Fractals are geometric “objects” with self-similar structures on infinite scales, that is, there are exact or approximate copies of the entire object in pieces as small as desired.

The term fractal was coined by Benoît Mandelbrot in the 1970s, when computational resources made it possible to see the richness and beauty of these figures, beyond the examples and discoveries of the early 20th century, made by Helge von Koch, Gaston Julia, Pierre Fatou and others. Aside from their importance in mathematics, fractals serve as a tool in imitating natural settings and plants. For example, the image below shows a Romanesque broccoli!

Image: AVM, at Wikimedia Commons.

We can also observe this type of phenomenon by making a dizzying zoom from the observation of a fixed point in a certain image, which is called the Mandelbrot Set, defined in the theory of Dynamical Systems. To see this incredible zoom, as well as many others, just enter the Maths Town channel.

Another way to see the formation of fractals is using mirror balls, like the one in the figure below. Note that, by placing them in a certain position and using colored lights, we can see the pattern repeating itself "infinite" times: just bring the lights closer to the mirrored balls and watch the fractals proliferate closely!


At the beginning of the 20th century, several examples of fractal sets appeared, which served as counterexamples (or tests) in topology and, more particularly, in works related to the concept of dimension.

Karl Menger, in 1926, proposed this “sponge”, which is nothing more than a three-dimensional generalization of the Sierpinski Carpet, conceived 10 years earlier by Waclaw Sierpinski.

It starts with the partition of a cube into 27 cubes with a third of its size and the removal of 7 of these smaller cubes (the middle one and the middle 6 of the faces). For each remaining cube, the same procedure applies, and so on to infinity. The resulting set is self-similar: its pieces, if enlarged correctly, are congruent to the whole figure.

The “level 2” origami block sponge on the right was made by student Carla Teodoro, within the MegaMenger project of building a level 4 sponge from collaborative work around the world.

In this video, Henry Segerman addresses the issue of building a fractal zoom from real footage. For this, he will need to solve some technical 'problems'!

This Wikipedia page gives a more general view of fractals, then gives some examples where they can be seen and even presents computational constructions of a specific fractal, in addition to showing some other more formal concepts.                                                        

This Super Interessante article brings the definition of fractals and one of their classic equations. In addition, it shows several practical examples in several areas where this knowledge can be used, such as medicine, art, computer graphics and even economics!

This another Wikipedia page talks about the lakes of Wada, another type of figure that can be understood as a fractal. In this article, the construction of these lakes and also of the Wada basins is explained, and it also makes a connection between these lakes and the chaos theory.

The Institute of Mathematics, Statistics and Scientific Computing at Unicamp has several interesting articles involving fractals! This article, given in the form of audios, it presents a study on the trajectory of sharks and the connection to fractals, showing that it follows a pattern comparable to the fractal curve!

This article, given in video form, shows how numerical methods to find the roots of certain polynomials allow the artistic production of fractals! It's a great video to use in Basic Education as it introduces a concept and wraps it with art, which is more tangible.                                                        

Finally, this article, which is a synopsis for a class, introduces the concept of Koch's Square, a fractal that allows students, from its construction, to recognize geometric progressions that follow the perimeter and area of the figures obtained. It's another very engaging topic for Basic Education!                                                        

Still thinking about Koch, there are also Koch's snowflake fractals, shown in this Khan Academy video, which shows how to build it and calculate its perimeter - which is infinite - and its area - finite! The area calculation is better explained in this video and in this other, also from Khan Academy.

This Wikipedia page talks about Menger's Sponges, giving an overview of them, showing how they are built, what are their properties and their formal definition. Finally, mention the MegaMenger project, for which the sponge shown above was made!

Finally, this MengerTec channel video shows the construction of a Menger Sponge very slowly, so that you can see the small cubes fitting together to form a much larger cube!