spreading from mathematics to art, but the most amazing thing is that, digging deeper, you come to the conclusion that it reflects the most basic esoteric principles of the universe.
Origin of the term Fractals are structures made up of parts that are similar to the whole. Translated from Latin, "fractus" means "crushed, broken, broken." In other words, this is the self-similarity of the whole to the particular within the framework of geometric figures. There is an exact science of studying and compiling fractals - fractasm.
The term "fractal" was introduced into mathematics by Benoit Maldenbrot in 1975, which is considered to be the year of birth of fractasm. In mathematics, fractals are understood as sets of points in Euclidean space that have a fractional metric dimension, or a metric dimension that is different from the topological one. And of course, like any other mathematical science, fractasm is saturated with many complex theoretical studies and formulas.
Fractals in fine arts Returning to the past, in the art of mankind, as well as in nature, one can easily find examples of the use of fractals. The striking works in this system are the drawing by Leonardo da Vinci " global flood”, engravings by Japanese artist Katsushika Hokusai and works by E. Escher are also a vivid example of fractality, and this list is endless.
Thus, the manifestation of fractality has gone beyond the scope of mathematical theory and has found its home in many areas of life, including vividly represented in the art of the twentieth century. new forms of art appear, the basis of which is fractal graphics.
Fractal expressionism or fractalage, in the amazing works of D. Nielsen, fractal monotypes from L. Livshits, fractal abstraction by V. Ribas, fractal realism by V. Useinov and A. Sundukov. Fractal paintings have become an integral part of the fine arts, which are exhibited all over the world. The fractal has become one of the most popular and sought-after phenomena in the post-modernism of our century.
Application of fractal theory in architecture Geometric fractals are used in architecture. The main representatives of this group are such objects as: the Peano curve, the Koch snowflake, the Sierpinski triangle, the Cantor dust, the Harter-Hateway "dragon", etc. All of them are obtained by repeating a certain sequence geometric constructions using dots and lines.
The fractals of this group are the most obvious. If we analyze the image data, we can distinguish the following properties of geometric fractals: an infinite set of geometric fractals covers limited area surfaces; the infinite set that makes up the fractal has the property of self-similarity; the lengths, areas and volumes of some fractals tend to infinity, others are equal to zero.
The Sierpinski Triangle The next way to get the Sierpinski triangle is even more similar to the usual scheme for constructing geometric fractals by replacing parts of the next iteration with a scaled fragment. Here, at each step, the segments that make up the broken line are replaced by a broken line of three links (it itself is obtained in the first iteration). You need to postpone this broken line alternately to the right, then to the left. It can be seen that already the eighth iteration is very close to the fractal, and the further, the closer the line will get to it. This fractal was described in 1915 by the Polish mathematician Vaclav Sierpinski. To get it, you need to take an (equilateral) triangle with an interior, draw middle lines in it and throw out the central one from the four small triangles formed. Further, these same actions must be repeated with each of the remaining three triangles, and so on.
Variants of the Sierpinski Triangle Carpet (square, napkin) Sierpinski. The square version was described by Vaclav Sierpinski in 1916. He managed to prove that any curve that can be drawn on the plane without self-intersections is homeomorphic to some subset of this holey square. Like a triangle, a square can be obtained from different designs. The classic way is shown on the right: dividing the square into 9 parts and throwing out the central part. Then the same is repeated for the remaining 8 squares, and so on.
Sierpinski's pyramid One of three-dimensional analogs of Sierpinski's triangle. It is built in a similar way, taking into account the three-dimensionality of what is happening: 5 copies of the initial pyramid, compressed by two times, make up the first iteration, its 5 copies make up the second iteration, and so on. The fractal dimension is equal to log25. The figure has zero volume (half of the volume is discarded at each step), but the surface area is preserved from iteration to iteration, and for the fractal it is the same as for the initial pyramid.
Sponge of Menger Generalization of the Sierpinski carpet into three-dimensional space. To build a sponge, you need an infinite repetition of the procedure: each of the cubes that make up the iteration is divided into 27 three times smaller cubes, from which the central one and its 6 neighbors are discarded. That is, each cube generates 20 new ones, three times smaller. Therefore, the fractal dimension is equal to log320. This fractal is a universal curve: any curve in three-dimensional space is homeomorphic to some subset of the sponge. The sponge has zero volume (since it is multiplied by 20/27 at each step), but has an infinitely large area.
It has long been no secret that objects that have the features of fractals are perceived by the human eye as the highest manifestation of harmony and beauty. Tree crowns and mountain ranges, unique patterns of snowflakes and "golden" spirals of sea shells and waves, crystals and corals - we are ready to endlessly contemplate them both in wildlife and on the canvases of artists.
Katsushika Hokusai. Big wave off Kanagawa.
A simplified scientific definition of a fractal (from the Latin fractus - “crushed, broken, broken”) is a set that has the property of self-similarity. This concept also denotes a self-similar geometric figure, each fragment of which is repeated with a decrease in its scale. Many systems of the human body have fractal characteristics: the structure of the circulatory system, the bronchial tree, neural networks.
