Billiard in a Reuleaux triangle
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- Опубликовано: 21 май 2021
- Seeing the billiard in a Reuleaux triangle was a wish by several viewers (including, I believe, Carmen, Bogdan, Auferen, Bluelightzero and Jonathan), so here it finally is! The Reuleaux triangle is a shape of constant width. You can build it by starting from an equilateral triangle, and then adding to each side a circular arc centered on the opposite corner. The angles of such a triangle measure 120°, like those of a hexagon, and we can see some similar features in the evolution of wave fronts. In addition, the curved faces produce a phenomenon that we have seen for the elliptical billiard, namely the fractal rainbow zigzags due to almost tangent trajectories colliding arbitrarily often with the boundary. As you cannot pave the plane with Reuleaux triangles, however, the wave fronts don't reconnect quite as perfectly as in the case of the hexagon.
Music: "Your Suggestions", by Unicorn heads@UnicornHeads
Check out this version by Share SailCG: • Billiard in a Reuleaux...
Current version of the C code used to make these animations: github.com/nilsberglund-orlea...
www.idpoisson.fr/berglund/sof...
Some outreach articles on mathematics:
images.math.cnrs.fr/_Berglund...
(in French, some with a Spanish translation) 
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Interesting how some portions of the circle get shredded to rainbow pieces and other parts continue to be segments until the end
Wow. Actually this is the best shape for now.
Glad you like it!
How bout that Bestagon
I would like to see a shape with curved and normal sides
See ruclips.net/video/q1Rmjhp0UcQ/видео.html
My favorite were the lemon and the bestagon but this one is easily one of the best too.
This channel is so underrated, glad i found it
Glad you enjoy it!
Woah this one is particularly beautiful
Glad you like it!
Particularly... particular... particle... haha
@@patrickhodson8715 pun intended 😉
It's awesome to think that add a little imperfections in the symmetry of shape bring to an uniform distribution of points instead of this beautiful draw
Not all perturbations of the shape will give a uniform distribution, though some of them, similar to the Bunimovich stadium, will.
You should try doing something in hyperbolic space. Or maybe have boundaries that change the speed of the particles.
Thanks for the suggestions. Hyperbolic space has been suggested by a few others. It is doable, but requires some work to implement computing geodesics. Changing the speed is also possible, though I'd have to think about what is physically intresting.
@@NilsBerglund Ooh, maybe something with refraction? Such that not only the speed, but also the angle of the particles is changed.
@@Selicre I did simulations with refraction for waves. I suppose I could try it with particles too, thanks for the idea!
Interesting how the converging behavior so quickly leads to noise
The noise seems to be due mainly to particles that are almost tangent to the boundary. This causes strong dispersion, as for Sinai billiards.
I think it would be cool if you did some simulations with reflection and refraction in different shapes/refractive index.
I already made a few, for instance ruclips.net/video/JLXAR2PYecI/видео.html and ruclips.net/video/Q8P4iL6ZafQ/видео.html
I have a few more in mind...
Love it and hate its lack of continuity at the same time.
Absolutely stunning...thank you.
You're welcome!
It's really cool how the image starts to look as if it is rotating at around the 2:30 mark
Dutch Rollercoasters
amazing, are you gonna do more reuleaux polygons
Thanks, I will probably do a few more now I have coded them, as well as similar shapes with sides curved inward, which are also quite interesting.
@@NilsBerglund
Splendid! Every new shape always seems to surprise us by how the reflections behave!
This one in particular did things I did not expect it to do!
Now I want a fractal pattern made out of these, surrounded by a fractal grating.
Glad you are back to this
I never stopped making these, I just alternate between different types of simulations.
I love this channel so much. How do you avoud floating point errors when generating the animation?
Thanks! I cannot avoid floating point errors, and you occasionally see their effect, for instance near corners of the billiard. But since I compute exact formulas for the intersection points and angles, these errors are much smaller than if I would compute them be some numerical approximation procedure.
@@NilsBerglund How large of an effect does using longer floats (80-bit? 128-bit?) or some sort of arbitrary precision type have for this sort of billiard?
@@animowany111 It all depends on how unstable the dynamics is. Chaotic billiards have a positive Lyapunov exponent, meaning round-off errors are multiplied by some roughly constant amount at each bounce. Higher precision will add a few bounces that can be reliably simulated, but not more.
great job!i like it!
Thanks a lot!
This one is my favorite so far
Glad you like it!
Best one so far
Thanks!
J'aime vraiment celui là !
Content que ça vous plaise!
Now I wanna see what it would do in other reuleaux shapes
I will probably make a few more soon.
Now I'm wondering what happens if you put a triangular wawe front in a circle.
i wonder how a 3d one will be
A concave one next.
