Table of contents:
- What is a fractal?
- Geometry and Algebra
- Fractal dimensions
- Science and art
- War and Peace
- Constructive (geometric) fractals
- In the language of mathematics: dynamic (algebraic) fractals
- Complex numbers
- Family of dragons
- Fractals and life
Fractals have been known for a century, have been well studied and have numerous applications in life. However, this phenomenon is based on a very simple idea: a multitude of shapes, infinite in beauty and variety, can be obtained from relatively simple structures using just two operations - copying and scaling.
What do a tree, a seashore, a cloud, or blood vessels in our hand have in common? At first glance, it may seem that all these objects have nothing in common. However, in fact, there is one property of structure inherent in all the listed objects: they are self-similar. From the branch, as well as from the trunk of the tree, there are smaller branches, from them - even smaller ones, etc., that is, the branch is like the whole tree.
The circulatory system is arranged in a similar way: arterioles depart from the arteries, and from them - the smallest capillaries through which oxygen enters the organs and tissues. Let's look at satellite images of the sea coast: we will see bays and peninsulas; let's take a look at it, but from a bird's eye view: we will see bays and capes; Now let's imagine that we are standing on the beach and looking at our feet: there are always pebbles that protrude into the water further than the rest.
That is, the coastline remains similar to itself when zoomed in. The American (though raised in France) mathematician Benoit Mandelbrot called this property of objects fractality, and such objects themselves - fractals (from the Latin fractus - broken).
What is a fractal?
This concept has no strict definition. Therefore, the word "fractal" is not a mathematical term. Typically, a fractal is a geometric figure that satisfies one or more of the following properties: • It has a complex structure at any magnification (as opposed to, for example, a straight line, any part of which is the simplest geometric figure - a line segment). • Is (approximately) self-similar. • Has a fractional Hausdorff (fractal) dimension, which is greater than the topological one. • Can be built with recursive procedures.
Geometry and Algebra
The study of fractals at the turn of the 19th and 20th centuries was rather episodic than systematic, because earlier mathematicians mainly studied "good" objects that were amenable to research using general methods and theories. In 1872, the German mathematician Karl Weierstrass constructs an example of a continuous function that is nowhere differentiable. However, its construction was entirely abstract and difficult to perceive.
Therefore, in 1904, the Swede Helge von Koch invented a continuous curve, which has no tangent anywhere, and it is quite simple to draw. It turned out that it has the properties of a fractal. One of the variants of this curve is called the "Koch snowflake".
The ideas of self-similarity of figures were picked up by the Frenchman Paul Pierre Levy, the future mentor of Benoit Mandelbrot. In 1938, he published his article "Plane and spatial curves and surfaces, consisting of parts similar to the whole", which describes another fractal - the Lévy C-curve. All of these above fractals can be conditionally attributed to one class of constructive (geometric) fractals.
Another class is dynamic (algebraic) fractals, which include the Mandelbrot set. The first studies in this direction began at the beginning of the 20th century and are associated with the names of the French mathematicians Gaston Julia and Pierre Fatou.In 1918, Julia's almost two-hundred-page memoir, devoted to iterations of complex rational functions, was published, in which Julia's sets were described - a whole family of fractals closely related to the Mandelbrot set. This work was awarded the prize of the French Academy, but it did not contain a single illustration, so it was impossible to appreciate the beauty of the objects discovered.
Despite the fact that this work glorified Julia among the mathematicians of the time, it was quickly forgotten. It wasn't until half a century later that computers came to attention again: it was they who made the wealth and beauty of the world of fractals visible.
As you know, the dimension (number of measurements) of a geometric figure is the number of coordinates required to determine the position of a point lying on this figure.
For example, the position of a point on a curve is determined by one coordinate, on a surface (not necessarily a plane) by two coordinates, in three-dimensional space by three coordinates.
