What do a tree, a coast, a cloud or blood vessel have in our hands? At first glance, nothing seems to unite all these objects. However, in fact, all of these objects have an inherent structural property: they are self-similar. From branches, and from trunks, smaller processes, from them - even smaller, etc., that is, branches are like the whole tree. At this point, they resemble one of the most beautiful mathematical objects—the fractal.
In a similar way, the circulatory system is arranged: arterioles leave the arteries and enter the smallest capillaries of the organs and tissues from them - oxygen . Let's look at the satellite image of the coast: we will see the bay and the peninsula; let's see it, but from the aerial view: we will see the bay and the promontory; now imagine we stand on the beach and look at our feet: there will always be pebbles that get more water into the water than other pebbles. That is, the increasingly large coastline is still similar to itself. This property of an object is called fractal by the American (although he grew up in France) mathematician Benoit Mandelbrot, which itself is called fractal (from Latin fractal - broken).
What is fractal?
If you look at a lot of fractals, you can see that they have many differences. These differences are observed not only in the shapes of the figures that make up the fractals, but also in the representations of these sets. Therefore, there are differences between geometric, algebra and random fractals. Let's discuss each of them in more detail.
Geometric Fractal
This is the fractal type we are most familiar with. They are built on any geometry by segmenting its parts and transforming them. Examples include the L system. They were originally designed to simulate biological cell systems, but can also be applied to other branch systems.
Pythagoras trees are one of the simplest examples of geometric fractals
algebraic fractals
algebraic fractals are built on the basis of mathematical formulas - if you build a graph on a coordinate plane, they can be converted to geometric formulas. Algebraic fractals include the Mandeblot fractals, Julia and Newton basins. They are all built on a set of complex numbers, which consist of real numbers and imaginary parts. It is just that the fractals of Mandeblot and Julia are based on complex squares, while the Newton Basin is based on cubes.
This is what Newton's pool looks like - fractals are built on many complex cubes
random fractals
This fractal is built on the basis of mathematical formulas, but during the construction process, the parameters in it change randomly. This leads to the emergence of strange forms, very similar to natural forms. Unlike geometry and some algebra, random fractals can only be constructed using a computing mechanism. Illustration of
random fractals. As you can see, they can be asymmetrical and very strange.
Geometric and algebra
At the turn of the nineteenth and twentieth centuries, the study of fractals was more contingent than systematic, because early mathematicians focused on "good" objects that could be studied using general methods and theoretically. In 1872, the German mathematician Karl Weierstrass established an example of the continuous function , which is differentiable anywhere. However, its construction is completely abstract and difficult to understand. Therefore, in 1904, the Swedish Helge von Koch proposed a continuous curve with no tangent anywhere and was easy to draw. It turns out that it has fractal properties. A variant of this curve is called "Koch Snowflake".
What is fractal? The concept of
has no strict definition. Therefore, the word "fractal" is not a mathematical term. Typically, fractals are geometric figures that satisfy one or more of the following properties: • Have complex structures at any scaling (for example, unlike straight lines, any part of it is the simplest geometry - line segment). • (approximately) self-similar.
What do a tree, a coast, a cloud or blood vessel have in our hands? At first glance, nothing seems to unite all these objects. However, in fact, all of these objects have an inherent structural property: they are self-similar. From branches, and from trunks, smaller processes, from them - even smaller, etc., that is, branches are like the whole tree. At this point, they resemble one of the most beautiful mathematical objects—the fractal.
In a similar way, the circulatory system is arranged: arterioles leave the arteries and enter the smallest capillaries of the organs and tissues from them - oxygen . Let's look at the satellite image of the coast: we will see the bay and the peninsula; let's see it, but from the aerial view: we will see the bay and the promontory; now imagine we stand on the beach and look at our feet: there will always be pebbles that get more water into the water than other pebbles. That is, the increasingly large coastline is still similar to itself. This property of an object is called fractal by the American (although he grew up in France) mathematician Benoit Mandelbrot, which itself is called fractal (from Latin fractal - broken).
What is fractal?
If you look at a lot of fractals, you can see that they have many differences. These differences are observed not only in the shapes of the figures that make up the fractals, but also in the representations of these sets. Therefore, there are differences between geometric, algebra and random fractals. Let's discuss each of them in more detail.
