Yellow's great escape: trajectory of one particle in a Tokarsky unilluminable room
Channel: Nils Berglund
Date: 29/11/2021

Some viewers of the video https://youtu.be/Hu1-oDVFwSc showing trajectories in a Tokarsky room complained that the animation stopped just before the yellow trajectory seemed to be about to escape its regular pattern in the left part of the room. So I reran the simulation, and it turns out that shortly after the end of the previous simulation, the trajectory gets absorbed in a corner. So here I changed the initial angle of the yellow trajectory very slightly, to obtain a more interesting result. One thing to notice is that the trajectory takes only finitely many directions - I leave it an exercise to work out how many.  
The Tokarsky room has been constructed in relation with the illumination problem. The illumination problem asks the following question: assume you have a room with mirrored walls. Is it always possible to place a light source in such a way that no dark corners remain in the room? Of course, the room has to be "in one piece" (connected, as we say in mathematics): it should not consist of several separate rooms. 
The problem was formulated by Ernst Straus in the 1950s, and first solved by Roger Penrose in 1958. He constructed a room that cannot be illuminated completely, wherever you put the light source. The room is built with four half-ellipses connected by straight parts, see for instance https://youtu.be/dWMuQt8Bdvk
A second example, containing only straight walls, was found by Tokarsky in 1995. The solution works in the approximation of geometric optics, meaning that light travels in straight lines. Unlike Penrose's solution, it leaves only one single point in the dark, provided one considers that any ray hitting a vertex of the polygon disappears. To make the effect visible, the vertices of the polygon have been replaced by absorbing circles in this simulation. The two small circles in the video indicate the position of the source of light, and the spot that is left dark.

Render time: 2 minutes 25 seconds
Color scheme: Parula (but only one color)
https://www.mathworks.com/help/matlab/ref/colormap.html

Music:  Triangle by Audionautix is licensed under a Creative Commons Attribution 4.0 licence. https://creativecommons.org/licenses/by/4.0/
Artist: http://audionautix.com/

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)