A lighthouse beam in a von Koch snowflake fractal
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- Опубліковано 6 лип 2024
- Like the video • Billiard in a von Koch... this simulation shows a mathematical billiard in 6 different levels of approximation of a von Koch snowflake fractal:
Level 0: 0:00
Level 1: 1:00
Level 2: 2:00
Level 3: 3:00
Level 4: 4:00
Level 5: 5:00
For each level, a light beam starts from the center, with a slowly changing angle. Whenever it reaches the boundary of the approximate fractal, it is reflected as on a perfect mirror, the color of the beam changes slightly, and its luminosity decreases. In total, 36 reflections are shown for each starting angle. To avoid numerical problems, the corners have been replaced by very small absorbing circles.
The jumps in the trajectory occur whenever it encounters a corner where the boundary is concave. As the level increases, the behavior of the trajectory becomes more and more erratic, suggesting that there is no well-defined limit for the billiard dynamics as one approaches the limit of a genuine fractal boundary.
After making this simulation, I found that this question has been investigated by Michel L Lapidus and Robert Niemeyer, see for instance arxiv.org/abs/0912.3948
There is no definite answer on existence of a limit, but some results on families of periodic trajectories and some conjectures.
Render time: 5 minutes
Color scheme: Parula, initially from Matlab
www.mathworks.com/help/matlab...
Music: "Song for Michael" by Magic In The Other@MagicInTheOther
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) - Наука та технологія
On top of the math properties this works as a visual demonstration of why roughness and porosity, at the right scales, make materials absorb more light. At higher levels the beam constantly gets trapped in tiny surface features where at lower levels it would have bounced around the whole space while still bright.
yeah I was about to comment on how it's interesting that the light becomes trapped in certain sections as the fractal deepens, that's such a good analogy
Except for snow and neutron reflectors
@@1495978707 ironically enough for exceptions, this fractal is snowflake crystal-like.
So would that be better or worse?
Cool how the laser can be going mental but the initial beam doesn't even look like it's moving
Infinite fractals probably diverge no matter how small the change is in the initial angle, do they? Just like the double pendulum.
It's called chaos. Big differences from very small starting differences
It's kinda like a gear train.
Any time the laser hits a corner, it reflects in a very different direction.
And it's why it makes doing precision shots in a game so damn hard lol
The interesting thing about a Koch snowflake at iteration infinity is that it would have no straight barriers for the ray to bounce off of. It would be only corners, infinitely many corners
Each barrier is infinitesimally small after infinite iterations, if the line is rotating on a circular trajectory then it is in my opinion intuitive to assume it "bounces" at every point on the circle.
Infinitely many corners yes, but also infinitely many infinitesimally long lines. Would be interesting to see a function of the lengths between every intersection of the lines, see how chaotic it gets, or if there is a pattern.
It's like the Weierstrass function but slightly better looking.
what im hearing is that the beams would always bounce straight back like the dvd logo at a corner hence
@@theshuman100only if the angle bisector of the corner is parallel with the incoming ray
At that point, the number of corners the ray "sees" depends on the Wavelength of the ray. Which probably explains most of how "color" works. A surface has a microscopic shape which allows some wavelengths in, scatters others randomly, and reflects a narrow range coherently?
Nicely synchronized with the music! And this is super interesting to watch.
Yay, thank you!
Pretty sure it's not synchronized, that's an illusion.
Would make a really cool visualizer though, if the speed of the beam's rotation was tied to bpm or something.
@@NilsBerglund wait has it been synchronised somehow or is the rate of rotation constant?
NGL, the music puts me in the mindset of "Hotel California," by The Eagles....
I tried to wake up early but the guitar rocked me back to sleep
So cool how each reflection traces the inner surface faster, the beam flashing around faster yet as it dims
It is really cool when the beam bounces in such a way as to perfectly line up with the source (like it's feeding it'self). Almost like it's finding the hidden solution to some complicated math problem and it makes me wonder: does every shape (even one with a fractal dimension; level infinity) have at least one angle where this effect will manifest? And that's with inf angle resolution (i.e. mathworld).
