[Epilepsy warning] Trajectory length on log scale in approximate von Koch snowflake fractals
Channel: Nils Berglund
Date: 08/07/2022

It put an epilepsy warning in the title, because a commenter suggested the flashing graphics might have such an effect on sensitive people. 

This variant of the video https://youtu.be/YF8kJJhDgBc shows a light beam in various approximations of a von Koch snowflake fractal. The beam is reflected on the sides of the snowflake, and absorbed whenever it hits one of the small discs which have been placed at the corners. At level 0, the radius of the discs is equal to 0.01 times the circumradius of the equilateral triangle, and then the radius is divided by 3 at each new level.

The plot shows that number of reflections of the beam until it is absorbed, as a function of the initial angle of the beam. Its dependence on the angle looks quite chaotic, but I don't know if its properties have been studied. 
The video has 7 parts, showing each a higher level of approximation of the snowflake fractal:
Level 0: 0:00
Level 1: 0:52
Level 2: 1:44
Level 3: 2:36
Level 4: 3:28
Level 5: 4:20
Level 6: 5:12
Some questions related to the behavior of trajectories as the level increases have been investigated by Michel L Lapidus and Robert Niemeyer, see for instance https://arxiv.org/abs/0912.3948 There is no definite answer on existence of a limit, but some results on families of periodic trajectories and some conjectures. 

Render time: 6 minutes 57 seconds
Color scheme: Parula, originally from Matlab
https://www.mathworks.com/help/matlab/ref/colormap.html

Music: "Blue Mood" by Robert Munzinger@yunglogos

Current version of the C code used to make these animations: 
https://github.com/nilsberglund-orleans/YouTube-simulations
https://www.idpoisson.fr/berglund/software.html

Some outreach articles on mathematics:
https://images.math.cnrs.fr/_Berglund-Nils-1343_.html
(in French, some with a Spanish translation)