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.
@jamesgillum9604
7 ай бұрын
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
@1495978707
7 ай бұрын
Except for snow and neutron reflectors
@corrinflakes9659
7 ай бұрын
@@1495978707 ironically enough for exceptions, this fractal is snowflake crystal-like.
@hgb3405
4 ай бұрын
So would that be better or worse?
@Jose-yt3qz
20 күн бұрын
Had an odd thought. What if Black Holes trap light due to the same principle?
@bobman929
2 жыл бұрын
Cool how the laser can be going mental but the initial beam doesn't even look like it's moving
@aa01blue38
2 жыл бұрын
Infinite fractals probably diverge no matter how small the change is in the initial angle, do they? Just like the double pendulum.
@theroboman727
2 жыл бұрын
It's called chaos. Big differences from very small starting differences
@lekoman
7 ай бұрын
It's kinda like a gear train.
@JamieElli
7 ай бұрын
Any time the laser hits a corner, it reflects in a very different direction.
@JamesTDG
7 ай бұрын
And it's why it makes doing precision shots in a game so damn hard lol
@RebekahSolWest
2 жыл бұрын
Nicely synchronized with the music! And this is super interesting to watch.
@NilsBerglund
2 жыл бұрын
Yay, thank you!
@Dark0neone
2 жыл бұрын
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.
@khanmaxfield7974
7 ай бұрын
@@NilsBerglund wait has it been synchronised somehow or is the rate of rotation constant?
@The_Legend47
7 ай бұрын
NGL, the music puts me in the mindset of "Hotel California," by The Eagles....
@empmachine
2 жыл бұрын
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!
@CuulX
2 жыл бұрын
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).
@NilsBerglund
2 жыл бұрын
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.
@ObjectsInMotion
2 жыл бұрын
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.
@zoltankurti
2 жыл бұрын
@@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.
@CuulX
2 жыл бұрын
@@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.
@fabiofdez
2 жыл бұрын
So cool how each reflection traces the inner surface faster, the beam flashing around faster yet as it dims
@JaelinBai
2 жыл бұрын
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
@hegelscat9423
2 жыл бұрын
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.
@seramuse888
2 жыл бұрын
It's like the Weierstrass function but slightly better looking.
@theshuman100
7 ай бұрын
what im hearing is that the beams would always bounce straight back like the dvd logo at a corner hence
@liamernst9626
7 ай бұрын
@@theshuman100only if the angle bisector of the corner is parallel with the incoming ray
@ShuRugal
7 ай бұрын
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?
@AesaKamar
2 жыл бұрын
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~
@NilsBerglund
2 жыл бұрын
Thanks! I didn't know I was a computational artist, but I like the sound of it ;)
@valuerie
6 ай бұрын
does "index a 2d space" with a "1d ray" mean the ray passes through all possible points in an area? vocab moment
@j.thomas1420
2 жыл бұрын
0° to 45° to investigate, and independently the level of the snowflake, we obtain every pattern reachable. Even the simple triangle has that property.
@williamhenby952
7 ай бұрын
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
@triPocoPi9576
4 ай бұрын
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.
@シロダサンダー
2 жыл бұрын
This would be great as the background for a clock... :)
@MirlitronOne
7 ай бұрын
Came for the ray tracing, stayed for the music.
@FadkinsDiet
8 күн бұрын
Making 10/4 time signature sound good
@davidallen5142
7 күн бұрын
Honestly, both were great at keeping our attention. If it was the ray tracing alone or the music alone, we wouldn’t have watched the whole video. The well-done editing held the video together. Two are better than one. If one falls, the other can help him up.
@darkpheonix77
2 жыл бұрын
I nominate this as the best screensaver ever.
@beckybascue7012
Жыл бұрын
It's impossible to feel more satisfied then how you feel when you realize each level is a perfect minute.
@Aniga34574
7 ай бұрын
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.
@NilsBerglund
7 ай бұрын
Thanks!
@entity07
4 ай бұрын
I wonder how the wavelength might change with the reflections, whether there would be a prismatic effect
@stevenclark2188
2 жыл бұрын
It's amazing just how few iterations that took to get very chaotic.
@asdfghyter
7 ай бұрын
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
@shadamethyst1258
2 жыл бұрын
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
@aaanimations_
7 ай бұрын
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.
@kliu1066
4 ай бұрын
@@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
@EarlOfMaladyCrescent
2 жыл бұрын
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!
@NilsBerglund
2 жыл бұрын
Thank you very much!
@maciejmatyka
2 жыл бұрын
It is interesting. As usual the question about details ;) - are you reflecting beam using segment/ray collision or is it pixel based solution?
@NilsBerglund
2 жыл бұрын
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.
