Fractal painting. Wonders of fractal graphics

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 enlarged, 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 rich 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 use a formula to find a new value. 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 is called the “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.

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 look a little into history. The term “fractal” appeared in 1975 thanks to the mathematician and creator of fractal geometry Benoit Mandelbrot. He made a huge contribution to the understanding of this phenomenon in nature and life. A lot of interesting information on this topic can be found in his famous book “Fractal Geometry of Nature”.

Now let's look at what a fractal is? 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 is a fractal.

If we take examples from nature, snowflakes, a winding coastline, and 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 of a larger size. The same can be seen in trees. From a large branch grows the same branch, but smaller, and from this branch grows an even smaller branch, etc. That is, branches of the same shape are repeated, decreasing in size. And this is a fractal - repeating self-similarity.

By the way, if we want to enlarge a picture with a fractal structure, then it will be “running in a circle”, since the fractal will increase indefinitely. We will see the same picture, despite the magnification. Infinity when increasing or decreasing is an amazing property of fractals.

How is a fractal constructed?

To draw a fractal, we will use the Sierpinski triangle. Proposed by the Polish mathematician Waclaw Sierpinski back in 1915, this fractal has become widely known and wonderfully illustrates the principle of constructing fractals. Here is a diagram of its construction:

An equilateral triangle is used as the main figure here. 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 division of each of these three triangles. We already get nine new figures, then twenty-seven... And so on ad infinitum. And this entire set is located inside the original triangle. Therefore, when you approach a picture in electronic form, a feeling of infinity arises.

Fractal graphics

So, what is fractal graphics? It is no coincidence that we examined the essence of a fractal and the principle of its construction, because this is what fractal graphics are based on. To create such a graphic image, artists use special editors. The fractal image in them is formed from parent objects and descendant objects and is calculated using mathematical formulas. Therefore, graphic files in these programs weigh little (unlike raster graphics). As an example of a fractal graphics editor, we can name ChaosPro. This is a free fractal generator that works in real time. Here are a number of interesting images generated in ChaosPro:

Using fractal geometry, you can generate the surface of water, clouds, mountains. You can use several coefficients to calculate surfaces of complex shapes. 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 the fine arts and simply pleasing to the eye?

Frosty patterns on the window, the intricate and unique shape of snowflakes, sparkling lightning in the night sky fascinate and captivate with their extraordinary beauty. However, few people know that all this is complex fractal structures.

Infinitely self-similar figures, each fragment of which is repeated as the scale decreases, are called fractals. The human vascular system, the alveolar system of an animal, the convolutions of sea shores, clouds in the sky, the contours of trees, antennas on the roofs of houses, the cell membrane and stellar 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 the 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 “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 to refer to any shapes that have any of the following properties:

Non-trivial structure on all scales. This property distinguishes fractals of such regular figures as a circle, ellipse, graph of a smooth function, etc.


Increase the scale of the 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 topological.


Construction is possible only with the help of a recursive procedure, that is, defining an object or action through oneself.

Thus, fractals can be divided into regular and irregular. The first are a mathematical abstraction, that is, a figment of the imagination. For example, the Koch snowflake or the Sierpinski triangle. The second type of fractals is the result of natural forces or human activity. N Regular fractals, unlike regular ones, retain the ability to self-similarity within limited limits.

Every day fractals are finding more and more applications in science and technology - they describe the real world as well as possible. We can give examples of fractal objects forever; they surround us everywhere. A fractal as a natural object is a vivid example of eternal continuous movement, formation and development.

Fractals have found wide application in computer graphics to construct images 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 IP address assignment system in the Netsukuku network 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 state 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.

The use of fractal geometry in the design of “fractal antennas” is very promising.
Currently, fractals have become actively used in nanotechnology. Fractals have become especially popular among traders. With their help, economists analyze the exchange rate of stock exchanges, financial 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 the artistic works, works with a textual, structural and semantic fractal nature were found.

/BDE mathematics/

The Julia set (in honor of the French mathematician Gaston Julia (1893-1978), who, together with Pierre Fatou, was the first to study fractals.His work was popularized by Benoit Mandelbrot in the 1970s)

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 - a non-self-intersecting continuous curve of infinite length. This curve is not tangent at any point.
Cantor set- a loose uncountable perfect set.
Menger sponge is an analogue of the Cantor set with the only difference that this fractal is constructed in three-dimensional space.
Triangle or Sierpinski carpetis also an analogue of the Cantor set on the plane.
Weierstrass and van der Waerden fractalsrepresent a non-differentiable continuous function.
Trajectory of Brownian particlesis also 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 forming a fract of curves on a plane. First of all, it is necessary to define an arbitrary polyline with a finite number of links, the so-called generator. Next, each link is replaced by a generating element, or rather a broken line, similar to a generator. As a result of this 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 obtain the third generation of the curve, the same algorithm is performed - each link is replaced by a reduced generating element. Thus, to obtain each subsequent generation, all links of the previous one are replaced by a reduced generatrix of elements. Then, the nth generation curve for any finite n is called a prefractal. In the case when n tends to infinity, the Koch curve becomes a fractal object.

