@@jakeck2893 actually if you pay enough attention you will see the circle always increase the diameter it just flip back when it touch a "surface" a line of a triangle will intersect with 2 points in the circle, not a "surface" and even if it flip back without a "surface" the circle we saw in the video never leave the triangle but a circle won't have enough diameter for a line which is continuously increase length
It happens periodically over and over again, e,g. at 0:36, 1:03, 2:11, 2:38, 2:46, and 3:12. I guess 1:36 stands out because the lines are yellow and there is a lot of them.
@@Roofluffer I think it has something to do with the fact that the angles are 30, 60, and 90 degrees. There are several paths of equal length where the point originating in the starting point will mirror a few times off edges and return back to the starting point in the same direction. And the fact that these paths have equal length means wavefronts sent in these directions will meet back there all at the same time.
For a good joke video, you should do one that starts in the middle of a parabola and just straightens out and goes offscreen as nothing else happens for the remaining several minutes of the video
The music makes these even better. And it fits very well. And it's interesting how the lines form n-gons at the same place where the original circle startet.
Thanks. I noticed that too - it means that at certain times, the big expanding circle in the plane will cross many images of the starting point under the reflection symmetries at the same time.
Would be nice to see how it would evolve starting from the circumcenter, as it would hit all vertices simultaniously. Although maybe if it's not regular they would start to desync
In this case, the circumcenter is the middle of the hypotenuse. You are right, the front will keep hitting all corners simultaneously. You can actually understand why by drawing the tiling generated by this triangle (which has a hexagonal symmetry).
Sometimes I feel like I'm part of an experiment. I stared at this with absolutely no commentary and when the music stop I went "no no NO" then it ended D: It's absolutely incredible on how simple videos can have such effects lol
but imagine a retro/indie game where you had to navigate a level without touching the lines, so it would incentivize finding paths quickly through levels before navigating them becomes impossible.
I have really been enjoying these, and would very much like if you did one for a heptagon? Not sure whether I would prefer it from the center or a bit to the side, but regardless, it would help for my D&D game XD
It’s a warm Thursday night... you’re in a void watching thousands of colorful lines bounce off of the walls of a triangle floating indecently in space. Cool jazz fills the void. You’re comfortable.
Do you happen to know how the combined length of the arc segments varies over time? I would think the angle measure remains the same, but it looks as if the the length is constantly increasing, and I bet it would be an interesting function that describes this growth
For many triangles, the curve would be cut into more and more pieces. An exception is the equilateral triangle. A general rule is that all angles should divide 180° to get a continuous curve.
Here I chose the point arbitrarily (it is the same as in the wave simulation). I might of course have accidentally used a point with special properties...
Am i correct in thinking this is done by using radial coordinates (instead of xy you have angle and length) and then using the mod/remainder function? Also, couldnt you unfold the mess and slowly reveal the circle?
Yes to your second question, you could piece all the little curves together to form a big circle. Regarding your first question, I suppose you could do something like that, though with some extra sign changes because of the reflections. What I do is slightly more involved, because I use the same code for non-polygonal domains such as the ellipse. The front is made of several thousand particles, each starting with speed 1 in a different direction, and I have the computer determine their trajectory by taking into account collisions with the boundary.
Knowing that the length of the circle has infinite points, can we prove that the circle's points will fill all the area within the triangle after a certain time?
It might be possible to prove something like this: given any strictly positive, but arbitrarily small real number epsilon, the points of the circle will pass within distance epsilon of any point in the triangle given enough time. In other words, if you give the circle a little thickness, it will fill the triangle at some point, whatever the thickness is, provided it is not zero.
What would change is that the front would hit certain parts of the triangle at the same time: the corners if you start from the circumcenter, the sides if you start from the incenter.
@@NilsBerglund Yes, of course. You are the "Number one" candidate for the award! After all, you are the one who made the video, the one who put beautiful music in the background, a lot of colorful circles. You even bothered for an amazing headline: Triangular the Circle !! So this is really a beautiful, and even very relaxing video, but just before the award committee announces you as the winner: Please, let us know what the mathematical rationale for the title? How did you manage to "triangular the circle"? What is the logical idea behind the video? You will receive the prize happily, only before it was important for us to draw your attention that you forgot to attach a mathematical explanation to the video. With regards The prize committee.