Geeze your videos can provoke thought (even in my not-yet-awake-morning-brain). You rock!
The koch snowflake (iteration infinity) only has lines of length 0 and corners everywhere. How do you define the bounce angle against a corner? Because that's the only thing that's left. Maybe as if the normal of the corner is the average of the two sides that make it (which can be computed from the iteration that introduced the corner).
Thanks - provoking thoughts is the best that can happen with these videos, I'm really glad when they do.
I'm not sure if much is known on fractal shapes. For convex shapes with a smooth boundary, the situation is well understood. A result due to Henri Poincaré and George Birkhoff, for instance, says that for any rational number p/q between 0 and 1, there exist at least two periodic trajectories of period q, turning p times around the boundary.
For the non-fractal shapes, every single angle eventually comes back to the origin, there is nothing special about the ones that line up here, they just line up in few cycles. The only reason you dont see the others come back to the origin is because the light ray is fading with each bounce. If it didn't, you'd see they all connect.
@@CuulX I would imagine the corners are countable, aren't they? If that's the case I would expect most lines to not hit any corners of the fractal.
@@zoltankurti excellent question! Since the fractal only has corners and no line segments and since it is connected, a line from the center at any real number angle must pass from the inside through some corner for the first time somewhere before it exits the fractal forever. So there has to be at least one corner for every real number between 0 and tau radians (or other angle system). Conclusion: there are uncountably many corners on the fractal.
On the other hand, the exact position should give you a specific corner. But non-rational numbers with their infinite decimal expansion can't correspond to a corner introduced at a specific iteration. So finding where the intersection is is similar to finding the last digit of a real number, impossible. So not possible for all real number angles, even though the intersection exists.
The "good" news is that while koch snowflake and the intersection at real numbers isn't computable, neither are any real numbers in a computer so you will never need to find the answer in a simulation of particles regardless of that fact. All numbers are the rational approximation when stored by a computer.
It is probably not possible to do symbolic math with real numbers to find things like the real number position of points, but not sure. The itarative process is nonanalytic by definition?
I think you guessed the countability of the set of corners since the iterative process shown usually to construct the fractal is presented in a countable way. If the contruction process is an uncountable one instead: Take one side of the snowflake and then the corner is the one that corresponds to the decimal expansion of the real number in base 4 for the 4 new line segment starts introduced at each decimal. The iteration count is then the same as the length of the decimal expansion and can be countably infinite. But it's clear that each real number addresses a unique and specific point. Same thing really from another perspective.
Fascinating how my intuition just cannot accept that the patterns on level 5 are just as deterministic and harmonious as on level 0.
It’s fascinating to see how fractals convert the smooth continuous movement of a rotating ray into discretized chaos reflecting all over the place seemingly randomly
It also interesting to think about how an infinitely iterated fractal can cause a 1d ray to index a 2d space
And kudos! You’re one of my favorite computational artists and this is one of my favorite videos~
Thanks! I didn't know I was a computational artist, but I like the sound of it ;)
does "index a 2d space" with a "1d ray" mean the ray passes through all possible points in an area? vocab moment
0° to 45° to investigate, and independently the level of the snowflake, we obtain every pattern reachable. Even the simple triangle has that property.
Obligatory nitpicking: it's 60°, not 45°. You might be thinking 8 degrees of symmetry, but triangles only have 6.
Other than that, yeah, every level of the Kock snowflake has 3 axes of reflective symmetry, and 120° rotational symmetry
I nominate this as the best screensaver ever.
An idea that came to mind watching this, have the walls of the fractal show the brightest color that hit that portion of it. This would help show us what from the light has the most focus on vs not per level, and would be giving a stark contrast as the light does die out before touching most of the walls in the higher levels.
Thanks!
I wonder how the wavelength might change with the reflections, whether there would be a prismatic effect
I like how it gets more chaotic at every iteration, and the beats with the music seems to almost match up in rhythm and tempo.
Do you put a lot of conscious effort into the selection of music? How does this selection process look like?
Sometimes I select a track I like first, and make a simulation matching its duration and mood.
