Bifurcation Diagram

Generating the famous fractal, which can be used to model populations with various cycles, generate pseudo-random numbers and determine one of nature’s fundamental constants, the Feigenbaum constant \delta.
The fractal nature comes from iteratively applying a simple function, f(x) = \lambda \cdot x \cdot (1-x) with 0 \leq \lambda \leq 4, and looking at its poles.
The resulting image looks mundane at first, when looking at 0 \leq \lambda \leq 3, though the last quarter section is where the interesting things are happening (hence the image below only shows the diagram for 2 \leq \lambda \leq 4).
From \lambda = 3 on, the diagram bifurcates, always doubling its number of poles, until it enters the beautiful realm of chaos and fractals.

Bifurcation Diagram lambda in range [2; 4]
For more on bifurcation, fractals and \delta, I refer to this Wikipedia entry and WolframMathworld.


# Python 2.7.7 Code
# Jonathan Frech, 24th of March 2017

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Haferman Carpet

The Haferman Carpet is a fractal, which kind of looks like a woven carpet. To generate it, you start with a single black pixel and apply in each cycle a set of rules.
In each generation every pixel in the carpet will be replaced by nine pixels according to the rules. A black pixel is represented by a 0, a white one by a 1.

The rules

  • 0 \rightarrow \left( \begin{array}{ccc}1&0&1\\0&1&0\\1&0&1\end{array} \right) \text{and } 1 \rightarrow \left( \begin{array}{ccc}0&0&0\\0&0&0\\0&0&0\end{array} \right)

6 iterations of the Haferman Carpet


# Python 2.7.7 Code
# Pygame 1.9.1 (for Python 2.7.7)
# Jonathan Frech 26th of February, 2016

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