A lighthouse beam in a 100,000-gon

This is a simulation of a billiard in a regular polygon with 100,000 sides. A trajectory is shown, starting at one corner, with an angle changing in the course of the simulation, is displayed. Whenever the trajectory returns to a small disc around the origin of the trajectory, it is absorbed. The plot to the right shows how many times the trajectory is reflected before being absorbed.
I originally wanted to do this for the billiard in a circle, which is computationally less expensive, but this would have required an adaptation of my code for billiards in circles and ellipses, to add absorbing circles. It seemed easier to just use a polygon with a large number of sides, and the computations turned out not to take long.
For a true circle, the trajectory returns to the point of origin whenever it forms a regular polygon, possibly star-shaped. This happens whenever the angle between two collisions with the boundary, when measured from the center of the circle, is a rational multiple of a full turn. This results in dips on the length plot.
Render time: 5 minutes 37 seconds
Compression: crf 23
Color scheme: Plasma by Nathaniel J. Smith and Stefan van der Walt
github.com/BID...
Music: "Traverse The Sky" by Asher Fulero‪@AsherFulero‬
Current version of the C code used to make these animations:
github.com/nil...
www.idpoisson....
Some outreach articles on mathematics:
images.math.cn...
(in French, some with a Spanish translation)
#billiard #polygon #polygon gon

[@jonasschwalb2787]

The combination of regularity and chaos of the length is fascinating

[@gyinagal]

Imagine each wall of the shape is numbered 0-99,999, with zero being the source. If the wall is shining at the wall whose number is n, then the number of reflections before the light ray returns to the starting point is 100,000/gcd(n, 100,000).
100,000 only has 36 divisors, half of which are less than 100, which means for most numbers, the gcd with 100,000 is very small, which is why it seems chaotic. And of course, if the numbers are relatively prime, then it reflects 100,000 times, hitting every single wall before returning home.

[@chrisgaming9567]

Would the results be significantly different for a 10,000-gon, 1000-gon, million-gon etc?

[@NilsBerglund]

What we see here is already very close to what should happen for a circle. So increasing the number of sides should not change the result much. Decreasing it will gradually reveal differences. I seem to remember that for 1000 sides, the result was noticeably different.

[@beanieduchet2928]

Marvelous- reminds me of Spirograph, my favorite childhood toy, Happy Friday !

[@jonasschwalb2787]

I recreated these in geogebra when I learned linear algebra at school. It's a fun little project, you might wanna give it a shot

[@Rhino-Flea]

Is there anything interesting to be said about the geometry of the ray self-intersections?

[@Rhino-Flea]

The Lonely Runner Conjecture called-it wanted Nils to shed some light on it, but the light just kept bouncing away...

[@hex-automata]

How much does the pattern change if you use a circle ("infinita-gon") instead?

[@NilsBerglund]

Not much, I think.

[@gyinagal]

@@NilsBerglundi think the fact that there’s a finite number of sides matters here. For example, if you added 3 more sides then it would never resolve into nice tidy shapes because 100,003 is prime. At every angle the light would reflect 100,003 times.

[@gyinagal]

@@NilsBerglundin fact, I think a true circle would be much more chaotic. Imagine the source of the light is labeled zero and every point on the wall corresponds to a real number between 0 and 1. I could be wrong but I think the light beam would return to the source when pointing at a rational number but reflect forever when pointing at an irrational number, never returning to the source at all.
That would be bizarre. I can’t even imagine what that would look like as the beam sweeps around the circle.

[@amityFinder2099]

the patterns would be similar but the length graph on the right would be much smoother and not steppy

[@gyinagal]

@@amityFinder2099I disagree, I think it would be discontinuous everywhere

[@Less_Than_StellarYT]

OH MY GOD

[@tuckermatis1572]

What's the difference between a circle and a regular megagon anyway lol

[@NilsBerglund]

The main difference is that I had the code at hand for this one, while for the circle I have not completely implemented the absorption by a disc 😁

[@michaelkhoo5846]

Very cool video. 100,000 is a "lakh" in Hindi, so this could be a lakhogon?

[@urosjarc3450]

NOTED: When the beam is joust right, you can summon the demons. :) 0:39