4  Fractals

Several self similar fractals can be iterated:

• Highway dragon
• Koch Curve
• Cantor Set
• Weierstrass Monster
• Sierpinski triangle
• Pythagoras Tree
• Sunflower

The Fractals Widget

The number of iteration steps can be set by the slider. The construction laws are self-explanatory if a low number of iteration steps is selected.

4.1 Calculate

The number of selected iterations is calculated and displayed.

4.2 Parameter settings for the different fractals

For the following fractals some parameters can be set to influence their generation.
If there are no parameter settings foreseen or necessary the buttons and input fields are de-activated.

4.2.1 Highway Dragon

The direction of the construction can be selected. The effect can be seen best if only a few iterations are selected.

4.2.2 Weierstrass Monster

The limit object of the Weierstrass Monster is a function which is continuous in \(\mathbb{R}\), but not differentiable at any point in \(\mathbb{R}\).
It is defined by \[ f(x) = \sum_{k=0}^N \lambda^{k(s-1)} \cdot sin(\lambda^{kt}) \] where N is the number of iterations. The values for \(λ\) and \(s\) can be selected.

Note: not every parameter combination leads to nice monsters.
The fractal dimension of the monster is estimated and displayed.
This is only a very rough estimation, because the calculation is done on the base of the Nth iteration by the curve length.

4.2.3 Sunflower

The configuration of seeds in a sunflower is simulated. Starting from a central point the radius of every step is increased by s while the angle is increased by \(φ\) given in degree.

4.2.4 PS print/Print

A Postscript output is generated and stored in <Path to Store Directory>/plots.
The PS-files are named automatically.