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Fractal charm: Space filling curves
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- Опубліковано 15 січ 2016
- A montage of space-filling curves meant as a supplement to the Hilbert curve video.
• Hilbert's Curve: Is in...
These animations are largely made using a custom python library, manim. See the FAQ comments here:
www.3blue1brow...
github.com/3b1...
github.com/Man...
You can find code for specific videos and projects here:
github.com/3b1...
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
Is it just me or is everyone here smart as hell and I just like the video because it’s satisfying
not just you.
True lol
Its not too hard to understand just look at the seed example:)
Fortnite
Yeah pretty much
The perfect loading screen.
I would never get any shit done lmao. But on the other hand my anxiety and depression are at an all time low!
Imagine getting stuck in the those fractal shapes
Loading tips!
Genius
Flow snake = snow flake! Just got that.
Leave it to a computer scientist to iterate a joke into the ground.
Tadesan lol
Flow🐍 is ❄flake.
Joke! XD
Good point comrade.
0:54 "This one has the most delightful name"
Me: "Snowflake curve?"
1:01: "FLOW SNAKE"
Me: 😮
Switch the first two letters
@@Steffy07 Snow flake :P
@@Steffy07 Who are you talking to?
@@NoriMori1992 you?
1:10
Triforce.
I know it's "Sierpiński triangle," but I'm a gamer. It's "Triforce" to me.
The title's great and all, until someone decides to read your internet history.
So underrated
uh oh
Curves!😏
O-O
I don't get it, I'm not too young, I just don't get it, someone please explain.
Goddamn why does nobody believe me
So the last one has an area of 0 when the seed angle is infitessimally close to 0, but when the angle is exactly 0 it turns to have a non-0 area? I need to lie down..
+Kabitu1 It is wild, isn't it? With fractals, rather than talking in terms of area, the more natural notion seems to be fractal dimension (and I guess the measure according to that fractal dimension).
+Kabitu1 And not only does it have a non-0 area, it has an infinite circumference.
+3Blue1Brown, I'm a little late to the party, but is the box dimension of the non-zero angled curve equal to one for theta equal to some small number, then jump to 2 for theta equal to zero? Or does the box dimension approach 2 as theta approaches 0, equal zero at theta equals zero?
+
Kabitu1 you might need to slow down
At theta=180° the final iteration is 1 dimentional, at thera=0° the final iteration is 2 dimensional. But for every angle in between the final iteration is neither 1D or 2D, but in between.
I wonder what function gives the dimension when the angle is the input.
I just worked it out. It’s D=ln(4)/ln(2(1+sin(θ/2))).
Well the dimension formula is D=ln(n)/ln(s) where n=the number of copies to make the original and s=the scaling ratio of those copies. Because the seed always has 4 evenly sized line segments, n=4. The scaling ratio will be the length of the seed relative to to the lines that make it up. That length is 2(1+sin(θ/2)). Putting it all together, you get D=ln(4)/ln(2(1+sin(θ/2))).
xenontesla122 wow that is amazing
@@xenontesla122
That's a neat idea. I'll try this on my students for an illustration of what fractal dimension is.
nobody:
bacteria every 10-20 min:
Flow Snake seems like a spoonerism for Snow Flake, and it looks much like the Koch Snowflake, but with a seemingly angular tilt.
Makes a bit of sense though, because a curve is kinda like a snake, and fractals kinda flow from segment to segment.
Sounds like an EDM composer nickname
Sometimes I feel that the math we have is little dubious, but can't find anything wrong in it. Everything is so nicely connected and still the some of the end results are so weird.
wait till you hear about Borwein integrals
For the first time in my life, I felt that mathematics is so beautiful.
Any mathematical "porn" gets me interested tbh
+Hwd405 Yor profile pic matches perfectly
***** gorgeous tbh
God i hope he didnt have a Spyro profile pic when he said that
Axrah at the time it was Jesy Nelson from Little Mix doing the "Balegdeh" face I think? I barely remember this comment
@@Hwd405 oh thank god, i also like how you reply to a comment to your 4 yo comment in under 1 hour
I haven't seen a video without explanations from Brown. I guess that such video is considered art. It comes from 3Blue1Brown, his educational work is a work of art.
I thought of a way to make a single 1D line that goes through all of 3D space. First you make a Hilbert curve for a slice of space, then you extend it to all of space, and then you unfold it. With this, now you have the entirety of 3D space on a 2D plane, in which you can draw a Hilbert curve, turning the entirety of 3D space into a single 1D line.
I love fractals and space filling curves because they are so beautiful; looks like a whole new universe, and I absolutely love to look and play with settings to create new ones. Is funny how simple math generates the most beautiful and amazing visions one could have, many times even by accident.
The bacteria when I drop my food for a millisecond:
Nobody:
UA-cam: Hey kid.. wanna see some sick shapes?
1.2 million people: Sure why not
trueee
This is what im thinking to fall a sleep i would think about a box and fill it with lines and curves until theres no more space left. Damn yt recommendation know me
トップダウンで結晶構造みたいなものが出来るのすごいな
My favourite channel is back! :D
And then he turned into more of himself. Funniest shit I've ever seen
You just invented a new genre of porn
(And I'm into it)
how
how is this porn?
@@Distantexplosion1076 porn = pleasure , this = pleasure pretty easy to understand
Dante Pedersen it’s a joke
@@lilyerwin8114 ok
The new Mother 4 PSI effects are looking great!
Flow Snake Has Got To Be The Best Names Ever Possibly Created And Nothing Will Beat It
Brilliant title
+electrocat1 Until someone's reading your internet history.