@wmlye1
2 жыл бұрын
@@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...
@NilsBerglund
2 жыл бұрын
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.
@brandonsaffell4100
2 жыл бұрын
@@NilsBerglund exactly is a pretty big word.
@NilsBerglund
2 жыл бұрын
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.
@theangrierbrit
7 ай бұрын
bro this new just shapes and beats level looks hard
@mariodistefano2973
2 жыл бұрын
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!!!
@R2Bl3nd
2 жыл бұрын
The very last iteration really looked like a laser at a rave or something
@NilsBerglund
2 жыл бұрын
Pink Floyd (or at least David and Richard) playing Echoes in Gdansk ;)
@wmlye1
2 жыл бұрын
The first video on the "suggested next" grid after this played was kzitem.info/news/bejne/lYqBqnt5e5R1Zag, which has a laser light rave style thumbnail. The KZitem algorithm strikes again!
@NilsBerglund
2 жыл бұрын
Cool! Indeed, some of these "lighthouse beam" sims remind me of David Gilmour and Rick Wright playing PF's Echoes at Gdansk: kzitem.info/news/bejne/poOky3efcniQiIY
@wmlye1
2 жыл бұрын
@@NilsBerglund For years, the most popular show at our local planetarium was "Laser Floyd".
@entity07
4 ай бұрын
A bit of afterimage could be a cool effect to overlay. It would replicate what your eyes do naturally with a bright light
@ReverseDFatE
2 жыл бұрын
Hmm, I wonder what would happen in the snoflake corners would be rounded slightly.
@Rensra
Жыл бұрын
I played this on double speed, just for fun, and the groove of this song magnified alongside the visuals. Dig!
@hjdbr1094
2 жыл бұрын
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?
@NilsBerglund
2 жыл бұрын
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.
@spankeyfish
2 жыл бұрын
the music adds a psychedelic air to the animation, I can imagine hippies dropping acid and watching it on a loop for hours
@lamaost8487
4 ай бұрын
I just saw the thumbnail and was mentally prepared for a great Trial, but this is also cool
@Xabraxus
7 ай бұрын
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.
@howtoappearincompletely9739
7 ай бұрын
That was mesmerising, and you chose the perfect backing music for it. :-)
@lightningfirst689
Жыл бұрын
Looks a bit like Animusic. Like the beam is what's creating the music, especially level 1.
@akaelalias4478
2 жыл бұрын
This is one of my favourite videos of yours!
@NilsBerglund
2 жыл бұрын
Thank you! :)
@Worldahurt
5 ай бұрын
Good luck at the billiards tournament !
@kingghoul2324
7 ай бұрын
Me, trying to get a headshot in Steamworld Heist:
@thermitty_qxr5276
2 жыл бұрын
Its interesting to see that even if the origin moves slowly the laser will move more faster than the last lines.
@ares395
7 ай бұрын
The more bounces away you are the more 'unstable' the beam is by jumping around
@Xezlec
5 ай бұрын
I'd love to see what happens if the corners are all rounded! It'd transition "gradually" instead of suddenly
@TheDeepDiveLLC
7 ай бұрын
What I see when I'm trying to fall asleep
@DreadedEgg
2 жыл бұрын
gosh you are really coming into your own with this channel
@vincemarenger7122
2 жыл бұрын
Imagine being inside a mirror covered room shaped like this.
@lkahfi
2 жыл бұрын
Nice transition!
@NilsBerglund
2 жыл бұрын
Thanks!
@SynthRockViking
7 ай бұрын
The Ancients moved mountains with vibrationsss
@rmarbertin8131
6 ай бұрын
I had an idea to do something like this, but to hear reverberations, in fractals. Easier said than done though.
@cmos905
2 жыл бұрын
mesmerizing, i loved it!
@NilsBerglund
2 жыл бұрын
Yay, thank you!
@elecboy5126
2 жыл бұрын
I liked the description more than the video
@00vulture
7 ай бұрын
Now this is what I think Euclides's head looked like
@coconutcute712
6 күн бұрын
Now do it again with the lighthouse starting at a corner of the level 0 triangle
@streincorp87
6 күн бұрын
Level 2 seems to be synchronized with the music 👏
@NilsBerglund
5 күн бұрын
I'm taking advantage of our brain's ability at detecting patterns. This is why coincidental matches or near-matches between the image and sound are perceived more strongly.
@SirWulfrick
2 жыл бұрын
I have the sudden urge to get stoned and watch this on a 3hr loop. @.@
@zebraforceone
7 күн бұрын
I wonder what it would sound like if we map frequency to the sides. Only one way to find out!
@lougarcia1485
2 жыл бұрын
Mathematical walkthrough, 3, 6, 9,@ a time. Lucky freakin humans!!
