Squaring the circle
Vložit
- čas přidán 27. 04. 2021
- The evolution of a wave front in a square domain, in the approximation of geometric optics (that is, without any dispersion or interference between the waves). In fact, the evolution of the curve is quite simple to obtain: take a circle with radius proportional to time, and as soon as it leaves the square, fold back, as often as needed, all parts of the circle that are outside the square. A more realistic simulation with dispersion and interference is here: • Low sea in the square ... - you can try comparing the shape of the wave fronts in both simulations.
Music: Down The Street Blues, by Unicorn Heads@UnicornHeads
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) - Věda a technologie
I love thinking about stuff like this, like how if you think about it the lines are just getting straighter because the circle is still getting bigger and bigger but it’s all just squished into this little box
Exactly my thought! All the linesegments have the same curvature. What a revelation. Basically you could fold it up to circle.
mindustry gamer
@@7rodo ayyy
@@Mechanarian ayyy
@@thatreallycooluser9663 ayyy
I love how that one line becomes more and more straight as it hits the edge
Makes sense, if the square weren't there it would just be a circle expanding forever, and a really big circle looks like a straight line close-up
Wow good points. I have not thought about that. But yeah, its actually just one single line...
That's the outer edge of the circle.
@@alexpotts6520 Exactly! since the circle started in the middle it's basicaly still ther, just being folded over and over again to fit in the square.
@@coolbionicle And as another person stated in the comments, if you combine all the lines, you get a very big circle
Would love to see what this would look like when the start is in one of the corners instead of the middle
Same
It would look exactly the same as any of the 4 quarters of this square
Would that change the focal point?
@@denyraw why?
@@derekbauer2125 because all the corners of the square are the same
Fun fact: if you connect all the lines, they still form a circle. The circle is there. It is just very big.
Aaaand now I have _"All these squares make a circle, all these squares make a circle, all these..."_ stuck in my head again
That's what happens when youchug a literal gallon of LSD
@@chamberkingston7609 and we still don't know where he even _got_ that milk jug
...and now we never will...
Kami: mr popo
Mr popo: BITCH DONT TELL ME WHAT TO DO!!!
The cool thing is that if you let your eyes drift it sometimes looks like a rotating cylinder.
holy shit you're right
Darn
It's like having an infinite grid of squares like this with a circle expanding from the middle of each one, except the circles pass through all the boundaries without reflecting. This video is looking at one of the squares
Whoa, you’re right.
This is my favorite one yet, the geometric patterns are soo pretty
Glad you like it!
4:50 is an accurate depiction of modern art
nice
a beautiful stunning display of man's ability to find beauty in the abstract?
@1SL So you mean being high?
Mathematical?
Pfft, I don’t think so!
I wonder if, when given enough time, it'll go back to how it started
It won't. As the circle grows larger its edges become straighter. So there's no way to go back to having a small curved edge rather than a long straight-ish edge
Indeed, it won't go back, since the length of the curve increases linearly with time. However, for an approximation like here that has finitely many pixels, Poincaré recurrence tells you that you will at some point go back - but the time required for that can be extremely long, something like the Nth power of 2 where N is the number of pixels.
Nope, never. The number of lines only increases, never decreases. At the start there was 1 line that formed the circle. Then there were 8 lines when the circle first reflected. It will never go back to just 1 line.
@@NilsBerglund Poincare recurrence only holds if f is measure-preserving - I don't think it is in this case
The dynamics is measure-preserving in phase space, that is, you have to consider both position and velocity (or momentum)
This is what it would look like if every idle TV's logo bounced on the corner every time
This is how I'm gonna imagine 4d rotation from now on
i think im addicted to this..
How do you deal with the points of the circle that hit the corners?
That's a good point. One can show that for polygons with angles in Pi/n (n = 4 here), there is a well-defined limit when approaching the corner from either side. For right angles, the particle is simply turned by Pi: it reverses its velocity. For other angles, however, the limit is not necessarily well-defined, and strange things may happen...
I love how eventually it just produces an anti-circle. So many gunk on the outside that it produces a near-round shape while having pure serenity and blankness in the middle.
4:43 i just love how well defined that polygon in the middle is, it’s so close to a circle!
U earned a sub! I hope this will start in somewhere else than the middle thought. Good video! :D
Thanks for subbing
@@NilsBerglund your welcome! :3
if you take a sine + cosine wave and put it on one of those funky self-folding waveshapers and gradually increase volume without clipping its pretty much just this on xy oscilloscope mode
A nice demonstration of a spherical wave being approximated by a plane wave substantially far away from the source.
