Fractal Explorer

Fractal Explorer

Concept of iteration


Geometrical Aspect

As you now know fractals can be simplified because they consist of parts which are similar to themselfes or because they are scale invariant. With this knowlegde the complex stuctures of most of the fractals can be calculated easily. This is achieved extracting the construction rules out of the fractal and then apply them over and over again.

I.e. the Sierpinski triangle:

The construction of the first level Sierpinski triangle is easy: Take a regular triangle and connect the middle-points of the sides. Now there are four new triangles. Just cut the middle triangle out. But even if the number of triangles increases dramatically the rules don´t change. After an infinite number of repetitions of the operation the result is the Sierpinski triangle.

An object which exists of infinite triangles with an infinite small area can be reduced to that simple rule.


Algebraical Aspect

The concept of iteration does not only work for geometrical object, but it also works for numbers. In that case one needs an iteration formula. To iterate one has to apply a start value to the formula and then apply the formula to the result.

A very simple iteration formula would be Zn+1 = Zn * 2.

For a start value of 1 the results would be:

  1. 1
  2. 2
  3. 4
  4. 8

and so on. Even though this series of numbers is neither chaotic nor fractal it shows the basic principal of iterating numbers over formulas. For fractal series one needs complex numbers.