@socalacura1338
6 ай бұрын
If we took this as a limiting procedure, could we infer that as # of iterations --> inf, that the amount of light diffused with each reflection approaches the amount of lumens the initial source produced, aka no light will be reflected anywhere?
@NilsBerglund
6 ай бұрын
I'm not sure anything precise is known on this. There are results for the heat equation, on the so-called harmonic measure of certain fractals, that make similar statements. But I don't know if anything similar is known for the wave equation.
@socalacura1338
6 ай бұрын
@@NilsBerglund Thank you so much for the timely response! I absolutely love fractal math, but I never really thought about it in the context of reflecting light/heat, so seeing that visual was very stunning!
@drsatan7554
2 жыл бұрын
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?
@NilsBerglund
2 жыл бұрын
I would need a source with a licence allowing reuse, otherwise the video may be taken down for copyright reasons.
@rv706
2 жыл бұрын
Oh, the Koch's fractal is such a massive snowflake!
@leahl5007
5 ай бұрын
If you keep increasing the stages, eventually you’ll render the Steamed Hams scene from the Simpsons 👍
@Skeptical_Numbat
2 жыл бұрын
Utterly fascinating. A wonderful demonstration of how - given the right initial conditions & sufficient time - symmetry (order) can appear from chaos. The music is excellent & fits the subject brilliantly. Kudos to the artists & developer.
@sirgog
2 жыл бұрын
I'd be interested to see level 6 or 7 but with the rotation being extremely, extremely slow - perhaps one arcminute per minute.
@NilsBerglund
2 жыл бұрын
At some point, you hit the limits of numerical precision. For instance in the video kzitem.info/news/bejne/zamVnGqagn9qYGk making the laser turn more slowly turns the motion in something quite jerky.
@sirgog
2 жыл бұрын
@@NilsBerglund That's fascinating
@Harry_Ballzonya
15 күн бұрын
This video is 7 hours and 54 minutes too short
@amazingfireboy1848
2 жыл бұрын
Yet another episode of I have no idea what's going on but it's really cool.
@amazingfireboy1848
7 ай бұрын
Oh hey, I feel like I've seen you before. How are you?
@sergio_circoloide
7 ай бұрын
is... is that... ok, i might have brain damage. this is definitely not ceroba's theme
@GaryFerrao
2 жыл бұрын
I just came for the corners.
@TheRealAnsontp
2 жыл бұрын
So this is what dreams look like from 3rd person-
@yqisq6966
7 ай бұрын
Music is so cool.
@cheydinal5401
2 жыл бұрын
0:53 Like a Pink Floyd Album cover
@Jam_MG
6 ай бұрын
0:54 is my favourite frame
@oncedidactic
2 жыл бұрын
I would love to see a heat map of this
@Vfulncchl
2 жыл бұрын
For some reason I really wanted it to go faster
@drsmoto3400
2 жыл бұрын
good musical accompaniment the combined visuals and music remind me of pink floyd
@hymnsfordisco
2 жыл бұрын
A transition from specularity to diffusion
@Phriedah
2 жыл бұрын
If you were to map out which parts of the infinite fractal were touched by the lazer as you swept a full 360, do you think there would be any parts of the infinite fractal that are never 'touched'?
@NilsBerglund
2 жыл бұрын
Not if you start in the center (I checked). It may be the case that different parts of the boundary receive different amounts of energy, though.
@franmedina2096
2 жыл бұрын
I wish I had this level of understanding of the universe
@feinstruktur
7 күн бұрын
beautiful
@Stierguy1
2 жыл бұрын
I conjecture that in the true koch snowflake, the lighthouse beam never crosses the lighthouse except at integer intervals of pi/3
@ikitclaw7146
2 жыл бұрын
If its a lighthouse beam shouldnt it disperse over distance like a lighthous beam does? wudnt laser be more accurate here? with the coherent "light" source used?
@NilsBerglund
2 жыл бұрын
You're right about dispersion. In fact, even a laser beam will disperse, only more slowly. The term "lighthouse beam" refers here to the fact that its direction rotates at constant speed. It was suggested in a viewer's comment.
@ikitclaw7146
2 жыл бұрын
@@NilsBerglund ahh i see, still mesmerizing to watch! lol
@NguyenTrungHieu536
4 ай бұрын
Vincent van Gogh
@Hailfire08
2 жыл бұрын
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?
@NilsBerglund
2 жыл бұрын
I'm not sure how to simulate those, but I can try to find out.