Geometric Fractal
This is the fractal type we are most familiar with. They are built on any geometry by segmenting its parts and transforming them. Examples include the L system. They were originally designed to simulate biological cell systems, but can also be applied to other branch systems.
Pythagoras trees are one of the simplest examples of geometric fractals
algebraic fractals
algebraic fractals are built on the basis of mathematical formulas - if you build a graph on a coordinate plane, they can be converted to geometric formulas. Algebraic fractals include the Mandeblot fractals, Julia and Newton basins. They are all built on a set of complex numbers, which consist of real numbers and imaginary parts. It is just that the fractals of Mandeblot and Julia are based on complex squares, while the Newton Basin is based on cubes.
This is what Newton's pool looks like - fractals are built on many complex cubes
random fractals
This fractal is built on the basis of mathematical formulas, but during the construction process, the parameters in it change randomly. This leads to the emergence of strange forms, very similar to natural forms. Unlike geometry and some algebra, random fractals can only be constructed using a computing mechanism. Illustration of
random fractals. As you can see, they can be asymmetrical and very strange.
Geometric and algebra
At the turn of the nineteenth and twentieth centuries, the study of fractals was more contingent than systematic, because early mathematicians focused on "good" objects that could be studied using general methods and theoretically. In 1872, the German mathematician Karl Weierstrass established an example of the continuous function , which is differentiable anywhere. However, its construction is completely abstract and difficult to understand. Therefore, in 1904, the Swedish Helge von Koch proposed a continuous curve with no tangent anywhere and was easy to draw. It turns out that it has fractal properties. A variant of this curve is called "Koch Snowflake".
What is fractal? The concept of
has no strict definition. Therefore, the word "fractal" is not a mathematical term. Typically, fractals are geometric figures that satisfy one or more of the following properties: • Have complex structures at any scaling (for example, unlike straight lines, any part of it is the simplest geometry - line segment). • (approximately) self-similar.• Fractal Hausdorf (fractal) dimensions with greater than topology dimensions. • Can be constructed through recursive processes.
The idea of self-similarity of characters was accepted by the Frenchman Paul Pierre Levy, the future mentor of Benoit Mandebrot. In 1938, his article "Plane and Space Curves and Surfaces composed of Whole Parts", which describes another fractal, the Levy C Curve. All of these fractals described above can be conditionally attributed to a class of constructive (geometric) fractals.
Many natural objects, such as plant stems, are also fractals with a limited number of repeating elements.
Another category is dynamic (algebraic) fractals, which belongs to the Mandeblot collection. The first study of this area began in the early twentieth century and was related to the names of French mathematicians Gaston Julia and Pierre Fatu. In 1918, Julia published nearly two hundred pages of memoirs about iteration of complex rational functions, describing the Julia collection, an entire fractal family closely related to the Mandeblott collection. This work won the Academy of France award, but it does not contain an illustration, so it cannot appreciate the beauty of the items found. Although this work made Julia famous among mathematicians of the time, it was quickly forgotten. Half a century later, with the advent of computers, people's attention turned to it again: it was they that made the richness and beauty of the fractal world visible.
Fractal dimension
As you know, the size (number of dimensions) of a geometric figure is the number of coordinates required to determine the position of the point located on this figure.
For example, the position of points on a curve is determined by one coordinate, on a surface (not necessarily a plane) by two coordinates, and in three dimensions.
From a more general mathematical perspective, the dimension can be determined by: the increase in linear dimensions, for example, twice, for one-dimensional (from a topological perspective) object (segment) causes a triple in size (length) for one-dimensional (from a topological perspective), for two-dimensional (square) for the same linear dimension increase in size (area) and 8 times for three-dimensional (cube). That is to say, "real" (so-called Hausdorf) -dimensional can be calculated as the ratio of the logarithm of the increase in the "size" of the object to the logarithm of its linear size. That is, for segment D=log(2)/log(2)=1, for plane D=log(4)/log(2)=2, and for volume D=log(8)/log(2)=3.
Now let's calculate the dimensions of the Koch curve, for which the structure of the curve, a single segment is divided into three equal parts and replaced by an equilateral triangle without the segment. When the linear size of the smallest segment triples, the length of the Koch curve increases in logarithm (4)/logarithm (3) ~ 1.26. In other words, the dimension of the Koch curve is fraction!