@@tamirerez2547 The title is a play-on-words based on a previous video called "Squaring the circle", which is impossible as we all know. There is more in the description on the physical meaning of the simulation: it shows the evolution of a circular wave front in an idealized fluid (without dissipation or interference), when it is reflected on the boundaries of the triangular container.
@@NilsBerglund So it is just a colourful GAME witn circles? As the days when we were kids in the kindergarten? With a paper and lot of colour markers... Now I like it! No need to imaginery fluid... reflection of a wave... triangular... Why didn't you say A GAME WITH CIRCLES from the beginning? Very nice video!! 😉
I guess different people see different things in these videos: some see the maths/physics, others see a game, still others the aesthetic aspect. All these ways of looking at them are fine, of course!
In fact all the lines you see at any time are "new", so one would have to decide on a criterion saying which parts to highlight. But I did something with a fading past here: czcams.com/video/2NLtpcyuRO8/video.html
You can try working it out by reflecting the billiard boundary instead of the wave front: look at a growing triangle in a plane tiled with triangles defining the billiard.
The total length of the curve grows proportionally to time. At any time, you could reassemble all the pieces of circular arcs to form a big circle, whose radius grows linearly with time.
All the pieces of curve can be put together to form a big circle of radius growing proportionally to time. So you won't get a small circle again. Since there are finitely many pixels in the simulation, after an extremely long time the video would start showing a periodic pattern (but which may have a very long period).
I did not code it very carefully here, but you can show that for angles of the form 180°/n for some integer n (here n is 2, 3 and 6 for the three corners), there is a well-defined limit when you hit the boundary closer and closer to the corner from either side. Depending on whether n is even or odd, you either make a turn by 180°, or you are reflected with respect to the angle's bisector. The best way of seeing this is to "unfold" the corner, let particles move on straight lines, and then fold the trajectory back to the actual corner.
If you mean whether the line will get spread uniformly over the whole triangle: I'm not sure. The system is not ergodic in terms of both position and velocity, but in might be in terms of position only.
It would also give a closed curve. Since the triangle here is half an equilateral one, the outcome would not be too different, though it would look more symmetrical if you start from the center. I will try something like that once I have extended my code to more polygons.
What if you circle the triangle tho
hm, you might be onto something here
Is it even possible tho?
@@w_ldan i think so. if you join together the lines in the vid, you get one big circle. if that's the case, the inverse can be done: one big triangle
@@jakeck2893 i didn't have a brain big enough to dissagree with you so yea sure, im agree with you.
@@jakeck2893 actually
if you pay enough attention
you will see the circle always increase the diameter it just flip back when it touch a "surface"
a line of a triangle will intersect with 2 points in the circle, not a "surface"
and even if it flip back without a "surface"
the circle we saw in the video never leave the triangle but a circle won't have enough diameter for a line which is continuously increase length
That moment at 1:36 when you see another circle formed in the same place as the original one 🤯🤯🤯🤯
And the music lined up perfectly
It happens periodically over and over again, e,g. at 0:36, 1:03, 2:11, 2:38, 2:46, and 3:12. I guess 1:36 stands out because the lines are yellow and there is a lot of them.
@@kasuha Wow that’s pretty interesting
@@kasuha there's a pattern in those times... but im not sure what 'type' of math
@@Roofluffer I think it has something to do with the fact that the angles are 30, 60, and 90 degrees. There are several paths of equal length where the point originating in the starting point will mirror a few times off edges and return back to the starting point in the same direction. And the fact that these paths have equal length means wavefronts sent in these directions will meet back there all at the same time.
You've heard of squaring the circle and now Triangulating the circle, get ready for circling the circle
This has already been done: czcams.com/video/17DHYxcfpuk/video.html
This is funny lol 😂😂
Sus moment
1:35 is too perfect with the music
For a good joke video, you should do one that starts in the middle of a parabola and just straightens out and goes offscreen as nothing else happens for the remaining several minutes of the video
I like how the colors change
Very satisfying
try coke
@@jhheight can’t afford it :(
The music makes these even better. And it fits very well.
And it's interesting how the lines form n-gons at the same place where the original circle startet.
