often brilliant discoveries made in science, to radically change our lives.For example, the invention of the vaccine can save a lot of people, and the creation of new weapons leads to murder.Just yesterday (on a scale of history) a man "tamed" electricity, and today can not imagine my life without him.However, there are openings that is called, remain in shadow, and despite the fact that they also have some effect on our life.One of these discoveries was the fractal.Most people have not even heard of such a thing and will not be able to explain its meaning.In this article we try to deal with the question of what a fractal, consider the meaning of this term from the perspective of science and nature.
order in chaos
To understand what a fractal should begin debriefing from the perspective of mathematics, but before going into the sciences, we are a little philosophize.Each person is inherent natural curiosity, thanks to which he knows the world.Often in his quest of knowledge he tries to operate the logic of judgment.Thus, analyzing the processes taking place around, he tries to calculate the relationship and bring certain laws.The greatest minds of the world are busy these tasks.Roughly speaking, our scientists are looking for patterns where there are none, and should not be.Yet even in the chaos of a connection between certain events.It is this connection and acts fractal.As an example, consider a broken branch, which is dragged along the road.If we carefully look at it, we see it with all its branches and twigs like a tree herself.It is this similarity to a separate part of a unit indicates the so-called principle of recursive self-similarity.Fractals can be found in nature all the time, as many inorganic and organic forms are formed similarly.This cloud, and sea shells and snail shells, and the trees, and even the blood system.The list is endless.All these random shapes with ease describes a fractal algorithm.Here we come to that, to see what a fractal from the position of the exact sciences.
little dry facts
word "fractal" from Latin translates as "partial", "separation", "fragmented", and as for the content of this term, the wording as such does not exist.Usually it is interpreted as a self-similar set, part of the whole, which is repeated in their structure at the micro level.This term coined in the seventies of the twentieth century, Benoit Mandelbrot, who is recognized as the father of fractal geometry.Today, the term is meant fractal graphic image of a structure that at a larger scale will be similar to myself.However, the mathematical basis for the creation of this theory has been laid before the birth of the Mandelbrot set, but she could not develop until electronic computers.
History, or How it all began
At the turn of 19-20 centuries, the study of the nature of fractals was of sporadic.This is due to the fact that mathematicians prefer to study objects that are amenable to a study based on general theories and methods.In 1872, German mathematician Weierstrass constructed an example of continuous functions nowhere differentiable.However, this construction proved to be entirely abstract and difficult to understand.Then I went to the Swede Helge von Koch, who in 1904 built a continuous curve, not having anywhere tangent.It's pretty easy to draw, and, as it turned out, it is characterized by fractal properties.One of the variants of the curve was named after the author - "Koch snowflake."Further, the idea of self-similarity of figures developed future mentor B. Mandelbrot Frenchman Paul Levy.In 1938, he published an article entitled "Plane and space curves and surfaces consisting of parts similar to the whole."In it, he described a new species - the C-curve Levy.All the above figures are conditional to such a form as geometric fractals.
dynamic or algebraic fractals
This class is the Mandelbrot set.The first explorers in this direction was the French mathematician Pierre Fatou and Gaston Julia.In 1918, Julia published work, which was based on the study of complex iterations of rational functions.Here he describes a family of fractals, which are closely related to the Mandelbrot set.Despite the fact that the author of this work is glorified among mathematicians about it was quickly forgotten.Only half a century later thanks to computers work Julia got a second life.Computers allowed to make visible for everyone the beauty and richness of the world of fractals, which could "see" the mathematics, displaying them in terms of functions.Mandelbrot was the first to use the computer to perform calculations (manual that amount is impossible to draw), allowed to build the image of these figures.
Man spatial imagination
Mandelbrot began his academic career at the research center of IBM.Studying the possibility of data transmission over long distances, scientists are faced with the fact of large losses, which arose because of the noise.Benoit was looking for ways to solve this problem.Looking through the results of the measurements, he noticed a strange pattern, namely noise charts look the same in a different time scale.A similar pattern was observed for the period of one day, and for seven days or hours.Benoit Mandelbrot himself used to say that it does not work with formulas, and plays with the images.This distinguished scientist imaginative thinking, any algebraic problem he translated into a geometric area where the correct answer is obvious.So it is no wonder that such a man is rich spatial reasoning, and became the father of fractal geometry.After the realization of this figure can only come when studying pictures and ponders the meaning of those strange twists forming a pattern.Fractal patterns have identical elements, but has similar at any scale.
