Trajectory length on log scale in Tokarsky's paper windmill room
Channel: Nils Berglund
Date: 2022-06-29

This variant of the video https://youtu.be/p5Yl_z4GEHw shows a rotating beam inTokarsky's room with a non-self-illuminable point, with smaller absorbers in the corners, and a logarithmic scale on the length plot.
The beam emitted from the center of the room is reflected whenever it encounters a wall, and absorbed when it hits a small disc around one of the corners. The inset shows a magnification of a square region around the center of the room, which allows to see how the beam avoids the central point.
To quote George Tokarsky, "if you lit a match in the center of the room, you would not be able to see its reflection. Or, to put it another way, you cannot see yourself in a mirror from the center of the room." One can also view this as a billiard problem: if you shoot a ball from the center of a pool table having this shape, the ball will never return to its starting point.
The graph at the bottom right shows a plot of the number of reflections of the beam until it is absorbed, as a function of the starting angle (in degrees). The absorbing discs at the corners are smaller than in the simulation https://youtu.be/p5Yl_z4GEHw , which is why trajectories tend to be longer. The number of reflections is shown here on a logarithmic scale, which makes the complicated structure of the curve more visible.
If the absorbing discs at the corners were infinitely small, the length of a trajectory would be infinite whenever the slope of the beam is irrational. When the slope is a rational p/q, the length is finite, and depends mainly on how large p and q are (p and q are assumed to be mutually prime numbers). The page http://blog.zacharyabel.com/2012/09/how-unilluminating/ provides some background on the procedure of unfolding the trajectory, which can be used to analyze the length of the trajectory: This length is close to the distance of the first point on the lattice of integer points (n,m) encountered by the beam.

Check out these visualizations of unilluminable rooms made by George and his students: https://youtu.be/5ILv2YsYHkI and https://youtu.be/PxT6BRyWFdk - they come with instructions on how to create your own unilluminable room.

Render time: 1 minutes 37 seconds
Color scheme: Parula, originally from Matlab
https://www.mathworks.com/help/matlab/ref/colormap.html

Music: Cop A Feel 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)