But usually, I make the simulation first, and then select the music. In that case I start by looking for tracks that have approximately the right length, and listen to a few of them (typically between 2 and 10 tracks) while playing the video. The YT audio library lists track according to genre (classical, jazz, pop, ambient, and so on) and mood (calm, sad, happy, inspirational, dramatic, funky, ...), which helps narrowing the selection. Also, by now, I have identified a number of artists I tend to like, though I also look for new ones.
For each track, I try to decide whether it goes well with the simulation. I pay attention to the general mood, things like rhythm, changes in tempo/key/style, and whether they match the simulation. When several tracks are similar, and the video has several parts, I tend to select the track that best matches the changes between parts.
This would be great as the background for a clock... :)
I can see this being used as a visual proof that a square-approximated circle (used in the false proof that pi=4) doesn't have a second derivative and thus isn't a circle
I like this idea, that the square's angles don't match the circle's, but the thing is the squares mostly reflect both horizontally and vertically, and as the area of the square approaches 0, the reflections get more and more accurate.
I'm probably completely off-topic.
With this said, I think pi = 3.14159265358979... not 4.
@@aaanimations_which is why OP mentioned it is a false proof, and that you cant say a square is a circle and pi = 4. You can use this as a visualisation for the proof despite it being false
It's amazing just how few iterations that took to get very chaotic.
Not only is this frikin cool, its a good representation of the butterfly effect
It's impossible to feel more satisfied then how you feel when you realize each level is a perfect minute.
bro this new just shapes and beats level looks hard
i want to see how frequently each each edge of the snowflake is hit, like, every time the light hits it, it’s color get closer to some other color. i would love that.
Came for the ray tracing, stayed for the music.
Amazing! Those rebounded lines are jumping about all over the place, particularly on the higher levels! & as usual, brilliant music to go with the cool visuals!
Thank you very much!
One of my favorites yet. Thank you!!
Glad you like it!
this is also a nice illustration of a chaotic system! even the tiniest change in the input angle makes the final parts of the beam jump around erratically
The very last iteration really looked like a laser at a rave or something
Pink Floyd (or at least David and Richard) playing Echoes in Gdansk ;)
The first video on the "suggested next" grid after this played was ua-cam.com/video/4TKDGCBbD2s/v-deo.html, which has a laser light rave style thumbnail. The UA-cam algorithm strikes again!
Cool! Indeed, some of these "lighthouse beam" sims remind me of David Gilmour and Rick Wright playing PF's Echoes at Gdansk: ua-cam.com/video/EMneCi9F_UQ/v-deo.html
@@NilsBerglund For years, the most popular show at our local planetarium was "Laser Floyd".
A bit of afterimage could be a cool effect to overlay. It would replicate what your eyes do naturally with a bright light
Fantastic simulation! Very very instructive on how equations behave in every situation and at the limits! would be interesting also if the fractal would be in 3D!!!
ok so holy cow i love this video like
A) this is just a really cool concept in and of itself
B) Its shown AMAZINGLY, with the gradual decrease in the lines brightness to show # of reflections, and so higher levels dont seem too chaotic with the reflection changing rapidly
C) also the music chosen for this?? fits so freaking well!!
this video makes my math brain very happy
Amazing project, and an amazing result.
Thank you very much!
the music adds a psychedelic air to the animation, I can imagine hippies dropping acid and watching it on a loop for hours
Your music taste for this one is exquisite
I found the syncopated beats go well with the visuals.
I just saw the thumbnail and was mentally prepared for a great Trial, but this is also cool
i wonder what is possible in real life with this, light shows, music and to even everything around us
Good luck at the billiards tournament !
That was mesmerising, and you chose the perfect backing music for it. :-)
Me, trying to get a headshot in Steamworld Heist:
It is interesting. As usual the question about details ;) - are you reflecting beam using segment/ray collision or is it pixel based solution?
I compute the collisions exactly. The boundary is a polygonal line, with about 3000 sides at level 5. The code now uses a structure containing data on each side of the polygonal line, and intersections with all sides are computed by solving an equation of degree 1. The the earliest intersection is computed, and used to iterate the map.
