They are very crazy geometric figures, produced through mathematical equations that can be interpreted as shapes and colors by computer programs. Its main feature is self-similarity. “They contain, within themselves, smaller copies of themselves. These copies, in turn, contain even smaller copies and so on”, explains Eduardo Colli, professor at the Institute of Mathematics and Statistics of the University of São Paulo (IME-USP). Fractals are linked to areas of physics and mathematics called Dynamical Systems and Chaos Theory, because their equations are used to describe phenomena that, despite appearing random, obey certain rules – such as the flow of rivers. They are not explained by Euclidean geometry (the one you learn in school), as they have a fractional dimension. “This fraction is related to the quantity and magnification scale of the copies of the figure contained within the figure”, says Colli. Another feature is that they have infinite complexity: a zoom in on an image detail reveals new details.
• Immerse yourself in a Mandelbrot Set fractal in the video abr.io/mandelbrot
• Build your own fractal! Just type any number in the form at the abre.io/fractal link and see the result
CLASSIC EXAMPLE Graphically, a good example of a fractal is the Koch Curve, which refers to a snowflake. The procedure to create it is simple and repeatable: add triangles to the perimeter of an initial triangle
We unravel one of the classic fractal equations
Solve this one!
There are several types of equations capable of generating these figures. One of the most well-known sets of fractals is that of the Polish mathematician Benoît Mandelbrot, who helped to popularize them from 1975 onwards. Although it requires an advanced level of mathematical knowledge, this equation is reasonably simple compared to others. Look:
The origin
It is a complex number, which represents the starting point of the figure. It is discovered by another equation, which adds a real and an imaginary part
The progress
By repeatedly applying and developing this function, called iteration, it is possible to find out “where the point goes”
The direction
It is a point established within the complex plane. It is formed by adding two other numbers, sometimes referred to as “coordinates” in a fractal image.
The theory in practice Where researchers have found fractals
Medicine
The structure of the lung and the ramifications of neurons refer to these figures. Among other benefits, understanding the development of fractals can help predict the evolution of diseases such as cancer, facilitating early diagnosis.
Art
The Englishman Phil Jackson released, in 1998, the album Organized Chaos, which transformed mathematical calculations into fractal music. And psychedelic figures like the ones in this article have already become an exhibition, even at the Museu da Imagem e do Som (MIS) in São Paulo
CG
Some types have been used as the basis of digital animations. They help create textures, simulate vegetation or build complex landscapes. Apollo 13 (1995) and Titanic (1997) are some films that applied this feature
Geography
The folds of the rock layers that form the soil are created by even smaller folds, like a fractal. By defining these standards by computer, it is possible to study soil instability and prevent catastrophes such as the one in the mountainous region of Rio de Janeiro
Economy
The fractal concept is used to understand the behavior of the Stock Exchange. The variation in the value of the stock in a trading day is similar to the variation in a week, a month, a year or a decade. With this, it is possible to make more accurate statistics
Fontes Eduardo Colli, professor at the Institute of Mathematics and Statistics of the University of São Paulo (IME-USP); Fábio Kon, professor at the Department of Computer Science at IME-USP; Nilson Jorge Baldovinotti, member of the Research Group on Training Processes and Teaching Work of Mathematics Teachers at Unesp in Rio Claro (SP); The Fractal Geometry of Nature, by Benoit B. Mandelbrot; Yale University website; FractalArts.com website