What is a fractal? Fractals in nature. The wonders of fractal graphics What is fractal art
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.
Evolution of fractals
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 as its scale decreases.
Untitled Wang Fu XIV century
Fractals have long been firmly established in the fine arts, starting with the
in the summer of the civilizations of the Aztecs, Incas and Mayans, ancient Egyptian and ancient Roman.
Firstly, they are quite difficult to avoid when depicting wildlife, where
fractal-like shapes are found all the time.
Farewell on the Shen Zhou River, 15th 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 became most widespread in the directions
op art (optical art) and imp¬art (from the word impossible - impossible).
The first of them grew out of abstractionism in the 1950s, or, more precisely, spun off
from geometric abstraction. One of the pioneers of op art was Victor Vasarely -
French artist with Hungarian roots.
Klonopin
Guiva
But in the field of imp art, which is distinguished as an independent movement within
optical art, the Dutch artist Maurits Cornelis Escher became famous.
He used techniques based on mathematical principles to create his works.
Butterflies
Less and less
Escher became skilled at depicting “impossible figures”: creating optical illusions,
misleading viewers and causing the vestibular apparatus to strain.
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Article Nikolaeva E.V.. candidate of cultural studies, associate professor at Moscow State University of Design and Technology "RESEARCH OF FRACTALS IN FINE ART", published on the website Russian State Institute of Art Studies, perhaps one of the many scientific studies of fractals (see list of references), through which the author explains the nature of fractals translated into art.
It will be difficult for an inexperienced reader to understand the essence of the story the first time. I would like to note that, along with the great world fine arts Leonardo da Vinci, Katsushika Hokusai, Maurits Escher and other contemporary foreign artists, the author mentions our compatriot, the famous conceptual artist. I will give only a few fragments of this article and show the works mentioned in the article.
"Fractal" as a concept
As the ideas of fractal geometry of the French mathematician Benoit Mandelbrot, set out in a number of his works, among which the most famous "Fractal geometry of nature", gradually went beyond the scope of natural science discourse, "fractal" has become one of the most popular concepts in the post-postmodern research field. Fractal analysis has turned out to be a useful methodological tool in humanities mathematics: in the economics of stock markets, sociology, urban studies, in the so-called. mathematical history, synergetic concepts of culture, artmetrics…
Aesthetics of fractals
As examples classical "fractal" art B. Mandelbrot cited the frontispiece “God the Geometer” of the French “Biblical Moral Teaching in Pictures” of the 13th century, drawing Leonardo da Vinci "The Flood", engravings by a Japanese artist of the late 18th – early 19th centuries. Katsushika Hokusai "One Hundred Views of Mount Fuji" and work M. Escher (XX century). In his brief art history excursion dedicated to the long history of fractals, B. Mandelbrot gives a special place to the work of K. Hokusai, noting his amazing "feel for fractals" and the courage to turn to forms that were recognized by science much later. The work of Hokusai, according to Mandelbrot, may be “the best proof that fractal structures have been known to mankind since time immemorial, but they were described only through art.” Famous “The Great Wave” (The Great Wave or The Breaking Wave off Kanagawa)(Fig.5)… Since then, identifying and imitating the fractality of classical painting (Fig.6) has become a fascinating scientific and artistic practice (etc.)…
Gradually the concept "fractal art" went far beyond the scope of mathematical, algorithmic, digital art. The concept of fractality owes its emergence to such new forms of painting and media art as fractal expressionism or fractalage(“fractalage”, analogue fractal painting) Derek Nielsen(Derek K. Nielsen), fractal monotypes Lea Lifshits, fractal abstraction Victor Ribas, fractal realism Vyacheslav Useinova(Fig. 7) and Alexey Sundukov, fractal suprematism ( V. Ribas, S. Golovach, A. Rabotnov, A. Pettai and others.) (Fig. 8). . Fractal paintings of various compositional and semantic types, created by different media and software tools with varying degrees of skill, are now exhibited at numerous exhibition venues - virtual and real.
