Often, brilliant discoveries made in science can radically change our lives. For example, the invention of a vaccine can save many people, and the creation of new weapons leads to murder. Just yesterday (on the scale of history) a man “tamed” electricity, and today he can’t imagine his life without it. However, there are also such discoveries that, as they say, remain in the shadows, and despite the fact that they also have one or another impact on our lives. One of these discoveries was fractal. Most people have not even heard of such a concept and cannot explain its meaning. In this article, we will try to understand the question of what fractal is, and consider the meaning of this term from the standpoint of science and nature.
Order in chaos
In order to understand what a fractal is, it would be necessary to begin the analysis of flights from the perspective of mathematics, however, before delving into the exact sciences, we philosophize a little. Each person has a natural curiosity, thanks to which he learns the world around him. Often in his quest for knowledge, he tries to operate with logic in judgments. So, analyzing the processes that occur around, he tries to calculate the relationship and to deduce certain patterns. The largest minds on the planet are busy with these tasks. Roughly speaking, our scientists are looking for patterns where they do not exist, and it should not be. Nevertheless, even in chaos there is a connection between certain events. This fractal acts as this connection. As an example, consider a broken branch lying on the road. If you look closely at her, we will see that she, with all her branches and knots, looks like a tree herself. This similarity of a separate part with a single whole testifies to the so-called principle of recursive self-similarity. Fractals in nature can be found all the time, because many inorganic and organic forms are formed similarly. These are clouds, and sea shells, and snail shells, and tree crowns, and even the circulatory system. This list goes on and on. All these random forms are easily described by the fractal algorithm. Here we come to consider what fractal is from the perspective of the exact sciences.
Some dry facts
The word "fractal" from Latin is translated as "partial", "divided", "fragmented", and with regard to the content of this term, the wording as such does not exist. Usually it is interpreted as a self-similar set, a part of the whole, which is repeated by its structure at the micro level. This term was coined in the seventies of the twentieth century by Benoit Mandelbrot, who is recognized as the father of fractal geometry. Today, the concept of fractal means a graphic image of a certain structure, which at an enlarged scale will be similar to itself. However, the mathematical basis for creating this theory was laid 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 episodic in nature. This is due to the fact that mathematicians preferred to study objects that can be studied on the basis of general theories and methods. In 1872, the German mathematician C. Weierstrass constructed an example of a continuous function that is nowhere differentiable. However, this construction turned out to be entirely abstract and difficult to perceive. Then went the Swede Helge von Koch, who in 1904 built a continuous curve that has nowhere tangent. It is quite easy to draw, and as it turned out, it is characterized by 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, Frenchman Paul Levy. In 1938, he published an article entitled “Flat and Spatial Curves and Surfaces Consisting of Parts Similar to the Whole”. In it, he described a new kind - the Levy C-curve. All of the above figures conditionally relate to such a form as geometric fractals.
Dynamic or algebraic fractals
Many Mandelbrot belong to this class. The first researchers in this direction were 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, they quickly forgot about it. And only half a century later, thanks to computers, Julia's work received a second life. Computers made it possible for every person to see the beauty and richness of the fractal world that mathematicians could “see”, displaying them through functions. Mandelbrot was the first to use a computer to carry out calculations (it’s impossible to carry out such a volume manually), which made it possible to construct an image of these figures.
Spatial imagination man
Mandelbrot began his scientific career at the IBM Research Center. Studying the possibility of transmitting data over long distances, scientists are faced with the fact of large losses that occurred due to noise interference. Benoit was looking for solutions to this problem. Looking through the measurement results, he drew attention to a strange pattern, namely: 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 transferred any algebraic problem to the geometric field, where the correct answer is obvious. So it is not surprising
that such a person, distinguished by rich
spatial thinking, became the father of fractal geometry. Indeed, awareness of this figure can come only when you study the drawings and think about the meaning of these strange twists that make up the pattern. Fractal patterns do not have identical elements, but they have similarity at any scale.