From a more general mathematical point of view, you can define the dimension in this way: an increase in linear dimensions, say, twice, for one-dimensional (from a topological point of view) objects (segment) leads to an increase in size (length) twice, for two-dimensional (square) the same increase in linear dimensions leads to an increase in size (area) by 4 times, for three-dimensional (cube) - by 8 times. That is, the "real" (so-called Hausdorff) dimension can be calculated as the ratio of the logarithm of an increase in the "size" of an object to the logarithm of an increase in its linear size. That is, for the segment D = log (2) / log (2) = 1, for the plane D = log (4) / log (2) = 2, for the volume D = log (8) / log (2) = 3.
Let us now calculate the dimension of the Koch curve, for the construction of which the unit segment is divided into three equal parts and the middle interval is replaced by an equilateral triangle without this segment. With an increase in the linear dimensions of the minimum segment three times, the length of the Koch curve increases in log (4) / log (3) ~ 1, 26. That is, the dimension of the Koch curve is fractional!
Science and art
In 1982, Mandelbrot's book "The Fractal Geometry of Nature" was published, in which the author collected and systematized almost all the information available at that time about fractals and presented it in an easy and accessible manner. In his presentation, Mandelbrot made the main emphasis not on cumbersome formulas and mathematical constructions, but on the geometric intuition of the readers. Thanks to computer-generated illustrations and historical tales, with which the author skillfully diluted the scientific component of the monograph, the book became a bestseller, and fractals became known to the general public.
Their success among non-mathematicians is largely due to the fact that with the help of very simple constructions and formulas that a high school student can understand, images of amazing complexity and beauty are obtained. When personal computers became powerful enough, even a whole trend in art appeared - fractal painting, and almost any computer owner could do it. Now on the Internet, you can easily find many sites dedicated to this topic.
War and Peace
As noted above, one of the natural objects with fractal properties is the coastline. One interesting story is connected with it, or rather, with an attempt to measure its length, which formed the basis of Mandelbrot's scientific article, and is also described in his book "The Fractal Geometry of Nature".
This is an experiment that was staged by Lewis Richardson, a very talented and eccentric mathematician, physicist and meteorologist. One of the directions of his research was an attempt to find a mathematical description of the causes and likelihood of an armed conflict between the two countries. Among the parameters that he took into account was the length of the common border of the two warring countries.When he collected data for numerical experiments, he found that in different sources the data on the common border between Spain and Portugal are very different.
This prompted him to discover the following: the length of a country's borders depends on the ruler with which we measure them. The smaller the scale, the longer the border is. This is due to the fact that with a higher magnification it becomes possible to take into account more and more coastal bends, which were previously ignored due to the roughness of the measurements. And if, with each increase in scale, the previously unaccounted for bends of the lines will open, then it turns out that the length of the boundaries is infinite! True, in reality this does not happen - the accuracy of our measurements has a finite limit. This paradox is called the Richardson effect.
Constructive (geometric) fractals
The algorithm for constructing a constructive fractal in the general case is as follows. First of all, we need two suitable geometric shapes, let's call them a base and a fragment. At the first stage, the basis of the future fractal is depicted. Then some of its parts are replaced with a fragment taken at a suitable scale - this is the first iteration of construction. Then, the resulting figure again changes some parts into figures similar to a fragment, and so on. If we continue this process to infinity, then in the limit we get a fractal.
Let's consider this process using the Koch curve as an example. As a basis for the Koch curve, you can take any curve (for the "Koch snowflake" it is a triangle). But we will restrict ourselves to the simplest case - a segment. A fragment is a broken line shown at the top in the figure. After the first iteration of the algorithm, in this case, the initial segment will coincide with the fragment, then each of its constituent segments will be replaced by a broken line, similar to a fragment, etc. The figure shows the first four steps of this process.