Thanks. I noticed that too - it means that at certain times, the big expanding circle in the plane will cross many images of the starting point under the reflection symmetries at the same time.
Would be nice to see how it would evolve starting from the circumcenter, as it would hit all vertices simultaniously. Although maybe if it's not regular they would start to desync
In this case, the circumcenter is the middle of the hypotenuse. You are right, the front will keep hitting all corners simultaneously. You can actually understand why by drawing the tiling generated by this triangle (which has a hexagonal symmetry).
1:35
2:10
2:05 hexagram
When you realise these lines can always be connected to make one circle
I think that if you let them flow indefinitely they will sometime contract into a cricle again
@@chaotixman465 it's getting bigger and bigger so I doubt so
@@chaotixman465 theres no way in hell nope
@@CarmenLC ohh, I really wanted to see that
@@chaotixman465 length of the wavefront is proportional to the time, so it is literally impossible
please keep making these in dark mode! helps me fall asleep at night
That's great! I will use the dark background on more videos, then.
@@NilsBerglund thank you! :D
We did it boys, ○ is no more
Sometimes I feel like I'm part of an experiment. I stared at this with absolutely no commentary and when the music stop I went "no no NO" then it ended D:
It's absolutely incredible on how simple videos can have such effects lol
Came for the visuals, stayed for the music 👍🏻
This is it. This is the best video CZcams has ever recommended to me
Glad you like it, thanks!
but imagine a retro/indie game where you had to navigate a level without touching the lines, so it would incentivize finding paths quickly through levels before navigating them becomes impossible.
your music choices are always on point i’m living for that
Thanks! I do spend a bit of time selecting the music, so it's really nice to hear that it works.
This is beautiful. I've looked at this for 5 hours now
Glad you like it! Please remember taking a break from time to time...
@@NilsBerglund yes lol
is that a perfect hexagon at 0:36
with the yellow lines
the bestagon
not really....
No, because the lines are all arcs of a circle, not straight.
Woah, is that hexagon converges to a single dot?
Hexagons are the bestagons.
This is a whole another level!
This is beautiful in a way hard to describe
Thanks!
Came for the triangulation, stayed for the music
I really wish I can watch this in reverse. Seeing a ton of chaos becoming more organized over time
Try this one for something similar: czcams.com/video/SOtzBvNUd2A/video.html
Now triangulate the square.
The music makes me feel like I'm watching something I shouldn't.
I love the new dark mode look
This video felt super long for 3 minutes but was enjoyable the whole way through. There aren't many videos like that.
I have really been enjoying these, and would very much like if you did one for a heptagon?
Not sure whether I would prefer it from the center or a bit to the side, but regardless, it would help for my D&D game XD
Glad you liked it! A few more polygonal billiards are in preparation.
Please do it in a isosceles triangle where the circle starts exactly ib the middle. This would be so satisfying!
I put it on my list of things to do. I need to add a few features to my code to do that, but I expect to be able to get to it in a few days.
It’s crazy how at around 2:50, a circular shape begins to form and grow roughly where the circle originated
Like the lines form a shape of a circle (obviously not actually a circle) but something that seems to have equal sides in a round shape
The curves are all pieces of a large circle, of radius growing with time. You could piece them together to reconstruct the circle.
Aaaaah quedé con ganas de que se cubriera entero >:c exijo una versión con 2 círculos!
It’s a warm Thursday night... you’re in a void watching thousands of colorful lines bounce off of the walls of a triangle floating indecently in space. Cool jazz fills the void. You’re comfortable.
no
you even made this comment on Monday
Each segment will have the exact same radius no matter how much they are broken up
my entire nervous system when i stand up too fast:
This needs to be on Wallpaper Engine!
This one seems more chaotic than the square
Just vibing to the music
Looks like we came full circle once again. ;)
fascinating! I wonder if this in some form could be used for noise algorithms; it feels suited for that
When Pythagoras sees a right triangle:
Best music video ever (jokes aside this is so cool to watch!)
Thanks for watching!
MORE I NEED MORE
Do you happen to know how the combined length of the arc segments varies over time? I would think the angle measure remains the same, but it looks as if the the length is constantly increasing, and I bet it would be an interesting function that describes this growth
The combined length grows linearly with time, because all the arc segments can be recombined to give a circle, whose radius is proportional to time.
this was the original bouncing square tik tok
I don’t know much about this stuff,but is it possible to do this with a left triangle,instead of a right one,and get a different outcome?