Julia - Mandelbrot
One of the first drawings of this figure was a graphical interpretation of the set, which was born thanks to the work of Gaston Julia and was finalized by Mandelbrot. Gaston tried to imagine what a set looks like, built on the basis of a simple formula that is iterated by a feedback loop. Let us try to explain what has been said in human language, so to speak, on the fingers. For a specific numerical value, using the formula we find a new value. We substitute it into the formula and find the following. The result is a large numerical sequence. To represent such a multitude, this operation needs to be done a huge number of times: hundreds, thousands, millions. This is what Benoit did. He processed the sequence and transferred the results in graphical form. Subsequently, he painted 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 connected with the visualization of self-similar images, the first to adopt the principles and algorithms for constructing these unusual forms were artists. The first of these was the future founder of Pixar Studios Lauren Carpenter. While working on the presentation of prototypes of airplanes, he came up with the idea to use the image of mountains as a background. Today, almost every computer user will be able to cope with this task, and in the seventies of the last century computers were not able to carry out such processes, because there were no graphic editors and applications for three-dimensional graphics at that time. And here Lauren came across Mandelbrot's book Fractals: Form, Chance, and Dimension. In it, Benoit gave many examples, showing that there are fractals in nature (phyva), he described their various forms 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 developed in response to a flurry of criticism from his colleagues. They claimed that a fractal is just a beautiful picture that has no value, which is a by-product of the operation of electronic machines. Carpenter decided to try this method in practice. Having carefully studied 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 visualize a completely realistic image of the mountain landscape on his computer. And today this principle is widely used. As it turned out, creating fractals does not take much time and effort.
Carpenter's Decision
The principle used by Loren was simple. It consists in dividing larger geometric shapes into smaller elements, and those into smaller ones, and so on. Carpenter, using large triangles, split them into 4 small ones, and so on, until he got a realistic mountain landscape. Thus, he became the first artist to use the fractal algorithm in computer graphics to build the desired image. Today, this principle is used to simulate various realistic natural forms.
The first 3D visualization using a fractal algorithm
A few years later, Lauren applied his groundwork in a large-scale project - the animation video Vol Libre, shown at Siggraph in 1980. This video shocked many, and its creator was invited to work at Lucasfilm. Here, the animator was fully realized, he created three-dimensional landscapes (the whole planet) for the full-length film "Star Trek". Any modern program (Fractals) or an application for creating three-dimensional graphics (Terragen, Vue, Bryce) uses the same algorithm to model textures and surfaces.
Tom Beddard
In the past, a laser physicist, and now a digital artist and artist, Beddard created a series of very intriguing geometric shapes called Faberge fractals. Outwardly, they resemble the decorative eggs of a Russian jeweler; they have the same brilliant intricate pattern. Beddard used the template method to create his digital visualizations of models. The resulting products are striking in their beauty. Although many refuse to compare the handmade product with a computer program, it should be recognized that the resulting forms are unusually beautiful. The highlight is that anyone can build such a fractal using the WebGL software library. It allows you to explore in real time various fractal structures.
Fractals in nature
Few people pay attention, but these amazing figures are everywhere. Nature is created from self-similar figures, we just don’t notice it. Just look through a magnifying glass at our skin or a piece of wood, and we will see fractals. Or take, for example, pineapple or even a peacock's tail - they consist of similar figures. And the cabbage broccoli Romanescu is generally striking in its appearance, because it truly can be called a miracle of nature.
Musical pause
It turns out that fractals are not only geometric shapes, they can also be sounds. So, musician Jonathan Colton writes music using fractal algorithms. He claims that such a melody corresponds to natural harmony. The composer publishes all his works under a CreativeCommons Attribution-Noncommercial license, which provides for the free distribution, copying, transfer of works by others.
Fractal indicator
This technique has found very unexpected application. On its basis, a tool was created for analyzing the stock market, and, as a result, they began to use it on the Forex market. Now the fractal indicator is located on all trading platforms and is used in trading equipment, which is called a price breakthrough. Developed this technique 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, minimum extreme point.
Finally
So we examined what fractal is. It turns out that in the chaos that surrounds us, in fact, there are 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 you need to look at a different scale. We can say with confidence that fractals still have many secrets that we have yet to discover.