What is a fractal? Fractals in nature

Often the brilliant discoveries made in science are able to radically change our lives. For example, the invention of a vaccine can save a lot of people, and the creation of new weapons leads to murder. Literally yesterday (in the scale of history) a person "tamed" electricity, and today can not imagine his life without him. However, there are also such discoveries, which, they say, remain in the shadows, and despite the fact that they also have some influence on our lives. One of such discoveries was a fractal. Most people have not 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, consider the meaning of this term from the position of science and nature.

Order in Chaos

In order to understand what a fractal is, we should start the analysis of flights from the position of mathematics, but before delving into the exact sciences, we are slightly philosophical. Each person has a natural curiosity, through which he knows the world around him. Often in his quest for knowledge, he tries to operate with logic in his judgments. So, analyzing the processes that occur around, he tries to calculate the relationships and derive certain regularities. The greatest minds of the planet are engaged in solving these problems. Roughly speaking, our scientists are looking for patterns where they do not exist, and should not be. And yet, even in chaos, there is a connection between these or those events. That's the link that the fractal is. As an example, consider a broken branch lying on the road. If you look closely at it, we will see that it is like a tree with all its branches and knots. Here 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 very often, because many inorganic and organic forms are formed similarly. This is the clouds, and sea shells, and shells of snails, and the crowns of trees, and even the circulatory system. This list can be continued indefinitely. All these random forms easily describe a fractal algorithm. Here we have come to consider what a fractal is from the position of exact sciences.

A few dry facts

The word "fractal" in Latin is translated as "partial", "divided", "fragmented", and as to the content of this term, there is no formulation as such. Usually it is treated as a self-similar set, part of the whole, which is repeated by its structure at the micro level. This term was invented in the seventies of the twentieth century by Benoit Mandelbrot, who is recognized as the father of fractal geometry. Today, the concept of a fractal implies a graphic representation of a structure that, like an enlarged scale, 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 reference, or How it all began

At the turn of the 19th and 20th centuries, the study of the nature of fractals was episodic. 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 K. 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. Next came the Swede Helge von Koch, who in 1904 built a continuous curve, which nowhere has a tangent. It is fairly easy to draw, and as it turned out, it is characterized by fractal properties. One of the variants of this curve was named in honor of its author - "snowflake Koch". Further, the idea of the self-similarity of the figures was developed by the future instructor of B. Mandelbrot, Frenchman Paul Levy. In 1938 he published an article "Flat and spatial curves and surfaces consisting of parts like the whole." In it he described a new species - Levy's C-curve. All of the above figures are conditionally related to the form, as geometric fractals.

Dynamic, or algebraic fractals

This class includes the Mandelbrot set. The first researchers of 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. The computers allowed to make visible to each person that beauty and wealth of the world of fractals that could "see" mathematicians, displaying them through functions. Mandelbrot was the first who used a computer to perform calculations (such a volume can not be manually done), which allowed to construct an image of these figures.

A man with spatial imagination

Mandelbrot began his scientific career at IBM's research center. Studying the possibilities of data transmission 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 at the results of measurements, he drew attention to a strange pattern, namely: the noise graphs looked the same in different time scales. A similar picture was observed both for a period of one day, and for seven days or for an hour. Benois Mandelbrot himself often said that he does not work with formulas, but plays with pictures. This scientist differed in figurative thinking, he translated any algebraic problem into a geometric domain, where the correct answer is obvious. So it is not surprising that such a person, who is rich in spatial thinking, became the father of fractal geometry. After all, awareness of this figure can only come when you study the drawings and ponder the meaning of these strange twists that form the pattern. Fractal patterns do not have identical elements, but they have a similarity at any scale.

Julia - Mandelbrot

One of the first drawings of this figure was the graphic interpretation of the set, which was born thanks to the works of Gaston Julia and was modified by Mandelbrot. Gaston tried to imagine how the set looks like, built on the basis of a simple formula, which is lost by the feedback loop. Let's try to explain what has been said with human language, so to speak, on your fingers. For a particular numerical value, we use the formula to 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 set, it is required 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 a graphic form. Subsequently, he painted 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 connected with the visualization of self-similar images, the first to adopt the principles and algorithms for constructing these unusual forms are the artists. The first of these was the future founder of the Pixar studio Lauren Carpenter. Working on the presentation of aircraft prototypes, it occurred to him to use the image of mountains as a background. Today, almost every computer user can handle this task, and in the seventies of the last century computers could not perform such processes, because there were no graphics editors and applications for 3D graphics at that time. And so Loren caught Mandelbrot's book "Fractals: Form, Accident and Dimension." In it, Benoit gave many examples, showing that there are fractals in nature (fyva), he described their diverse form and argued 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 argued that the fractal - it's just a beautiful picture, which has no value, which is a by-product of the work of electronic machines. Carpenter decided to test 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 visualize a very realistic image of the mountain landscape on his computer. And today this principle is widely used. As it turned out, the creation of fractals does not take much time and effort.

Carpenter's solution

The principle used by Lauren was simple. It consists in dividing the larger geometric figures into smaller elements, and those into smaller ones of the same size, and so on. Carpenter, using large triangles, crushed them into 4 small ones, and so on, until he got a realistic mountain landscape. Thus, he became the first artist who applied the fractal algorithm in computer graphics to build the required image. Today, this principle is used to simulate various realistic natural forms.

The first 3D-visualization on a fractal algorithm

A few years later, Lauren used his work in a large-scale project - the animation clip Vol Libre, shown on the Siggraph in 1980. This video shocked many, and its creator was invited to work in Lucasfilm. Here the animator was able to realize to the fullest extent, 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 for modeling textures and surfaces.

Tom Beddard

In the past, a laser physicist, now a digital master and artist, Beddard created a series of very intriguing geometric figures, which he called Faberge fractals. Outwardly, they resemble the decorative eggs of a Russian jeweler, with the same brilliant intricate pattern. Beddard used a template method to create his own digital visualizations of models. The received products amaze with the beauty. Although many refuse to compare the product handmade with a computer program, it should be recognized that the forms obtained are unusually beautiful. A highlight is that anyone can build such a fractal using the WebGL software library. It allows you to explore in real time different fractal structures.

Fractals in nature

Few people pay attention, but these amazing figures are everywhere. Nature is created from self-similar figures, we just do not notice it. It is enough to look through a magnifying glass on 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. A variety of broccoli broccoli Romanescu generally amazes its appearance, because it truly can be called a miracle of nature.

Musical pause

It turns out that fractals are not only geometric figures, they can 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 the CreativeCommons Attribution-Noncommercial license, which provides for the free distribution, copying, transfer of works by other persons.

Indicator-fractal

This technique has found a very unexpected application. On its basis, a tool for analyzing the stock market market has been created, and as a result, it was used in the Forex market. Now the indicator-fractal is on all trading platforms and is used in trading equipment, which is called a price break. Developed this technique Bill Williams. As the author comments on his invention, this algorithm is a combination of several "candles", in which the central 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, in fact, there are ideal forms. Nature is the best architect, the ideal builder and engineer. It is arranged very logically, and if we can not find the law, this does not mean that it does not exist. Maybe you need to look for it on a different scale. We can say with confidence that fractals store many more secrets that we have yet to discover.