Julia - Mandelbrot
One of the first pictures of the figure has a graphical interpretation of the set, which was born thanks to the work of Gaston Julia and Mandelbrot was finalized.Gaston tried to imagine what it looks like a lot, built on the basis of a simple formula, which is iterated feedback loop.Let us try to explain the spoken human language, so to speak, on his fingers.For a specific numerical value using the formula we find new meaning.Substituting it into formula and find the following.The result is a large number sequence.To represent such a set is required to do this operation a great number of times: hundreds, thousands, millions.This is done Benoit.He handled the sequence and transferred the results in graphical form.Later he painted the resulting shape (each color corresponds to a certain number of iterations).This graphic image was named "Mandelbrot fractal."
L. Carpenter: art, created by nature
theory of fractals quickly found practical application.Since it is very closely linked with the visualization of self-similar images, the first who has adopted the principles and algorithms for constructing these unusual shapes have become artists.The first of them was the future founder of Pixar Lauren Carpenter.Working on the presentation of the prototype aircraft, he came up with the idea to use the image as the background of the mountains.Today, such a task can handle practically every computer user, and in the seventies of last century, computers have been unable to carry out such processes, as image editors and applications for three-dimensional graphics at the time was not yet.And Lauren got Mandelbrot's book "Fractals: Form, and the dimension of an accident."It Benoit led many examples showing that there are fractal in nature (fyva), he described their various forms and prove that they are easily described mathematical expressions.This analogy mathematician argued the usefulness of his theory developed in response to a barrage of criticism from his colleagues.They argued that the fractal - it's just a pretty picture, which has no value, which is a by-product of the electronic machines.Carpenter decided to try it in practice.Having carefully reviewed the book, the future animator began looking for a way of realization of fractal geometry in computer graphics.It took him three days to fully visualize a realistic image of a mountain landscape on your computer.Today, this principle is widely used.As it turned out, the creation of fractals do not take a lot of time and effort.
Solution Carpenter
principle used Lauren, was simple.It is to split the larger geometric shapes into small elements, and those - of similar smaller and so on.Carpenter using large triangles, crush them into 4 small, and so on, as long as he did not get a realistic mountain landscape.Thus, he became the first artist who used the fractal algorithm in computer graphics to build the desired image.Today, this principle is used to simulate a variety of realistic natural forms.
first 3D-visualization of a fractal algorithm
Within a few years, Lauren has applied its experience in large-scale project - the animated video Vol Libre, shown at Siggraph 1980.This video has shocked many people, and its creator was invited to work at Lucasfilm.There animator could be realized to the full, he created a three-dimensional landscape (the whole planet) for the feature film "Star Trek".Every modern program ("Fractals") or an application for creating three-dimensional graphics (Terragen, Vue, Bryce) uses all the same algorithm for the simulation of textures and surfaces.
Tom Beddard
In the past, laser physics, and now digital-maker and artist, Beddard created a number of very intriguing geometric shapes called fractals are Faberge.Outwardly, they look like miniature eggs Russian jeweler, they are the same brilliant intricate pattern.Beddard generic method used to create digital renderings of their models.These products are striking for its beauty.Although many refuse to compare the product handmade by a computer program, but it should be recognized that the resulting forms unusually beautiful.The highlight is that to build this fractal can anyone using the software library WebGL.It allows you to explore a variety of real-time fractal structures.
Fractals in Nature
few people pay attention, but these amazing figures are everywhere.Nature created from self-similar shapes, but we do not notice.Just look through the magnifying glass on our skin or a piece of wood, and we will see fractals.Or take, for example, pineapple or even a peacock's tail - they are composed of similar figures.A variety of broccoli Romanescu generally affects their appearance, because it truly can be called a miracle of nature.
pauses
turns, fractals - is not only geometric shapes, they can be and sounds.So, musician Jonathan Coulton writes music via fractal algorithms.He argues that such a melody corresponds to the natural harmony.Composer publishes all his works licensed under CreativeCommons Attribution-Noncommercial, which provides for free distribution, copying, transmission of works by others.
indicator fractal
This technique has been very unexpected application.On the basis of its established tool for analyzing the stock exchange market, and as a result, it began to be applied on the market "Forex".Now the fractal indicator is on all trading platforms and is used in the trade technology, which is called the price breakout.He developed the technique Bill Williams.As commented on his invention of the author, this algorithm is a combination of several "candles", which reflects the central maximum or, conversely, a minimum extreme point.
In conclusion
So we looked at what the fractal.It turns out that in the chaos that surrounds us, in fact, there are ideal forms.Nature is the best architect, builder and engineer perfect.She arranged a very logical, and if we can not find a pattern, it does not mean that it is not.Maybe you need to look in a different scale.It is safe to say that fractals store has a lot of secrets that we have yet to discover.