Let's turn to another method of constructing a fractal object. To create it, you need to change the rules of construction: let the forming element be two equal segments connected at right angles. In the zeroth 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 shifts. Subsequent generations are built according to the rule: the first link on the left is replaced with a formative element in such a way that the middle of the link is shifted to the left of the direction of movement. Next, the replacement of links alternates. The limiting fractal curve constructed according to this rule is called the Harter-Haithway dragon.

In computer graphics, geometric fractals are used to simulate images of trees, bushes, mountain ranges, and 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. Upon returning 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 working 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) is a term introduced by Benoit Mandelbrot in 1975. There is still no strict mathematical definition of fractal sets. ABOUT n was able to generalize and systematize “unpleasant” sets and build a beautiful and intuitive theory. He discovered the wonderful world of fractals, the beauty and depth of which sometimes amaze the imagination and delight scientists, artists, philosophers... Mandelbrot's work was stimulated by advanced computer technologies, which made it possible to generate, visualize and explore various sets.

Japanese physicist Yasunari Watanaba created a computer program that draws beautiful fractal patterns. A 12-month calendar was presented at the international conference "Mathematics and Art" in Suzdal.

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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. Velazquez, J. Pollock, M Escher, H. Gries, J. Ball, S. Dali, L. Wayne, G. Klimt, Van Gogh, P. Filonov, A. Rodchenko, etc.).

It has long been no secret that objects with signs 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 living nature and on the canvases of artists.

Katsushika Hokusai. Big wave in 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 as its scale decreases. Many systems of the human body have fractal characteristics: the structure of the circulatory system, the bronchial tree, and 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, using 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 incomprehensible popularity of the works of the American abstract artist to many.

With the meticulousness characteristic of scientists, Richard Taylor began to calculate the fractal dimension of Pollock's paintings. So he established that this value varied from a value close to unity in 1943 to a coefficient of 1.72 in 1954. The physicist suggests using this indicator to date and confirm the authenticity of works, because, according to his data, as well as the research of other scientists, fractal analysis can help identify a fake with a guarantee of up to 93 percent.

To more accurately study the influence of fractal art on humans, Taylor used the electroencephalography (EEG) method, which makes it possible to record the slightest changes in the function of the cerebral cortex and deep brain systems. He showed that contemplation of fractal patterns is accompanied by a significant reduction in stress levels and even speeds up the body's recovery after surgery.

Evolution of fractals

Fractals have long been firmly established in the visual arts, starting with the civilizations of the Aztecs, Incas and Mayans, ancient Egyptian and ancient Romans, which have sunk into oblivion. Firstly, they are quite difficult to avoid when depicting living nature, where fractal-like forms are found all the time.

Some of the earliest and most pronounced examples of fractal painting are the landscape traditions of ancient and medieval China.

In the 20th century, fractal structures became most widespread in the directions of op art (optical art) and imp-art (from the word impossible). The first of them grew out of abstractionism in the 1950s; 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 movement within optical art, the Dutch artist Maurits Cornelis Escher became famous. He used techniques based on mathematical principles to create his works.

Escher became skilled at depicting “impossible figures”: creating optical illusions that mislead viewers and force the vestibular apparatus to tense.

Fractal complexity and the artist's brain

So, looking at fractals leaves a noticeable mark on a person’s brain activity, which is even recorded by special equipment. But there is also an inverse relationship: the artist’s mental and mental health can affect the quantity and quality of fractal compositions in his works.

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 out of it. He continued to portray felines even when he was admitted to a psychiatric hospital with progressive schizophrenia.

Here something incredible began to happen to his paintings: they bloomed with acidic psychedelic colors, and the cats gradually evolved into marvelous 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 Wain, one could assume that the transformation of the artist’s style was 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 that lead to the decline of cognitive functions and dementia, there is an inverse relationship. This was the case with Willem de Kooning, who was diagnosed with Alzheimer's disease in 1982. As Richard Taylor noted in his scientific publication, 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 studying such diseases.

This is what complex fractal structures looked like in Willem de Kooning's early paintings of the 1940s.

And so - late works, written during the period of illness. According to Taylor, there is a peace in them that was missing from the artist’s canvases during his creative heyday.

Fractal painting of new times

Today, creating fractal patterns is not particularly difficult. There are many computer programs that allow you to synthesize them in countless quantities with corresponding artistic value. But there are still authors working in this field the old fashioned way, using tangible rather than digital means. One worthy of note is Greg Dunn, a doctor of neurobiology at the University of Pennsylvania.

Greg Dunn. Hippocampus II, 2010

Firstly, 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 example."

Greg Dunn. Synaptogenesis, 2001

If you still can’t order one of the works of an American neuroscientist in order to constantly have an anti-stress picture before your eyes, just bookmark this article and return to it whenever the level of cortisol (“stress hormone”) in the blood begins to go off scale and cause discomfort.

Top 10 fractal painting artists from Arthive

Vincent Van Gogh

Piet Mondrian

Mikalojus Konstantinas Ciurlionis

Paul Klee

Salvador Dali