Abstract and keywords
Abstract (English):
Fractals are geometric objects, each part of which is similar to the whole object, so that if we take a part and increase its size to the size of the whole object, it would be impossible to notice a difference. In other words, fractals are sets having scale invariance. In mathematics, they are associated primarily with non-differentiable functions. The concept of "fractal" (from the Latin "Fractus" meaning «broken») had been introduced by Benoit Mandelbrot (1924–2010), French and American mathematician, physicist, and economist. Mandelbrot had found that seemingly arbitrary fluctuations in price of goods have a certain tendency to change: it turned out that daily fluctuations are symmetrical with long-term price fluctuations. In fact, Benoit Mandelbrot applied his recursive (fractal) method to solve the problem. Since the last quarter of the nineteenth century, a large number of fractal curves and flat objects have been created; and methods for their application have been developed. From geometrical point of view, the most interesting fractals are "Koch snowflake" and "Pythagoras Tree". Two classes of analogues of the volumetric fractals were created with modern three-dimensional modeling program: "Fractals of growth” – like Pythagoras Tree, “Fractals of separation” – like Koch snowflake; the primary classification was developed, their properties were studied. Empiric data was processed with basic arithmetic calculations as well as with computer software. Among other things, for fractals of separation the task was to create an object with an infinite surface area, which in the future might acquire great importance for the development of the chemical and other industries.

fractal, fractal curve, monomers, Pythagoras tree, Koch snowflake, fractals of growth, fractals of separation.

История фракталов

Фракталы – геометрические объекты, каждая часть которых подобна целому, так что если взять часть и увеличить до размеров целого, разницы заметить будет невозможно. Иными словами, фракталы – множества, обладающие масштабной инвариантностью [1]. В математике же фракталы, прежде всего, тесно связаны с недифференцируемыми функциями. Так, до XIX в. математики имели дело только с функциями, которые задают гладкие кривые [8]. Однако 18 июля 1872 г. Карл Вейерштрасс [12] в Королевской Академии наук Пруссии представил работу, в которой было показано, что для натурального числа a и числа 0 < b < 1 ряд не дифференцируем (рис. 1).


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