Pollock treatment
Richard Taylor from the University of Oregon has been studying fractal structures in general and specifically in painting since 1999. In particular, on the example of the paintings of his compatriot Jackson Pollock. Using computer analysis of the patterns from which the paintings are woven, the scientist found that they have qualities inherent in natural fractal phenomena, such as coastlines, For example. It is to this factor that the researcher is inclined to attribute the popularity of the works of the American abstract artist, incomprehensible to many.
With the meticulousness characteristic of scientists, Richard Taylor began to calculate the fractal dimension of Pollock's paintings. So he found that this value changed from a value close to one in 1943 to a factor of 1.72 in 1954. The physicist suggests using this indicator to date and authenticate works, because, according to his data, as well as the studies of other scientists, fractal analysis can help determine a fake with a guarantee of up to 93 percent.
For a more accurate study of the influence of fractal art on a person, Taylor used the method of electroencephalography (EEG), which allows recording the slightest changes in the function of the cerebral cortex and deep brain systems. He showed that the contemplation of fractal patterns is accompanied by a significant reduction in stress levels and even accelerates the recovery of the body after surgery.
The evolution of fractals
Fractals have long and firmly settled in the visual arts, starting with the civilizations of the Aztecs, Incas and Maya, ancient Egyptian and ancient Roman that have sunk into oblivion. Firstly, they are quite difficult to avoid when depicting wildlife, where fractal-like forms are found all the time.
One of the earliest and most pronounced examples of fractal painting is the landscape traditions of ancient and medieval China.
In the 20th century, fractal structures were most widely used in the areas of op-art (optical art) and imp-art (from the word impossible - impossible). The first of them grew in the 1950s from abstractionism, more precisely, it spun off from geometric abstraction. One of the pioneers of op art was Victor Vasarely, a French artist with Hungarian roots.
But in the field of imp-art, which is distinguished as an independent trend within optical art, the Dutch artist Maurits Cornelis Escher became famous. He applied techniques based on mathematical principles in the creation of works.
Escher has mastered the art of depicting "impossible figures": creating optical illusions that mislead viewers and make the vestibular apparatus tense.
Fractal Complexity and the Artist's Brain
So, the examination of fractals leaves a noticeable trace in the brain activity of a person, which is even fixed by special equipment. But there is also an inverse relationship: the mental and mental health of the artist can affect the quantity and quality of fractal compositions in his work.
One of the textbook examples is the biography of the Englishman Louis Wayne, who, after the death of his wife from cancer, just three years after the wedding, became interested in drawing anthropomorphic cats, and made a good career on this. He continued to portray cats, even when he ended up in a psychiatric hospital with progressive schizophrenia.
Here, something incredible began to happen to his paintings: they blossomed with acidic psychedelic colors, and cats gradually evolved into wondrous fractal structures. And if the discovery of the psychotropic properties of LSD had not been accidentally discovered by the chemist Albert Hofmann 4 years after the death of Louis Wayne, one could assume that the transformation of the artist's style is the result of an experimental treatment for schizophrenia, in which this substance was actually used, but only a couple of decades later.
As for diseases leading to cognitive decline and dementia, there is Feedback. This was the case with Willem de Kooning, who was diagnosed with Alzheimer's in 1982. As noted in his scientific publication by Richard Taylor, discussed above, the fractal complexity of his abstract paintings rapidly decreased in proportion to how the artist's dementia progressed. An analysis of the work of seven artists with various neurological problems showed the potential of art research as a new tool for the study of such diseases.
This is how complex fractal heaps looked on early paintings Willem de Kooning in the 1940s.
And so - later works written during the period of illness. According to Taylor, there is peace in them, which was not enough on the artist's canvases at the time of his creative heyday.
Fractal painting of the new time
Today, the creation of fractal patterns is not difficult. There are many computer programs that allow you to synthesize them in myriad quantities with appropriate artistic value. But there are still authors who work in this field the old fashioned way, using not digital, but quite tangible means. One of the most noteworthy is Greg Dunn, Ph.D. in neuroscience from the University of Pennsylvania.
Greg Dunn. Hippocampus II, 2010
First, for inspiration, he uses samples from the sphere of his immediate subject of study - various cells, departments and processes of the brain, the terminological designations of which coincide with the names of the paintings.
Greg Dunn. Cortex columns, 2014
Secondly, the scientist uses non-trivial materials and techniques: aluminum plates, metal powder, gold, enamel, mica, ink, and so on. On his page, he admits: I admire Japanese, Chinese and Korean masters of painting, their self-sufficiency and simplicity. I'm trying to follow their lead."
Greg Dunn. Synaptogenesis, 2001
If you still can’t order one of the works of an American neuroscientist to constantly have an anti-stress picture in front of your eyes, just bookmark this article and return to it whenever the level of cortisol (“stress hormone”) in the blood starts to go off scale and cause discomfort.
Top 10 fractal painting artists from Arthive
Vincent Van Gogh
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Road with cypresses |
Piet Mondrian
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Church of St. James, Winterswijk |
Farm in Duvendrecht in the evening |
Mikalojus Konstantinas Ciurlionis
Paul Klee
Salvador Dali
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Galatea with spheres |