The shadow of a non-existent house. V. Useinov. 2003., oil on canvas
Fractality as a quantitative and qualitative characteristic in fine arts
The first methodological tool borrowed by the humanities from fractal geometry was fractal dimension. Unlike urban studies, in which the calculated value of the fractal dimension of urban areas has not yet been converted into categories of artistic description of space, in art history a way has been found to create correlations between a work of fine art and its fractal dimension. Thus, according to data from special experiments, a certain value of the fractal dimension of a pictorial image (perhaps 1.5) may correspond to the aesthetic preferences of viewers. Or changes in the value of the fractal dimension may correlate with different periods of the artist’s creativity, increasing, for example, in Jackson Pollock(Fig. 9) from a value close to 1 in 1943 to 1.72 in 1954, which is proposed as an objective basis for dating and authenticating his work. Or the fractal dimension and its dynamics over time can serve as a characteristic of an entire artistic era, for example, early Chinese landscape painting.
Jackson Pollack. Convergence -1952-1024×621
...In general, fractal imagery is analyzed from one of two inverse positions: 1) characteristics are given that allow fractal computer graphics to be classified as art, or 2) fractal structures are identified in works of traditional art of different eras and movements (D. Velasquez, J. Pollock, M Escher, H. Gries, J. Ball, S. Dali, L. Wayne, G. Klimt, Van Gogh, P. Filonov, A. Rodchenko, etc.).
Presentation on the topic: Fractals in art and architecture Prepared by 10th grade student Varchenkov Vadim Valerievich, head - Stiplina Galina Nikolaevna Municipal budgetary educational institution "Secondary School 9" Tel.: , Safonovo, Smolensk region 2014 Nomination: "Mathematical models of real processes in nature and society"
Fractal is a mathematical term, has complex precise calculations and is built on precise mathematical principles, and is widely used in computer graphics and the construction of many computer processes. Now, the use of fractals extends from mathematics to art, but the most surprising thing is that digging deeper, you come to the conclusion that it reflects the most basic esoteric principles of the structure of the universe.
Origin of the term Fractals are structures consisting 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 composing fractals - fractasm.
The term “fractal” itself 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 different from the topological one. And of course, like any other mathematical science, fractasm is full of many complex theoretical studies and formulas.
Fractals in fine arts Returning to the past, in the art of mankind, as in nature, one can easily find examples of the use of fractals. Vivid works in this system are Leonardo da Vinci’s drawing “The Flood”, engravings by the Japanese artist Katsushika Hokusai and the works of E. Escher are also a striking example of fractality and the list can be continued endlessly.
Thus, manifestations of fractality went beyond the scope of mathematical theory and found a home in many areas of life, including clearly represented in the art of the twentieth century. New forms of art are emerging, 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 fine art, which participate in exhibitions around the world. Fractal has become one of the popular and sought-after phenomena in 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: Peano curve, Koch snowflake, Sierpinski triangle, Cantor dust, Harter-Hateway “dragon”, etc. All of them are obtained by repeating a certain sequence of geometric constructions using points and lines.
Fractals of this group are the most visual. If we analyze the image data, we can identify the following properties of geometric fractals: an infinite set of geometric fractals covers a limited surface area; 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, while others are equal to zero.
Sierpinski triangle The next way to obtain a 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 with a broken line of three links (it itself is obtained in the first iteration). You need to lay this broken line alternately to the right and then to the left. It can be seen that already the eighth iteration is very close to the fractal, and the further it goes, the closer the line will get to it. This fractal was described in 1915 by the Polish mathematician Waclaw 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 of the four small triangles formed. Then the same steps must be repeated with each of the remaining three triangles, etc.
Variants of the Sierpinski Triangle Carpet (square, napkin) by Sierpinski. The square version was described by Wacław Sierpinski in 1916. He managed to prove that any curve that can be drawn on a plane without self-intersections is homeomorphic to some subset of this holey square. Like a triangle, a square can be made from different designs. On the right is the classic method: dividing the square into 9 parts and throwing away the central part. Then the same is repeated for the remaining 8 squares, etc.
Sierpinski's pyramid One of the three-dimensional analogues of the Sierpinski triangle. It is constructed in a similar way, taking into account the three-dimensionality of what is happening: 5 copies of the initial pyramid, compressed by half, make up the first iteration, its 5 copies will make up the second iteration, etc. The fractal dimension is equal to log25. The figure has zero volume (at each step half the volume is thrown away), but the surface area is maintained from iteration to iteration, and for the fractal it is the same as for the initial pyramid.
Menger sponge Generalization of the Sierpinski carpet into three-dimensional space. To build a sponge, you need an endless 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 thrown out. That is, each cube generates 20 new ones, three times smaller. Therefore the fractal dimension is 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 at each step it is multiplied by 20/27), but it has an infinitely large area.
