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Geometric objects known as fractals used to be called 'beasts of mathematics' or 'outcasts'. Today, knowledge about them allows scientists to develop telecommunications, data security and entertainment (when the creators of cinema special effects use them), says Professor Andrzej Katunin from the Silesian University of Technology in Gliwice.
In mathematics, fractals are Euclidean space subsets presented in a graphic form as very complex objects with specific features. One of them is self-similarity, which means that a fractal viewed on a micro and macro scale looks the same. “The idea of the fractal can be compared to the cauliflower, which is built of florets, and those are built of smaller florets, and so on. A less common simple model of fractals can be the Russian nested doll: identical dolls of decreasing size placed one inside another,” says Katunin, an engineer, science promoter and mathematics enthusiast.
Why would people be interested in fractals? “Many are attracted by their visual aesthetics. They also have many applications. Freaks and beasts, as the fractal was called a hundred years ago, have been used to create modern tools, giving rise to advanced scientific research in the field of system dynamics, applicable almost in every field of science. Without fractals we would not have many modern achievements in various areas of life,” Katunin adds.
He points out that Benoit Mandelbrot was responsible for the turning point, the moment, from which fractals began to be viewed as a tool. Born in Warsaw (1924), the mathematician is considered the father of fractal geometry, the breakthrough being his research on fluctuations of cotton prices. When looking at data, Mandelbrot did not take into account days, weeks or months, but a much longer period: over a century. And although at the beginning he expected that prices would be randomly shaped, when analysing the data, he noticed patterns and similarities, which led him to develop tools for price analysis and even forecasting. “Today in economic sciences, fractal analysis is already a classic tool, widely used to study the stock exchange, price fluctuations, currency behaviours, etc.', Katunin says.
Fractals have many more applications. We carry one of the fractals in our pockets, says Katunin. It's a mobile phone, or more precisely its antenna. In various parts of the world, for example in the US or Europe, there are different GSM standards that determine the frequencies on which the phone sends and receives the signal. Thanks to its fractal shape, the antenna works in all bands, regardless of the part of the world in which the user is.
Without fractals, there would be no spectacular special effects in the cinema. Knowledge of these forms first helped create realistic visualizations of landscapes in films. More advanced landscape visualizations of one of the planets appeared, for example, in the second Star Trek movie, and later, for example, in The Last Starfighter and Apollo 13 (where fractal algorithms were used to visualise the Moon surface, or The Perfect Storm, in which fractals were used to simulate Hurricane Grace. Fractal algorithms also used to generate lava in the famous lightsaber fight between Obi-Wan Kenobi and Anakin Skywarker in Star Wars: Episode III (2005). Fractal effects can also be found in Luc Besson's film Lucy (2014), the 2nd parts Guardians of the Galaxy (2017) and Doctor Strange (2016), which is full of fractals, and the effects based on them are breathtaking, says Professor Andrzej Katunin, author of the bookFraktale. Matematyczne potwory, które odmieniły postrzeganie świata ('Fractals. Mathematical beasts that changed the perception of the world').
He notes that fractals even entered biology and medicine, as they help to understand the construction and operation of many systems and structures, for example the nervous system, lungs or bone tissue. Thanks to the fractal structure, they gain exceptional features, such as combined stiffness and lightness.
Cryptography is developing dynamically thanks to fractals. “In data encryption, especially sensitive data, it is important to generate keys, for example for bank accounts, credit cards and to secure operations performed with these cards. Multidimensional fractals in hypercomplex spaces are used there', says Katunin.
Based on the knowledge related to fractal geometry, Andrzej Katunin investigated damage to the aircraft structures and detects damages that are invisible for classic measuring devices. “Parts of aircrafts made of composite materials, for example parts of plating, steering, vertical and horizontal plane stabilizers, require monitoring and periodic inspection. This can be done with various methods, from ordinary visual inspections to various types of measurements. If there is even a very small crack, this means locally lowering the stiffness of the material. In measurements, we detect it by seeing the inconsistency of the surface or the incoherence of the signal that we obtain from it. A special algorithm, associated with a fractal dimension, can detect this inconsistency. I reveal damage that is not visible in signals from classic measuring devices. I also work on improving the sensitivity of these methods, so that it is possible to detect smaller damage,” he says.
Fractal shapes can also be found in nature. “Based on the research we have, it can be concluded that this is related to the common phenomenon of self-organization. In the micro-scale, the rules of self-organization govern, for example, the growth of bacteria or mould, the free cultures of which resemble fractals. In the mezo-scale, the shapes of marine currents, river beds, clouds or lightning are fractal shapes. The process of mountain building, shaped by many factors (tectonic movements, the activity of the earth's nucleus, volcanoes, etc.) also has fractal features. In the macro-scale, galaxy clusters have fractal structures,” says Katunin.
People copy the fractals found in nature to engineering solutions, architecture and art. Katunin continues: “Fractal features can be found in sacral architecture, for example in a temple in India. In ancient times, books were written that could be considered guides with instructions for building such temples. These fractals are there! This is also related to the symbolism used in specific cultures. Some Hindu temples, especially in northern India, are based on motifs called mandalas, which also have fractal features. They were created by the former inhabitants of India, based on the way they understood the structure of the universe. Fractal features appear in architecture precisely because people see them in nature. And we simply like it, which may be due to their infinite complexity.
“Fractal structures can be seen in the most mundane situations: in winter in the shapes of frost on the window, smoke from the chimney, steam from the kettle. Also when conducting research, because rust on aircraft parts are has fractal shapes. It is surprising to discover fractals in ever new places. My wife, a scientist, caught the 'fractal infection' from me and also sees them.” (PAP)
Anna Ślązak
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