Fractals are the images in which you can zoom indefinitely and you will find the same figure repeated at different scales over and over again. It is both a natural and a mathematical (and why not an artistic) concept. I have liked fractals for a long time because of that fascination that produces zooming in on an image and being found, and that no matter how much you advance you can always go a little deeper. A slightly more formal definition of a fractal can be found on wikipedia:
A fractal is a geometric object whose basic structure, fragmented or irregular, is repeated at different scales. The term was proposed by the mathematician Benoît Mandelbrot in 1975 and derives from the Latin fractus, which means broken or fractured. Many natural structures are fractal-like. The key mathematical property of a genuinely fractal object is that its fractal metric dimension is a non-integer.
Fractals in mathematics
There are many algorithms and mathematical processes to manufacture fractals from equations and formulas, however, among all of them, the famous Mandelbrot fractal stands out, its mathematical explanation reads like this:
The Mandelbrot set It was proposed in the 1970s, but it was not until a decade later that it could be represented graphically with a computer. This set is defined from any number “c”, which defines the following sequence:
For different values of “c”, we obtain different sequences. If the sequence is bounded, “c” belongs to the Mandelbrot set, and if not, it is excluded. For example, for c = 1 we get: 0, 1, 2, 5, 26, 677, etc. (0, 1 = 02 + 1, 2 = 12 + 1, 5 = 22 + 1, etc.) For c = -0.5 we get 0, -0.5, -0.25, -0.4375, -0.30859375, -0.404769897, etc. In this way, c = -0.5 belongs to the set and c = 1 does not.
If we also consider complex numbers, we obtain the following figure:
Here I have gotten an animated GIF that takes lots of time to load where you can see how this beautiful fractal is zoomed indefinitely, so wait for it to load completely and enjoy.
Fractals in nature
In nature, fractals can be found everywhere, for example in the leaves of trees or in the internal structures of some vegetables and fruits.
Other examples of fractals
Here I leave other figures that behave like fractals and that are very amazing