Details

The algorithm for computing Platonic quantum fractals in general: define vertices of a regular polyhedron (there are five Platonic solids). Choose a fixed parameter , , which defines the "sharpness" of the fractal. Then select a random starting point on the unit sphere. Randomly select a vertex index ; let (one of the possible vertices). Apply the transformation

.

This is a Möbius transformation. It gives a new starting point on the sphere. Iterate the algorithm 10000 or more times.

Plot the first two coordinates of each point. The Möbius transformation theoretically maps the unit sphere onto itself. Yet after many iterations due to numerical inaccuracies, the norm can start to diverge. Therefore, it is better to normalize after each transformation.

For more details about Platonic quantum fractals, see [1] and [2].

References

[1] A. Jadczyk and R. Oberg, "Quantum Jumps, EEQT and the Five Platonic Fractals." arxiv.org/abs/quant-ph/0204056.

[2] A. Jadczyk, Quantum Fractals: From Heisenberg's Uncertainty to Barnsley's Fractality, New Jersey: World Scientific 2014.