@@NilsBerglund I'm assuming you are using a zero-width ray; do you have any special cases if the ray were to precisely hit a point between segments, or do you just pick the first segment in the list? This now has me thinking about what if we had a non-zero width ray and split the ray into two smaller rays whenever the two sides of the ray hit different segments...
Here I chose to just kill the ray whenever it hits a corner of the boundary. For special angles, for instance those that divide 180°, one can define a limiting behavior. For 60° corners as occur here, the ray should be reflected with respect to the bisector of the angle. But you don't have a well-defined limit for general angles.
@@NilsBerglund exactly is a pretty big word.
By "exactly" I mean that there is an exact formula for the intersection between trajectory and boundary. The computer of course makes round-off errors when evaluating it. But this remains more precise, and more importantly much faster, than if one were to estimate the intersection coordinates by some approximation scheme, say Newton's method. That would be required for boundaries with more complicated equations, e.g. involving sines or exponentials.
I played this on double speed, just for fun, and the groove of this song magnified alongside the visuals.
Dig!
This is one of my favourite videos of yours!
Thank you! :)
Level 4 felt like lightning strikes
It makes sense, but I hadn't guessed that at higher levels, the light beam more easily gets trapped
It looks like a collision error in a game lmao
Hmm, I wonder what would happen in the snoflake corners would be rounded slightly.
There is no dark side of the moon. As a matter of fact, it's all dark.
Very Cool.
What I see when I'm trying to fall asleep
Great ideia!
Looks a bit like Animusic. Like the beam is what's creating the music, especially level 1.
Even a lighthouse beam has non-zero width. And spreads. Adding even a small angle to the beam would make the math harder, but instead of the discontinuous jumps you'd get a lot of ambient light and volumetric reflections. Especially if the corner sharpness is reduced from infinite.
Love the 10/8 soundtrack!
Yes, I love these non-standard time signature (as in PF's "Money"). And the syncopated beats go well with the erratic motion of the beam, I think.
Why does it feel like the fate of the universe depends on some bouncing lines 💀💀
Oh, the Koch's fractal is such a massive snowflake!
The more bounces away you are the more 'unstable' the beam is by jumping around
Imagine being inside a mirror covered room shaped like this.
Nice transition!
Thanks!
The Ancients moved mountains with vibrationsss
Have you chosen the music deliberately to have similar fractal structures?
It's eerie how different aspects of it synch up with the different phases.
I usually spend some time selecting the music, yes. Here I found that the slightly syncopated drumming matched the beam's jumps rather well. I think I also fixed the total length of the published version of the simulation after having found that track.
I love your videos!
Each time I see one of these, I start trying to figure out how to generate a soundtrack to match it. The level snowflakes seem to jump around at about the right pace for a rhythm of some kinds, and it seems there should be some way to convert the chaos to a melody: maybe extract the dominant frequency components and process them in some way? It would be as much art as math but could be mesmerizing if done well.
It's in interesting idea. I don't know enough about sound generation to do it myself at this point, but maybe someone more familiar with that wants to give it a try.
mesmerizing, i loved it!
Yay, thank you!
Now this is what I think Euclides's head looked like
I wonder if all the simulations of this have the same self intersection points, surely at some point it gets so precise that it gets calculated differently by different processors? I wonder where exactly that cut off point is past the floating point.
Its interesting to see that even if the origin moves slowly the laser will move more faster than the last lines.
gosh you are really coming into your own with this channel
I'd love to see what happens if the corners are all rounded! It'd transition "gradually" instead of suddenly
Quick question, if you fix the initial ray and keep increasing the level of the snowflake, does it converge to a determined pattern of reflexions? i.e., could you find the pattern of reflexions for any initial ray in a real Koch snowflake?
The article the description links to has some results in that direction. I think you have to adapt the angle a bit to the generation of the approximation. But then you get sequences of periodic patterns at every generation.
is... is that... ok, i might have brain damage. this is definitely not ceroba's theme
I liked the description more than the video
I just came for the corners.
There are moments when you wish you had three, even four, hands. This is one when I wish I had more eyes...
brilliant
This music reminds me of Rimworld!
I dont want to be reminded of Rimworld, I have exams this week! 😢
Impresionante, me encanta ❤
Yet another insightful video.