Science and Art
In 1982, Mandeblot's book "Natural Fractal Geometry" was published. In the book, the author collected and systematically described almost all the information about fractals at that time and presented it in a simple and easy-to-understand way. In his speech, Mandeblot mainly emphasized not the emphasis on formulas and mathematical structures, but the reader's geometry intuition. Thanks to illustrations and historical stories obtained with the help of computers, the author cleverly diluted the scientific components of the monograph, which became a bestseller and fractals became publicly known. Their success among non-mathematicians is largely due to the fact that images of amazing complexity and beauty can be obtained with the help of very simple structures and formulas that high school students can understand. When personal computers become quite powerful, art even has a full direction – fractal painting, which almost any computer owner can do. Now on the internet, you can easily find many websites dedicated to this topic.
Operating sequence required to construct Koch fractals
War and Peace
As mentioned above, one of the natural objects with fractal properties is the coastline. With this, or rather, trying to measure its length, there is an interesting story that forms the basis of Mandeblot's scientific articles and is also described in his book "Natural Fractal Geometry."
We are talking about an experiment conducted by Lewis Richardson, a very talented and quirky mathematician, physicist and meteorologist. One of his research directions is to try to find a mathematical description of the causes and possibilities of armed conflict between the two countries. It seems that what does fractal have to do with this?
But one of the parameters scientists consider is the length of the common boundary of the two belligerent countries. When he collected numerical experimental data, he found that among different sources, data on the common boundary between Spain and Portugal were very different. This prompted him to make the following discovery: The length of the country's borders depends on the rulers we use to measure them. The smaller the scale, the longer the boundaries obtained, because with greater increase, it is possible to consider new bends of more and more banks that have been previously ignored due to the roughness of the measurement. And if the previously unexplained line bends are opened every time you zoom, it turns out that the length of the boundary is infinite! Just like mathematical fractals. However, this actually didn't happen - there was a limited limitation on the accuracy of our measurements. This paradox is called the Richardson effect. The example of
Geometric Fractals illustrates how to combine multiple repeating elements into one object at the same time
construct (geometric) fractal
construct constructing algorithms for constructing fractals are usually as follows. First, we need two suitable geometry , which we call base and fragment. In the first phase, the basis of future fractals is depicted. Then replace some of its parts with fragments of appropriate proportions - this is the first iteration of the construction. Then, in the resulting figure, some parts are changed again to shapes similar to fragments, etc. If you continue the process indefinitely, then in the limit you will get a fractal.
Consider this process using the Koch curve as an example. Any curve can be used as the basis for the Koch curve (for the "Koch Snowflake", it is a triangle). But we will limit ourselves to the simplest case, namely, market segmentation. The fragments are damaged, depicted at the top of the picture. After the first iteration of the algorithm, in this case the original fragment will coincide with the fragment, then each of its constituent fragments itself will be replaced by a broken fragment-like fragment, and so on, the figure shows the first four steps of this process.
fractals can even be found in plants. For example, it is the fruit of Romanesque cabbage
In mathematical language : Dynamic (algebraic) fractal
This type of fractal appears in the study of nonlinear dynamic systems (hence the name). The behavior of this system can be described by the complex nonlinear function ( polynomial )f(z). Let's take some starting point z0 on the complex plane (see sidebar). Now consider such an infinite sequence of numbers on the complex plane, each sequence of numbers obtained from the previous sequence: z0, z1 = f (z0), z2 = f (z1), ...zinc +1 = f (zinc). Depending on the starting point z0, such sequences can behave differently: tending to infinity at n - ∞; converging to a certain endpoint; periodically taking a series of fixed values; more complex options are possible.
Complex
Complex is a number composed of two parts: real number and imaginary number , that is, the sum of the form of x + iy (here x and y are real numbers). i is the so-called imaginary unit, that is, a number that satisfies the equation i^2 = -1. On complex numbers, the main mathematical operations are defined - addition, multiplication, division, subtraction (only comparison operations are not defined). To display complex numbers, geometric representations are usually used - on a plane (called complex numbers), the actual part is deposited along the abscissa axis, the imaginary part is deposited along the ordinate, and points with Cartesian x and y coordinates will correspond to the complex number.
