What is a fractal?
In short fractal is a shape that has certain special characteristics. Interesting in fractals is, that new characteristics can be found infinitely.
The mathematics related to fractals got birth in the 1600s by work of famous German mathematician Gottfried Willhelm Leibniz. The actual mathematics in fractals saw daylight in the 1800s , but proper research on fractals started after efficient enough computers, that wasn’t possible in the 1800s (at all).
When the researchers first generated one of the most famous fractals, the Mandelbrot set, they thought they had made a mistake, because the result was so surprising. This reflects the role of efficient computers in research on fractals. Fractals became popular in the 80’s.
Fractals in the Nature
Fractals may sound something that can be only found in mathematics, but in the nature there can be found shapes that are fractal-alike. Some examples of these are tree branches, clouds, lightnings, fern’s leafs and coast lines.
Fascinating in the fractals of the nature is, that there can be found infinities in the nature! At least paradoxes about how long something is. The famous example of this is the length of the coast line of Great Britain. Benoit Mandelbrot came into conclusion that the length of the coast line of Great Britain is in fact infinite. This is known as the coast line paradox.
Though one view to this paradox is, that the mathematical world and the empirical world (our every day world) are separate. In an old Finnish book (published in 1969), it says that, in the mathematical world is infinite amount of points, but our empirical world does not have infinite amount of points no matter what is the measured length.
Fascinating thing in fractals (in mathematical world) is, that length can be infinite, but area related to that length is still finite.
Where fractals can be used?
Fractals themselves are unbelievable psychedelic art at their best. But the fractals as art can represent whole landscapes. Because of the artistic nature of fractals, for example the movie industry in Hollywood has used fractals. Fractals can make the special effects more realistic. As an example of movie, where fractals have been used in the special effects, Star Wars episode III is a good example.
Fractals are also used in economics, social sciences and medical sciences. For example the AIDS virus has been modelled by fractal geometry.
It may sound surprising, but the brain has also fractal-alike characteristics. Some people even speculate that the mind of humans would be somehow infinite! In one book on chaos theory for example a psychologist suggested that human’s memory would be something where concept infinite could be assigned.
Fractals are used also in data compression.
One of the most famous fractals is the Mandelbrot set and also the Julia set. The Mandelbrot set and Julia set are related to each other. The Sierpinski Gasket is also very famous fractal.
For some reason I’m fascinated by the Cantor’s set. The Cantor’s set is fractal that is constructed from the real number interval [0,1]. This interval is divided into three line segments that have the same length so that the middle segment in removed. The remaining segments are again divided to three segments of the same length and again the middle segments of the construction are removed. This is continued infinitely and the remaining points of segments belong to the Cantor set.
This was quite non-mathematical explanation of how the Cantor set is constructed, but I hope you got the idea. Georg Cantor was interested in, what happens, if this dividing and removing is continued infinitely.
Let the last fractal I mention be the L-Systems (Lindenmeyer system). L-Systems look like plants and have been used to simulate the growth and structures of plants. Aristid Lindenmeyer developed the L-systems in the year of 1968.