Normal people: **watches in normal speed**
Me an intellectual: **watches in 2x speed**
i think it would be interesting to have an animation at the end of the video that shows you linking all these lines together to recreate the original but enlarged circle
Thanks, I'll look into it. Some videos like czcams.com/video/JLywMk-AGPo/video.html give an idea by looking at a particular angle.
This is incomprehensible, thanks
ive asked for this video before, and you delivered! thank you so much
No problem!!
I love when that single drawn line of circle went chaotic real quick
Really interesting & great choice of music!
Glad you liked it!
Is there any chance that you could give the lines transparency so that when the colour of the overlap is the average of the overlapping line colours, and symmetry is better preserved?
It would be hard to do "by hand", but OpenGL might have some functions allowing to do that. I'll have to look into it, thanks for the suggestion!
This is very calming.
This is so hypnotic
Ngl I wanted the video to last longer, I wanted to keep watching the evolution
This is sO cool
Glad you enjoy it! The length is often limited by issues of numerical precision, time needed to make them, and memory. But I plan to make more software available, so people can make their own versions.
Do you like how my circle desintegrates into square?
I appreciate the length of this video
"All these squares make a circle, All these squares make a circle..."-Mr. Popo, DBZ abridged
First contact: 0:08
Corners pinch: 0:11
Four-Leaf Clover/Petunia petals: 0:15
Spanish Cross: 0:21
Propellers: 0:34
NICE octagon: 0:50
(and my brain shuts off right around there. :T)
CZcams recommendations: Squaring the circle?
Me: why would I- ... interesting... what about, circling the square🤔🤔?
Fascinating evolution!
I think this example of your work might be particularly effective if presented with imposed op-art style black and white dual shading. Not sure how easily you can do this, though.
Best wishes anyway.
i like how that one shape in the middle keeps adding more sides after each cycle. eventually it will look like a perfect circle
its interesting to see how, as the circle is just spreading out, the lines start to look straight, but they aren't, they are just smaller and smaller segments of an ever growing circle that is confined in a small space.
Welp, there's one of my questions answered, nice!
You're welcome!
If I ever make a video game, this'll be the loading icon.
This is much better than an ellipse
Do your simulations require big computational effort to run? I'd like to code something like this for fun :)
This one did not require much time to run, only quite a lot of memory (about 300Mb). I'm still improving the code for this one, but you will find links to the code I used in some similar simulations on my channel, see descriptions of billiards in an ellipse.
@@NilsBerglund thank you!
It is amazing!
Älskar hur bra den här blev
Beautiful
Thank you! Cheers!
ITS ADDICTING
ASMR for my eyes
I love it!
Thanks for watching!
I'd like to see this video in reverse.
Omg me toooo
im always super enamored with these videos. I make electronic music and would love to make some generative stuff to accompany these visuals. I see you post some of your stuff on github but what program/IDE are you using?
That sounds interesting! Why don't you get in touch via the contact info on my homepage, see "About" on my channel. The videos are generated by C code, using OpenGL and ffmpeg to generate the movie.
I love these videos
Thanks for watching!
These videos are so mesmerizing.
Not sure how hard it would be to do, but for cases like this one it could be interesting to see what happens if instead of changing colors when refracting, the initial circle was made of a gradient(/rainbow wheel) so you could see where each part of the circle is
For instance, in this case the lines/waves that are "most parallel" to the square edges would be the colors of the four cardinal points of the initial circle and lines with close slopes would likely have similar colors
That would not be hard to do, I can try sometime. A somewhat similar idea is used here: czcams.com/video/2NLtpcyuRO8/video.html
Way more interesting than circling the square
Some real Super Paper Mario vibes from this.
Ever notice how theres always at least 1 square in the middle at all times?
Sometimes more overlapping but still its cool
Now we just need a hyperbolic simulation and our collection is complete! 😅
Sooo mesmerising!! Could you circle the square now?
Why isn't the video from like 2011 it gives off that energy
Teacher: the test isnt confusing
the test:
finally, something with music
Square: What shape do you want to be, Circle?