In the language of mathematics: dynamic (algebraic) fractals
Fractals of this type arise in the study of nonlinear dynamical systems (hence the name). The behavior of such a system can be described by a complex nonlinear function (polynomial) f (z). Take some starting point z0 on the complex plane (see sidebar). Now consider such an infinite sequence of numbers on the complex plane, each of the following of which is obtained from the previous one: z0, z1 = f (z0), z2 = f (z1),… zn + 1 = f (zn).
Depending on the initial point z0, such a sequence can behave differently: tend to infinity as n -> ∞; converge to some end point; cyclically take a number of fixed values; more complex options are also possible.
A complex number is a number consisting of two parts - real and imaginary, that is, the formal sum x + iy (x and y here are real numbers). i is the so-called. imaginary unit, that is, that is, a number that satisfies the equation i ^ 2 = -1. The basic mathematical operations are defined over complex numbers - addition, multiplication, division, subtraction (only the comparison operation is not defined). To display complex numbers, a geometric representation is often used - on the plane (it is called complex), the real part is laid on the abscissa, and the imaginary part on the ordinate, while the complex number will correspond to a point with Cartesian coordinates x and y.
Thus, any point z of the complex plane has its own character of behavior during iterations of the function f (z), and the entire plane is divided into parts. In this case, the points lying on the boundaries of these parts have the following property: for an arbitrarily small displacement, the nature of their behavior changes sharply (such points are called bifurcation points). So, it turns out that sets of points with one specific type of behavior, as well as sets of bifurcation points, often have fractal properties. These are the Julia sets for the function f (z).
Family of dragons
By varying the base and fragment, you can get an amazing variety of constructive fractals.
Moreover, similar operations can be performed in three-dimensional space. Examples of volumetric fractals are Menger's sponge, Sierpinski pyramid and others.
The dragon family is also referred to as constructive fractals. Sometimes they are called by the name of the discoverers "dragons of the Highway Harter" (in their form they resemble Chinese dragons). There are several ways to plot this curve. The simplest and most intuitive of them is this: you need to take a sufficiently long strip of paper (the thinner the paper, the better), and fold it in half. Then bend it twice again in the same direction as the first time.
After several repetitions (usually after five to six folds, the strip becomes too thick to be neatly bent further), you need to unbend the strip back, and try to form 90˚ angles at the folds. Then the curve of the dragon will turn out in profile. Of course, this will only be an approximation, like all our attempts to depict fractal objects. The computer allows you to depict many more steps in this process, and the result is a very beautiful figure.
The Mandelbrot set is constructed in a slightly different way. Consider the function fc (z) = z ^ 2 + c, where c is a complex number. Let us construct a sequence of this function with z0 = 0, depending on the parameter c, it can diverge to infinity or remain bounded. Moreover, all the values of c for which this sequence is bounded form the Mandelbrot set. It was studied in detail by Mandelbrot himself and other mathematicians, who discovered many interesting properties of this set.
It is seen that the definitions of the Julia and Mandelbrot sets are similar to each other. In fact, these two sets are closely related. Namely, the Mandelbrot set is all the values of the complex parameter c for which the Julia set fc (z) is connected (a set is called connected if it cannot be split into two disjoint parts, with some additional conditions).
Fractals and life
Today, the theory of fractals is widely used in various fields of human activity. In addition to a purely scientific object for research and the already mentioned fractal painting, fractals are used in information theory to compress graphic data (here the self-similarity property of fractals is mainly used - after all, to remember a small fragment of a drawing and transformations with which you can get the rest of the parts, it takes much less memory than for storing the entire file).
By adding random perturbations to the formulas defining the fractal, one can obtain stochastic fractals that very plausibly convey some real objects - relief elements, the surface of water bodies, some plants, which is successfully used in physics, geography and computer graphics to achieve greater similarity of simulated objects with real. In electronics, antennas are produced that have a fractal shape. Taking up little space, they provide quite high-quality signal reception.
Economists use fractals to describe currency rate curves (a property discovered by Mandelbrot). This concludes this small excursion into the amazingly beautiful and diverse world of fractals.