For many triangles, the curve would be cut into more and more pieces. An exception is the equilateral triangle. A general rule is that all angles should divide 180° to get a continuous curve.
Mm,yes,i wish to watch this 12 years later
Pretty 👌
Thank you! Cheers!
This thing is hypnotic. If I watch it in bed, I sleep before it even reaches halfway😂😂
Sweet dreams!
2:28 what you see when you rub your eyes too hard 😂
Enchanted triangle
Sooooo... what's next? A circle in a hexagon? A snake eye?
Circulating the triangle next
What would a circle in the middle of an equilateral triangle be like?
We can look at that once I have extended my code to more polygonal domains.
Another name would be:
Growing Circle module fixed triangle
It was until 1:50 that CZcams’s video compression is starting to get destroyed XD
Fun fact: CZcams compression is sometimes shitty enough that if you upload and download a video the size of the file will go UP.
Caould you please run another circle with radius half inside right triangle with sides 1,1 and sqrt2?
Thanks, I'll put it on my list of things to try. If you want, you can also try running my code available here: www.idpoisson.fr/berglund/software.html
Where did you place the source? Was it on one of the centers or on an arbitrary point?
Here I chose the point arbitrarily (it is the same as in the wave simulation). I might of course have accidentally used a point with special properties...
Pentagoning the circle should be next.
All these squares make a circle
Am i correct in thinking this is done by using radial coordinates (instead of xy you have angle and length) and then using the mod/remainder function?
Also, couldnt you unfold the mess and slowly reveal the circle?
Yes to your second question, you could piece all the little curves together to form a big circle.
Regarding your first question, I suppose you could do something like that, though with some extra sign changes because of the reflections. What I do is slightly more involved, because I use the same code for non-polygonal domains such as the ellipse. The front is made of several thousand particles, each starting with speed 1 in a different direction, and I have the computer determine their trajectory by taking into account collisions with the boundary.
Knowing that the length of the circle has infinite points, can we prove that the circle's points will fill all the area within the triangle after a certain time?
It might be possible to prove something like this: given any strictly positive, but arbitrarily small real number epsilon, the points of the circle will pass within distance epsilon of any point in the triangle given enough time. In other words, if you give the circle a little thickness, it will fill the triangle at some point, whatever the thickness is, provided it is not zero.
❤❤❤
* Windows media player background intensifies *
The inside looks like it's rotating, and I'm not gonna tell you in what direction
Would it look much different if it started from the various centers of the triangle? My gut tells me the incenter might be worth showing.
What would change is that the front would hit certain parts of the triangle at the same time: the corners if you start from the circumcenter, the sides if you start from the incenter.
Beside of relaxing music and nice colours, 1000$ to the one who understand what this video is about...
Am I allowed to apply for the reward?
@@NilsBerglund
Yes, of course. You are the "Number one" candidate for the award! After all, you are the one who made the video, the one who put beautiful music in the background, a lot of colorful circles.
You even bothered for an amazing headline: Triangular the Circle !!
So this is really a beautiful, and even very relaxing video, but just before the award committee announces you as the winner:
Please, let us know what the mathematical rationale for the title? How did you manage to "triangular the circle"? What is the logical idea behind the video?
You will receive the prize happily, only before it was important for us to draw your attention that you forgot to attach a mathematical explanation to the video.
With regards
The prize committee.
@@tamirerez2547 The title is a play-on-words based on a previous video called "Squaring the circle", which is impossible as we all know. There is more in the description on the physical meaning of the simulation: it shows the evolution of a circular wave front in an idealized fluid (without dissipation or interference), when it is reflected on the boundaries of the triangular container.
@@NilsBerglund
So it is just a colourful GAME witn circles? As the days when we were kids in the kindergarten? With a paper and lot of colour markers...
Now I like it!
No need to imaginery fluid... reflection of a wave... triangular...
Why didn't you say A GAME WITH CIRCLES from the beginning?