Circle: YES
Square: I gotchu fam
I believe that, it will become a circle again after some time, Because Energy is constant of the system
Can you do this same thing but with an “infinite” plane of squares showing how the reflections are just an ever growing circle? Somewhat like Numberphiles video on the net of a dodecahedron but instead the net of an infinitely sided cube
That seems a good idea, yes, as it would explain that reflecting the wave is equivalent to reflecting the square on its boundaries without reflecting the wave...
i wonder how it might look to write the of the area containing the center of the square bound by the innermost curves and their intersections as a function of time. it must be continuous, and there is some periodicity inherent. could the integral of that function can be figured in an interesting way? even the pattern of local maxima must be interesting.
I'm not sure, but there may be a connection with Weyl's law ( en.wikipedia.org/wiki/Weyl_law ) and higher-order corrections to it, which would describe the number of lattice points in the growing circle.
What you’re calculator sees when you try to divide by 0
I am interested in the center position where it seems at times there is a rapid succession of what looks like a series of circles. It seems to have more circles repeating as time goes on.Then again maybe I am getting hypnotized and seeing things that aren’t there. Any way that’s what I see. The circles in the center seem to always reaper only to replaced by expanding squares.
I don't know if this has been considered, but it has the feel of an arithmetic problem to me.
I kept seeing asymmetrical movement, but of course that's just an optical illusion. Every frame maintains the initial four way radial symmetry it started with.
A game would lag so hard with this but it would actually be a challenge to avoid all of those
This looks like the projection of a rotating higher-dimensional polytope
I felt like I was going to have a stroke watching the ripples
How does it reflect off of the corners? Did you round them off ever so slightly so that there’d be a clear tangent line to work with? Or maybe just code in that the corner acts as though it has some specific tangent line when something bounces off of it?
You can show that there is a well-defined limit for the reflection of particles hitting the corner closer and closer on either side. These particles simply make half a turn. However I'm still working on eliminating a glitch in the code causing a particle to escape the square now and then...
so this was how earthbound battle backgrounds were made
Me : I hava an important exam tomorrow so I have to study from now
CZcams : wanna see a circle turning in to square?
Me :
TRES BELLE ANIMATION DE LIGNES MULTIPLES MULTICOLORES
ET TRES BELLE MUSIQUE AUSSI......
Merci beaucoup!
Alternate title: how to make a kaleidoscope
These videos are cool but you choose really awesome music for these videos
Thanks, I'm glad you like them both!
How did you do this demonstration? What programs did you use?
It's a C code, using OpenGL. I have not yet published the code for this one, but you will find code in the description of some other videos on this channel.
The ending is what a tesseract looks like
I would like to see this in reverse. Circling the square.
This gets crazy
This is cool. A SQUARED CIRCLE
Ok now circle the square
Hi! Awesome =) How many particles are in this simulation?
Thanks! This one uses 10,000 particles.
@@NilsBerglund Thanks for answering =) Have a good time!
minecraft is actually a game of circles. this video proves it
These things are so trippy. Can you do one with a black background and white color for the border.
Even more trippy at 2x speed
@@Some.username.idk.0, that’s what I’m doing lol
That's actually easy to do. I can try it on one of the next videos, thanks!
If you think about it, the radius of the circle continues to expand, and while this happens the circle stays contained in the square, and the bigger the radius of a circle the flatter a portion of the circumference appears to be, which is what's happening here.
Hands in the air,
Presidents, prime ministers
They said that we didn't care
But that isn't fair
Hands in the air,
Presidents, prime ministers
They said that we didn't care
But we're the square in the circle.
I love your videos, but I prefer much more the ones where chaos eventually emerges.
Maybe do one with more random shapes than square and ellipses, someday :)
Check out his videos on a stadium
As a 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 amount of lines increases every time it hits the corners.
Epic
so what time did the rest of you start disassociating?
So thats what people mean when they say "blink and you'll miss it"
So what does it basically solve? Just curious
Now we only have to find a way to do that with ruler and circle.
Love seeing the event horizon effect here, as the circle get bigger it only appears to straighten out
Wow cool
Interestingly all of those lines glued together would at every moment form an in size increasing circle.
If you pay attention towards the end, you'll see we've come full circle. ;)
Boooooooooooo puns
@@frimi8593 Thank you, thank you, I'll be here all night.
tv screen saver felt like this
Is this the same as if you had an infinite grid and superimposed all the grid-squares which have a piece of the circle in it as it grows?
Yes, except that some of the grid-squares are flipped upside-down or left-right.
Circle: **hits wall**
Me: I don’t get it this isn’t really working
Circle: **hits wall again**
Me: ok Einstein calm down