Benoit Mandelbrot is usually credited with discovering fractals in 1975. Mandelbrot was the one who invented the word. He was also the first to represent them visually. But some facts about fractals were known to mathematicians as far back as the 17th century.
Early Fractal Research
A fractal can be described as a pattern that is repeated at every scale. It cannot be replicated by classical geometry. The concept of self similarity was first brought up by the philosopher and mathematician Lebinz in the 17th century.
It wasn’t until 1872 before a function appropriate to be termed a fractal came into being. Karl Weierstrass showed an instance of a function that was continuous but could not be differentiated.
This definition was improved upon by Helge von Koch in 1904 when he defined it as a Koch curve. A study of the facts about fractals will show it is now called the Koch snowflake. In 1915, Wallow Sierpinski created a triangle and a carpet.
In 1938, Paul Levy came up with another fractal curve called the Levy C curve. Another important contributor was George Canto, who came up with the Cantor sets.
Other important researchers were Henri Poincare, Pierre Fatou and Gaston Julia. However these 19th and early 20th researchers didn’t have a computer where they could visualize their discoveries.
It wasn’t until Mandelbrot’s works that the fractal caught the public inertest. He was able to do this with the aid of computers. Using computer graphics he was able to go beyond the facts about fractals and show people what they looked like.
There are many types of fractals but many of them share similar traits. Most have a structure with small scales. Their shape cannot be described using traditional geometry, and it is self similar. Fractals have the Hausdoff dimension. It usually has a dimension bigger than the topological dimension.
No matter how it is magnified, the appearance is still the same. The closest representations to fractals in nature are clouds, coastlines and snow flakes. It should be noted that not all self similar objects can be classified as fractals.
Fractal as Computer Art
As facts about fractals emerged, computer software for producing them started appearing. The beauty of fractals made it appealing for people. Today it is used as an art form, background for Web pages and as works of art.
Generating fractals can be done in four ways. These include escape time fractals, the iterated functions systems, the random fractals and strange attractors. The escape time fractals are among the most well known. These include the Julia set, the Mandelbrot set and the Lyapunov fractal.
The iterated functions systems include the Menger sponge, the Peano curve and the Cantor set. The random fractals include the Brownian tree, the Brownian motion and fractal landscapes. Other fractals can be generated by computer software.
The facts about fractals may be confusing for some people, especially their principles. But even those not versed in arithmetic can appreciate the beauty of these images.