Types of Fractals
WHAT ARE THEY?
Our website uses a system in which we categorized all fractals into 14 types. Into the category of "nonstandard fractals" we grouped several fractals that did not fit into any of these. Obviously, it is not a true mathematical term.
Consider starting with a simple ring. Substitute this ring with a chain that has 20 rings, connected to each other in a circle. Now substitute each of those 20 rings with a chain. Then substitute each of the 400 rings with a chain, and continue iterating these substitutions. What you get is this fractal:
If you are familiar with knots, this kind of figure is called a link. The substitution of a figure with another figure is used generator iteration and is used in base-motif fractals. However, we considered this fractal nonstandard because, unlike base-motif fractals, it uses circles instead of line segments.
"STAR OF DAVID" FRACTAL
This fractal was created by our team and later turned out not to fit into any categories as well. It is also based on generator iteration where the figure substituted is not a line segment. In this fractal we start with an equilateral triangle. At every step of iteration we put a flipped triangle on top of every triangle. In other words, we substitute every triangle with a Star of David:
The fractal we get at the end is what we called the Star of David Fractal
CELLULAR AUTOMATA FRACTALS
Cellular automata is a system which uses a set of rules to simulate the growth of cells or organisms. Perhaps the most famous cellular automata which you might know is the game of Life. Sometimes, cellular automata can create patterns which turn out to be fractal. In fact, fractal cellular automata are very important in simulating the growth of bacteria. The pictures below are examples of cellular automata which are fractal in shape:
The most famous fractals of this type are probably the diffusion fractals, in which the spreading occurs from the center outwards. These fractals are also extremely useful in studying various kinds of diffusion:
Form a pattern of numbers by making a triangle, in which every number is the sum of the two numbers above it. What you get is the Pascalís Triangle, which is very famous Ė only not to fractals.
However, if we color all odd numbers in this triangle a different color...
... we get a fractal! Indeed, the pattern formed is the same as the one in the Sierpinski Triangle. Isn't it amazing with all the places where you can find fractals?
This fractal is described in detail in the lesson about chaos and fractals.
|Fractal Clock - this colorful applet displays a mystifying fractal clock. Based on fractal canopies.|
|Fractal Clusters - create fractals using your own shapes|