Therefore, any point z of the complex plane has its own behavior pattern in the iteration of the function f (z), and the entire plane is divided into multiple parts. At the same time, points located on these partial boundaries have the following properties: At any small displacement, their behavioral properties will change dramatically (these points are called bifurcation points).Therefore, it turns out that point sets and bifurcation point sets with one specific behavior type usually have fractal properties. These are the Julia sets of the function f(z).
Dragon tribe
By changing the cardinality and fragments, you can get various constructive fractals.
In addition, similar operations can be performed in three-dimensional space. Examples of volume fractals are "Mengel sponge", "Sherpings Foundation Pyramid", etc.
constructive fractals include the dragon family. Sometimes they are called "hehe hath dragon" by the discoverer (in their form they are similar to Chinese dragon ). There are several ways to construct this curve. The simplest and most intuitive thing about it is: you need to take a strip of paper that is long enough (the thinner the paper is better) and bend it in half. Then bend it halfway again, in the same direction as the first time. After a few repetitions (usually after five to six folds, the strip becomes too thick to bend further gently), you need to straighten the strip and try to form a 90° angle at the folds. Then in the configuration file, you will get the dragon curve. Of course, this is just an approximation, like all our attempts to depict fractal objects. The computer allows you to depict more steps in this process, and the result is a very beautiful graphic.
The construction of the Mandeblot collection is slightly different. Consider the function fc (z) = z^2+c, where c is a complex number. Let's build a sequence of this function with z0=0, depending on its parameters that can diverge to infinity or remain finite. In this case, all values c of this sequence are restricted to form the Mandeblot set. Mandeblot himself and other mathematicians studied it in detail, and they discovered many interesting properties of this set.
shows that the Julia and Mandeblot sets are similar in definitions. In fact, these two sets are closely related. That is, the Mandeblot set is all values of a complex parameter c, and the Julia set fc (z) is connected to that parameter c (if the set cannot be divided into two disjoint parts, it is called a join, but with some additional conditions).
fractal and life
Today, fractal theory is widely used in various fields of human activities. In addition to purely scientific objects used for research and already mentioned fractal painting, fractals are also used in information theory to compress graph data (the properties of fractal self-similarity are mainly used here - after all, remember a small fragment of drawing and transformation, you can use it to get the rest, much less memory than it takes to store the entire file). By adding random perturbations to the formula for specifying fractals, random fractals are obtained that very reasonably convey some real objects - relief elements, water surfaces, some plants, which are successfully used in physics, geographic and computer graphics to achieve greater similarity between simulated objects and real objects. In Radio Electronics , the produced antenna has a fractal shape. They take up little space and provide quite high-quality signal reception. Economists use fractals to describe currency fluctuations (a property discovered by Mandebrot). With this we will complete this small excursion into the amazing beauty and diversity of the fractal world.
We are talking about an experiment conducted by Lewis Richardson, a very talented and quirky mathematician, physicist and meteorologist. One of his research directions is to try to find a mathematical description of the causes and possibilities of armed conflict between the two countries. It seems that what does fractal have to do with this?
But one of the parameters scientists consider is the length of the common boundary of the two belligerent countries. When he collected numerical experimental data, he found that among different sources, data on the common boundary between Spain and Portugal were very different. This prompted him to make the following discovery: The length of the country's borders depends on the rulers we use to measure them. The smaller the scale, the longer the boundaries obtained, because with greater increase, it is possible to consider new bends of more and more banks that have been previously ignored due to the roughness of the measurement. And if the previously unexplained line bends are opened every time you zoom, it turns out that the length of the boundary is infinite! Just like mathematical fractals. However, this actually didn't happen - there was a limited limitation on the accuracy of our measurements. This paradox is called the Richardson effect. The example of
Geometric Fractals illustrates how to combine multiple repeating elements into one object at the same time
construct (geometric) fractal
construct constructing algorithms for constructing fractals are usually as follows. First, we need two suitable geometry , which we call base and fragment. In the first phase, the basis of future fractals is depicted. Then replace some of its parts with fragments of appropriate proportions - this is the first iteration of the construction. Then, in the resulting figure, some parts are changed again to shapes similar to fragments, etc. If you continue the process indefinitely, then in the limit you will get a fractal.
