A lighthouse beam in a Tokarsky room
Channel: Nils Berglund
Date: 04/12/2021

This is yet another representation of a Tokarsky unilluminable room, this time with a single trajectory starting with a slowly changing angle. For each angle, the trajectory is shown for up to 256 collisions, or until it is absorbed by one of the corners. The color of the beam changes after each collision, repeating every 32 collisions, and decreases in luminosity.  
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 To make the effect visible, the vertices of the polygon have been replaced by absorbing circles in this simulation.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 red and green circles in the room indicate the position of the source of light, and the spot that is left dark, and the square shows a close-up of the green circle.

Render time: 7 minutes
Color scheme: Viridis by Nathaniel J. Smith, Stefan van der Walt and Eric Firing
https://github.com/BIDS/colormap

Music: "Modern Situations" by the Unicorn Heads@UnicornHeads

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)