There will be one in a couple of days!
Try it from a corner rather than the center maybe (since the Reuleaux triangle contains parts of circles centered on the corners)? Although I don't know if the result would be interesting (it pretty much all depends on how it computes the result of a particle hitting a corner)
I can try, thanks for the idea!
I’m curious is there any math behind how these types of things work or do they just look cool and are fun to code? And if there is, is it something a second/third year math undergrad (like myself) would be able to follow?
Sure, there is quite a lot of math behind these. The Reuleaux triangle is not the easiest shape to start with. I would suggest you start with regular polygons (equilateral triangle, square, pentagon, etc, see ruclips.net/p/PLAZp3rbgWLo1ojqhBU9cY8VPPZpHCSCSk ) for the link between angles and continuous/discontinuous behavior at the corners. There are interesting links with the angles' value and whether the polygon paves the plane. Circles and ellipses also involve relatively simple math. Sinai billiards and the stadium are more difficult, but the basic mechanism generating instability should be understandable.
A dazzling pattern for today, hm? Neato!
Glad you like it!
mmm good
😋
5:48 Israel joined the show :D
I just had a thought... In a mixed billiard like this, any given origin point is going to have some proportion of angles that lead to a stable path, and some proportion of angles that lead to a chaotic path. I'm curious what a plot of that proportion would look like, as a heatmap.
That's a nice idea, thanks! This seems somewhat similar to computing the Lyapunov exponent averaged over all directions (Lyapunov exponents are a measure of the instability of trajectories). Maybe at some point I will show some plots in phase space, meaning here positions and velocities of collisions with the boundary.
You should make it so the starting circle goes through all the colors so we can see what portions originated where. Sorry if I'm a little hard to understand
I think I understand, you mean something like ruclips.net/video/A2okMHiNzMI/видео.html right?
@@NilsBerglund yes
Although the current one looks very pretty, too
reuleaux pentagon next?
There will be something similar in a couple of days!
woah funky o:
YAY
You’re gonna pay for this
wtf?
*You were talking?*
is there anyway to compile all of these animations into a xscreensaver type thing or a animation desktop for use with deskscapes?
Anyone knowing how to do that is welcome to contact me via GitHub...
Is the Realeaux triangle a mixed billiard like the lemon billiard? The trajectories near the corners appear fairly regular compared to those of the lemon billiard.
I would expect it is mixed, though I did not try to prove it. A way of approaching the question would be to study the stability of the periodic trajectory hitting the center of each circular arc. It has the form of an equilateral triangle, and I would bet it is stable, producing regular motion, like the horizontal orbit in the lemon. Good candidates for chaotic trajectories are those near unstable ("hyperbolic") periodic trajectories. There are general methods for showing they exist, though they may be quite rare.
How many particles did you use for this video?
There are 10 000 particles in this one.
Not long. About the same time as the duration of the video, plus a few minutes to build the mp4 file from individual images.
@@NilsBerglund Have you tried piping image data directly to ffmpeg? I've used the `rawvideo` codec for my own projects and it cut down the total render time quite a bit since there was no need to write the intermediate PNGs.
@@Selicre I haven't tried that, but was looking for something like that. Are you able to control exactly when an imaged is piped to ffmpeg? For the wave/Schrödinger simulations, I am using the openmp library, which makes the real time simulations speed up and slow down depending on which other processes are running.
@@NilsBerglund It's not realtime at all - it simply accepts a raw stream of bytes, with framerate, pixel format & resolution specified via command-line flags. When this data is passed to it is irrelevant.
Make anything has entered the chat
If the circle started near one of the corners of this triangle, would the waves on the sides be bigger?
Possibly - something of that kind happens for the ellipse: ruclips.net/video/0arOgc5iVIs/видео.html
So is it a chaotic system? Or is it just accumulating error in the float number operations?
Looks like all the mess happens in the "angles". It may be because of not existing tangent there.
It seems to be a mixed system, having both regular and chaotic trajectories. Most billiards seem to be of this type. But I don't have a proof for this - see my reply to Dovorans' comment on ways one could use to verify this.
Noice
Thanks
triangle
what kind of implications does this have in science? why would someone want to know the motion of a billiard ball in a rouleaux triangle? Just curious!
That's a very good question. I don't know if the billiard in a Reuleaux triangle in particular has a concrete application (though you never know), I made this animation because some viewers thought, rightly so, it would give nice visuals. However in general, mathematicians and physicists are interested in how the shape of the boundary (presence or not of corners, smoothness, etc) influence the dynamics: when is it regular, when is it chaotic, or a mixture of both? This yields insights into the theory of dynamical systems. Quantum billiards may also have some more practical applications, for instance for electronic components.