Consider this process using the Koch curve as an example. Any curve can be used as the basis for the Koch curve (for the "Koch Snowflake", it is a triangle). But we will limit ourselves to the simplest case, namely, market segmentation. The fragments are damaged, depicted at the top of the picture. After the first iteration of the algorithm, in this case the original fragment will coincide with the fragment, then each of its constituent fragments itself will be replaced by a broken fragment-like fragment, and so on, the figure shows the first four steps of this process.
fractals can even be found in plants. For example, it is the fruit of Romanesque cabbage
In mathematical language : Dynamic (algebraic) fractal
This type of fractal appears in the study of nonlinear dynamic systems (hence the name). The behavior of this system can be described by the complex nonlinear function ( polynomial )f(z). Let's take some starting point z0 on the complex plane (see sidebar). Now consider such an infinite sequence of numbers on the complex plane, each sequence of numbers obtained from the previous sequence: z0, z1 = f (z0), z2 = f (z1), ...zinc +1 = f (zinc). Depending on the starting point z0, such sequences can behave differently: tending to infinity at n - ∞; converging to a certain endpoint; periodically taking a series of fixed values; more complex options are possible.
Complex
Complex is a number composed of two parts: real number and imaginary number , that is, the sum of the form of x + iy (here x and y are real numbers). i is the so-called imaginary unit, that is, a number that satisfies the equation i^2 = -1. On complex numbers, the main mathematical operations are defined - addition, multiplication, division, subtraction (only comparison operations are not defined). To display complex numbers, geometric representations are usually used - on a plane (called complex numbers), the actual part is deposited along the abscissa axis, the imaginary part is deposited along the ordinate, and points with Cartesian x and y coordinates will correspond to the complex number.
Therefore, any point z of the complex plane has its own behavior pattern in the iteration of the function f (z), and the entire plane is divided into multiple parts. At the same time, points located on these partial boundaries have the following properties: At any small displacement, their behavioral properties will change dramatically (these points are called bifurcation points).Therefore, it turns out that point sets and bifurcation point sets with one specific behavior type usually have fractal properties. These are the Julia sets of the function f(z).
Dragon tribe
By changing the cardinality and fragments, you can get various constructive fractals.
In addition, similar operations can be performed in three-dimensional space. Examples of volume fractals are "Mengel sponge", "Sherpings Foundation Pyramid", etc.
constructive fractals include the dragon family. Sometimes they are called "hehe hath dragon" by the discoverer (in their form they are similar to Chinese dragon ). There are several ways to construct this curve. The simplest and most intuitive thing about it is: you need to take a strip of paper that is long enough (the thinner the paper is better) and bend it in half. Then bend it halfway again, in the same direction as the first time. After a few repetitions (usually after five to six folds, the strip becomes too thick to bend further gently), you need to straighten the strip and try to form a 90° angle at the folds. Then in the configuration file, you will get the dragon curve. Of course, this is just an approximation, like all our attempts to depict fractal objects. The computer allows you to depict more steps in this process, and the result is a very beautiful graphic.
The construction of the Mandeblot collection is slightly different. Consider the function fc (z) = z^2+c, where c is a complex number. Let's build a sequence of this function with z0=0, depending on its parameters that can diverge to infinity or remain finite. In this case, all values c of this sequence are restricted to form the Mandeblot set. Mandeblot himself and other mathematicians studied it in detail, and they discovered many interesting properties of this set.
shows that the Julia and Mandeblot sets are similar in definitions. In fact, these two sets are closely related. That is, the Mandeblot set is all values of a complex parameter c, and the Julia set fc (z) is connected to that parameter c (if the set cannot be divided into two disjoint parts, it is called a join, but with some additional conditions).
fractal and life
Today, fractal theory is widely used in various fields of human activities. In addition to purely scientific objects used for research and already mentioned fractal painting, fractals are also used in information theory to compress graph data (the properties of fractal self-similarity are mainly used here - after all, remember a small fragment of drawing and transformation, you can use it to get the rest, much less memory than it takes to store the entire file). By adding random perturbations to the formula for specifying fractals, random fractals are obtained that very reasonably convey some real objects - relief elements, water surfaces, some plants, which are successfully used in physics, geographic and computer graphics to achieve greater similarity between simulated objects and real objects. In Radio Electronics , the produced antenna has a fractal shape. They take up little space and provide quite high-quality signal reception. Economists use fractals to describe currency fluctuations (a property discovered by Mandebrot). With this we will complete this small excursion into the amazing beauty and diversity of the fractal world.
