Fractals are objects where each part is similar to the object as a whole. This means that the patterns of the entire figure are repeated in each part, only on a smaller size scale. Snowflakes are examples of fractals: each branch of the flake looks like the entire flake.
There is an area of mathematics that is dedicated to the study of fractals, called fractal geometry. Fractals form very beautiful geometric figures and generate patterns that can be used in cryptographic systems - systems that encode passwords. Using fractals makes passwords safer and harder to crack.
One of the first mathematicians to study fractals was the French Benoit Mandelbrot and studies in this area have advanced a lot with the computational resources available today. These features make it possible to enlarge the images for better visualization, in addition to identifying the patterns with which the images are reproduced. In fractals generated by computational methods, every bit of the fractal is exactly a copy of the original image and can be obtained from a specific equation, as can be seen in the images below, both are relative to the Mandelbrot sets and are watered from computers.
Graph of a Mandelbrot set
Graph of a variation of the Mandelbrot set
In nature, there are several examples of images that come very close to fractals, such as the leaves of a fern or the structure of broccoli. Note that each smaller leaf looks a lot like the whole leaf, and on each small leaf we have structures that are also very similar to the larger leaves. This reproduction is also visible in some species of broccoli, and especially in the Romanesque type, as can be seen in the images below.
Fern leaf: natural fractal
Romanesque broccoli: natural fractal
It is also possible to build fractals using only geometric resources. For example, starting from a triangle and dividing it into other smaller triangles, all of which are similar to each other.
by Franciely Guedes
Graduated in Mathematics