Any chance you could do a video with the Deja vu instrumental music, just to make it a touch more epic?
I would need a source with a licence allowing reuse, otherwise the video may be taken down for copyright reasons.
I have the sudden urge to get stoned and watch this on a 3hr loop. @.@
It might be better to only expose pixels in the beam slightly each time they are hit, and have each beam last for many more iterations.
Awesome
I had an idea to do something like this, but to hear reverberations, in fractals. Easier said than done though.
Mathematical walkthrough, 3, 6, 9,@ a time. Lucky freakin humans!!
I wonder what it may sound like if you map the edges of the snowflake to a tone being played when the edge is hit by the laser. As an octave spans 12 semitones it should be easy to evenly distribute one or more octaves around the figure.
Will it result in a sheer cacophony or will there be some kind of rhythm or distinctive chord progressions?
If you keep increasing the stages, eventually you’ll render the Steamed Hams scene from the Simpsons 👍
Hi Nils! Love your work! Did you by any chance play around trying to see which points got "touched" by the light? Obviously with some intensity threshold, and raytracing is hard. But it would be interesting to see if there's places the light didn't reach.
Thanks! I did something in that direction for a wave simulation, see ua-cam.com/video/GXPtzpZIXhQ/v-deo.html
I would expect the places where the average energy of the wave concentrate to have some similarity with what happens for particles, though I'm not sure.
Also, how do you always find new ways to reinterpret the movement of light? Like, how much is it your own ideas and how much is it finding new things in your research?
I have a lot of different sources of inspiration. My research and teaching make up a small part of it. A lot more comes from things I heard about here and there over the years, suggestions from colleagues, and ideas by commenters. Often these ideas remind me of things I had forgotten I knew about. Then it is usually not too hard to find some information online to fill in the missing information.
I wonder if (for higher level approximations) there are some dark spots that the laser never illuminates, even if it makes a complete, perfectly continuous circle.
That seems a bit unlikely for this particular shape, but I'm not sure. There are indeed polygonal "billiards" that are known to have unilluminable spots, see for instance ua-cam.com/video/U63o4enhQ4Q/v-deo.html
pretty!
You can going with this. Also you can spon the shape and keep the laser straight up.
I was surprised by the jumpiness when the bean passed vertices, but I'm hindsight it makes sense. The moments of resonance were really breathtaking.
I would love to see a heat map of this
Could this be used for cryptography, since it's hard to predict the starting position of the beam based on where it landed, as the levels increase?
In principle you can use chaotic systems for cryptography (look for "chaotic synchronization"). I'm not sure this particular system would be useful in practice.
great music, i like
Nice idea. I wonder if it is possible to do raytracing of full Koch snoflake (i.e. infinitely many branches), by calculating the branches on the fly. Should be possible, with a bit of trigonometry.
But as somebody else mentioned, Koch snowflake at infinty iterations, does not have any edges, just corners....
It is not straightforward to define the limit. But in the description, there is a link to an article looking at that problem, for instance by finding families of periodic trajectories that exist at any level.
Music is so cool.
0:53 Like a Pink Floyd Album cover
As a snowflake, I can confirm that is a fractal.
Yet another episode of I have no idea what's going on but it's really cool.
Oh hey, I feel like I've seen you before. How are you?
What about half iterations? Between the triangle and the six-pointed star there's a hexagon, etc.
I heard about bricks of negative refractive index materials forming 'perfect' lenses - could you do a wave simulation where a wave comes from a point, passes through a negative refractive index material, and focuses again to a point?
I'm not sure how to simulate those, but I can try to find out.
I wish I had this level of understanding of the universe
So this is what dreams look like from 3rd person-
For some reason I really wanted it to go faster
now i want to see a lasershow in a koch-snowflake shaped mirror room!
The laser uses one half mirror, so that one makes you 💭 think for sure.
Hi Nils! Do you think these simulations can help us "see" SRB measures an/or measures of maximal entropy for the billiard maps/flows?
I know SRB measures often occur in chaotic dissipative dynamical systems. When do they occur in conservative ones? Something to do with Aubry-Mather sets perhaps?