Despite the fact that fractals have been known to mankind for almost a hundred years and have been well studied during this time, there is no strict definition of them. Although this phenomenon is based on an extremely simple idea: obtaining complex geometric shapes by just two operations - copying and subsequent scaling.
So, a fractal is a mathematical set consisting of objects similar to this set. In other words, if we look at a small fragment of a fractal figure under magnification, it will look like a larger part of this figure, or even the figure as a whole. For a fractal, moreover, an increase in scale does not mean a simplification of the structure. Therefore, at all levels, we will see an equally complex picture.
Fractal properties
Based on the above definition, a fractal is usually represented as a geometric figure that satisfies one or more of the following properties:
- has a complex structure at any magnification;
- is approximately self-similar (parts are similar to the whole);
- has a fractional dimension that is more topological;
- can be constructed using a recursive method.
Fractals in the outside world
Despite the fact that the concept of "fractal" seems to be extremely abstract, in life you can come across many real-life and even practical examples of this phenomenon. Moreover, examples from the surrounding world should certainly be considered, for they will give a better understanding of the fractal and its features.
For example, antennas for various devices, the designs of which are executed by the fractal method, show their efficiency 20% higher than antennas of traditional design. In addition, the fractal antenna can operate with excellent performance simultaneously on a wide variety of frequencies. That is why modern mobile phones practically do not have external antennas of a classical device in their design - the latter are replaced by internal fractal ones, which are mounted directly on the printed circuit board of the phone.
Fractals have received great attention with the development of information technology. At present, algorithms have been developed for compressing various images using fractals, there are methods for constructing computer graphics objects (trees, mountain and sea surfaces) in a fractal way, as well as a fractal system for assigning IP addresses in some networks.
In economics, there is a way to use fractals when analyzing stock and currency quotes. Perhaps a reader trading in the Forex market has seen fractal analysis in action in a trading terminal or even applied it in practice.
Also, in addition to artificially man-made objects with fractal properties, there are also many similar objects in natural nature. Good examples of a fractal are corals, sea shells, some flowers and plants (broccoli, cauliflower), the circulatory system and bronchi of humans and animals, patterns formed on glass, natural crystals. These and many other objects have a pronounced fractal shape.