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.
Differentiable functions in a three-dimensional associative noncommutative algebra