Very nice video!! 😉
I guess different people see different things in these videos: some see the maths/physics, others see a game, still others the aesthetic aspect. All these ways of looking at them are fine, of course!
What if the lines got thinner and thinner as they went on, eventually fading out only leaving the "new" lines, before the field gets too crowded
In fact all the lines you see at any time are "new", so one would have to decide on a criterion saying which parts to highlight. But I did something with a fading past here: czcams.com/video/2NLtpcyuRO8/video.html
After so much triangulation, will those lines sum up to the circumference?
Yes, you can always reassemble all the circular arcs to make a large circle, whose radius grows proportionally to time
The circle reappears at 2:46 too.
what happens if you triangle the triangle? it will be normal on the first bounce but after that i cant visualise it in my head
You can try working it out by reflecting the billiard boundary instead of the wave front: look at a growing triangle in a plane tiled with triangles defining the billiard.
It's me trying to find stronghold
What if you circle the circle, but have the expanding circle start out off-centered in the outer circular wall?
See here: czcams.com/video/17DHYxcfpuk/video.html
@@NilsBerglund Ooo nice! I knew there was a chance you already had a video of that :D
how do you find these songs bro? they suit the videos so well :))
Go to CZcams Studio, and then to "Audio library" in the menu on the left
This video is in your recommended
Wooooo ahhh
Coolest thing is that he circle is just getting bigger a bigger, and if there were no walls, it would just be a circle 🤯
Next project: there are certain shapes that if a circle expands from one specific point, that area will never be hit. Do that with this.
So _this_ is what photons do with infinite mirrors...
a e s t h e t i c s
Im going to tell my kids that this is how remesh works...
Circulate the triangle now
When I hit my elbow on the edge of the table
My entire Nervous System:
Would the length of all the segments at the end equal the circumference of the circle if it wasn't trapped in the triangle?
The total length of the curve grows proportionally to time. At any time, you could reassemble all the pieces of circular arcs to form a big circle, whose radius grows linearly with time.
@@NilsBerglund thanks, your vids are great btw.
Noice
Now do it in an equilateral triangle. This isn't satisfying enough.
Sprinkle song.
(If you get the reference the you get a gold star 🌟)
I just watch this because it's satisfying, not for any math reasons
Which is perfectly all right!
por que?
Next try an equilateral triangle or icosceles triangle
Thanks, I'm pretty sure we will see some more polygons in a few days.
Could the circle be formed again after a large amount of time?
All the pieces of curve can be put together to form a big circle of radius growing proportionally to time. So you won't get a small circle again.
Since there are finitely many pixels in the simulation, after an extremely long time the video would start showing a periodic pattern (but which may have a very long period).
Imagine hexagon a circle
How is reflection defined when the wave hits a corner?
I did not code it very carefully here, but you can show that for angles of the form 180°/n for some integer n (here n is 2, 3 and 6 for the three corners), there is a well-defined limit when you hit the boundary closer and closer to the corner from either side. Depending on whether n is even or odd, you either make a turn by 180°, or you are reflected with respect to the angle's bisector. The best way of seeing this is to "unfold" the corner, let particles move on straight lines, and then fold the trajectory back to the actual corner.
Pls do it in a regular pentagon xD
I'm working on extending my code to regular polygons, so I guess we're gonna see some of them soon!
Забавно, что на 2.48 сформировалась сходящаяся и расходящаяся первоначальная огибающая круга
It would be better if you let the circle grow without the triangle and with triangle to show how big it could grow.
There have been some similar suggestions, yes. I plan to make a simulation like that. Thanks!
Does the principle of indifference hold?
If you mean whether the line will get spread uniformly over the whole triangle: I'm not sure. The system is not ergodic in terms of both position and velocity, but in might be in terms of position only.
@@NilsBerglund
I meant the topological space only in the position. It is very interesting!
Can someone tell what softwares to use to make such animations??
You can download the current version of my C code here: www.idpoisson.fr/berglund/software.html
What about an equilateral triangle?
It just like see a universe then it just increase itself and do it over again and just in one space....thousand universes in one spot.
but what if you used an equilateral triangle 👀
It would also give a closed curve. Since the triangle here is half an equilateral one, the outcome would not be too different, though it would look more symmetrical if you start from the center. I will try something like that once I have extended my code to more polygons.