@jaakkopontinen
2 жыл бұрын
1:31 has a Maker's Muse logo feel
@euchiron
2 жыл бұрын
Some of the flickering reminded me of plasma ball effects
@NilsBerglund
2 жыл бұрын
You can also see such effects in the video kzitem.info/news/bejne/t4iBzIGZaaSWdGU (snowflake) or kzitem.info/news/bejne/24J7nX-njox3h2k (Mandelbrot set).
@ewthmatth
2 жыл бұрын
So this qualifies as a chaotic system, right?
@NilsBerglund
2 жыл бұрын
It appears to be chaotic, yes. I don't know if this has been proved though. Billiards in polygons behave quite differently from billiards with concave boundaries, and the methods for proving they are chaotic are also very different.
@loggat3804
2 жыл бұрын
What the..!!! How the fluff is the triangle so much smoother than the hexagon????
@NilsBerglund
2 жыл бұрын
The difference is that the triangle has angles of 60°, which divide 180°. One can then show that the billiard flow is continuous at the angles, meaning that there are no jumps when the beam crosses an angle. For a hexagon, the angles of 120° do not divide 180°. There is still some "magic recombination" going on, owing to the fact that hexagons tile the plane, but jumps remain.
@unflexian
2 жыл бұрын
pretty!
@burkhardstackelberg1203
2 жыл бұрын
This is a lot of fluctuations! How would a circular wave behave here, how an expanding circle of particles? I think, they would look like a mostly-random caleidoscopic image very soon...
@NilsBerglund
2 жыл бұрын
See here for a "wave front" made of many particles: kzitem.info/news/bejne/mICYyWmcbppeaoI and here for a wave: kzitem.info/news/bejne/zrCNzJmmcYqHnKg
@dekade420
6 ай бұрын
so THIS is how anime characters move!
@pianochannel100
2 жыл бұрын
This would be cool if used to encrypt things :D
@Xayuap
2 жыл бұрын
at infinite maybe you just light it up all of it as the derivative won't be defined
@slavvaw
2 жыл бұрын
great music, i like
@nes32ify
7 ай бұрын
It's like the DVD logo X 1000
@xwtek3505
2 жыл бұрын
You can't have lightbeam reflecting off of this fractal, right? It's nowhere differentiable.
@NilsBerglund
2 жыл бұрын
That's right. The best you can hope for is to define a limiting dynamics on a sequence of better and better approximations of the fractal. There is a reference in the description to an article doing computations in that direction.
@psgp
7 ай бұрын
Uhhhh.... That moment... 3:59
@DrMustacho
4 күн бұрын
This is why reptiles are green
@persiancarpet5234
2 жыл бұрын
WE NEED MORE RECURSION
@ZackScriven
2 жыл бұрын
You aren’t going to explain what’s happening?
@NilsBerglund
2 жыл бұрын
See the description :)
@SysFan808
2 жыл бұрын
somehow, when i see the koch curve at the limit, i always see it as being able to cut in half and get it back the same as if ya cut it in quarters, but... i haven't really gotten my mind around it.
@NilsBerglund
2 жыл бұрын
That is how fractals work: every part is a smaller version of a larger part.
@SysFan808
Жыл бұрын
@@NilsBerglund yeah no yeah, it's just... there's two methods to this maddeningness: the "just look at it" and the "it's the same thing". with the "just look at it", you'd look at the koch snowflake not as a triangle of koch curves, but a hexagon of inward facing koch curves. and in the limit, it looks that way, but not? they clearly aren't the same. with the "it's the same thing", you realize doing the half method twice just results with the same thing. (whoda thunk) with each, there's still a bit of a confliction to whether it's actually true. and that's the maddening bit. enough freedom to wonder, but not enough to know, and it feels like noone's been here before.
@guillegeox
2 жыл бұрын
Impresionante, me encanta ❤
@Osama-Bon-Jovi-01
2 жыл бұрын
Awesome
@Xayuap
2 жыл бұрын
at infinite, maybe you just light it all up as the derivative won't be defined
@NilsBerglund
2 жыл бұрын
That's possible. See the description for a link to an article investigating this kind of question.
@CartoType
2 жыл бұрын
Very nice to see the reflections becoming more and more chaotic and frantically changing as the order of the fractal increased. At the limit the pattern will change an infinite number of ti mess in any finite time interval. But the music was jarring and discordant.
@v_pryadchenko
7 ай бұрын
Need more
@tsawy6
2 жыл бұрын
Interesting... Of courses the true Koch snowflake can't have the same process done to it because it's a limiting process, and this is /clearly/ a deeply discontinuous result
@NilsBerglund
2 жыл бұрын
That is right. And yet, people are investigating the question of how a limit could be defined. See the link to an article in the description.
@TarahVanessa
7 күн бұрын
People who get the math oh yes it’s just more math Me oHoHoHho cool triangle with lasers
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