+electrocat1 it's been changed *for some reason*
+Mormeemo_ Nooo. Now I wanna know what the original title was ...
Globalplayer1102 Fractal porn: Space filling curves
@@jumbochamploon2591 thanks
would you do a video on 3 dimensional fractals, ones that blurr the line between 2 and 3 dimensions instead of 1 and 2?
This is a sick idea.
Wait. After thinking about it, I'm not sure if it could geometrically work.
Neat
What would it look like
Vectors are useful tools for solving two-dimensional problems. Life, however, happens in three dimensions. To expand the use of vectors to more realistic applications, it is necessary to create a framework for describing three-dimensional space. For example, although a two-dimensional map is a useful tool for navigating from one place to another, in some cases the topography of the land is important. Does your planned route go through the mountains? Do you have to cross a river? To appreciate fully the impact of these geographic features, you must use three dimensions. This section presents a natural extension of the two-dimensional Cartesian coordinate plane into three dimensions.
7 years ago, under a thousand comments - how is the algorithm supposed to realise what a gem this is?
One more watch then bookmark it - whoops, nearly missed giving my upvote
You are extremely talented in making mathematical animations!
Great! Thanks!
He: Says something about science.
My brain at 3a.m.: Huh? It's lookin' cool
your videos are simply fantastic. I remember a long time ago I stumbled across space filling curves and Hausdorff dimension on wikipedia. Absolutely blew my mind. It's the crazy, completely mind-bending stuff like this that makes me want to become a mathematician. Anyway, what software do you use for the 2d animation? I am curious to try my hand at some illustrative animation.
The simplicity of this is nice.
I wish I wasn't alive anymore and my consciousness was just repeating 2-D shapes on a black background.
I don’t have any mathematical insight to offer here I just really love how that last fractal looks with small, nonzero values of theta.
1:02 It seemed to turn.
Who else is just watching because it's satisfying?
nothing more entertaining then watching shapes form at 8pm
i could watch this for hours and never get bored.
maths are amazing.
>every comment about the title is positive
>changes title
what the fuck?
+Chandler Gloyd I need to know what the title was! Could you please tell me?
***** Fractal porn: Space filling curves
Chandler Gloyd LOL! I see, thank you.
+Chandler Gloyd Perhaps the related videos were a little... inappropriate?
+JSQuareD I consider that a bonus.
name of the music ? :D
4 years later and nobody is here to answer his question. Is there no such thing as joy in this universe?
I was waiting answer for 2 years
Shazam doesn't pickup anything
Ohh wow... That's depressing
I guess we will never know... :sadface:
I literally shed a tear watching the end... math is just beautiful.
Back in 1975'ish the era of Mandelbrot and Feigenbaum et al fractals and bifurcations I reduced the basic arithmetic for Dragon Curves to bit-reverse-and-add-adjacent-indices which carries out in binary that familiar 'dragon', but also quinary (base 5) does exactly, likewise, except it does both ends at the same time, and, interesting of course, 2×5=10.
This video needs a sequel.
Agreed agreed
I wish there was some standalone software that would let you mess around with seeds and their resultant curves. Those animations are just so satisfying...
when doctor gave me ,spine anestheisa ,i saw amazing patterns , then felt everything is folding , no fear no self no other nothing exist at that point . nothing is real and everything is real. i felt great spiritually like i saw something all should see ,its a space between death and life , i know death is not that scary but as i start coming to my senses i felt love for everyone and everything in this world. its like booting up system and loading programs
This video really really strains my eyes for some reason
when i was baby, patterns used to scare me. i thought i was over it but it appears fractals have reignited that in me. weird.
never watched stuff like this and then youtube just decided to show this to Literally everyone
That's so beautiful. Today I shed a tear as I remember why I dropped math. I asked my math teacher: "Why should I learn math?" , he answered: "To no starve to death!". This video is the right answer he should have given. I'm 37, is there anyway I can learn math from 0? Is there time?
i hope this guy knows that very few people who watch his videos are actually interested in math or calculus or whatever this is about and they are just here because cool pattern makes human brain mesmerized (not saying his videos are bad)
Is there a tool to make these curves and test them out with seeds
+hama prgasc What's it called?
+XxCLIqGamerxX I answered this question ,but someone(idiot) deleted my comment.
AGAIN
you can do this in these software :
1. Processing
2. Structure Synth
3. Apophysis
Jetset Willy Yeah it says removed as spam, but thanks
thank you, I never knew those programs existed, just downloaded all 3. they rock
Robbie Devine Aslo VVVV will do the job.
vvvv.org
you can import Structure Synth code to VVVV via the integrated module and animate the whole.
watch?v=3h8PvAbTLAw
maybe you could do a video the Mandelbrot set. i would be interested to see that :)
Whoever animated this, deserves a plentiful raise.
Am I the only one who's here because it's satisfying? Not because it's educational?
i need more of this
Is there any way to quantify how quickly these transformations appear to "blend together" and start looking lie just a filled in object?
i.e. When I watch the Hilbert curve animation, the curve basically looks like a filled in square at the 9th animation. However, for the Peano curve, it looks like a filled in square after only 6 animations
I get that this is not a mathematically rigorous description, but I'm curious. These things are so interesting.
You can define the speed of convergence rigorously, describing how fast the distance between the n-th curve and the final curve decreases
It is mathematically rigourous. You're trying to find an n such that for all x > n, a certain area a following a curve PHCx would map out the unit square.
Why does his satisfy me deeply
The first filling curve you show us is like a big and bigger and bigger and bigger and bigger and bigger maze!
I am a 21 yo guy who hasn't actually got a proper knack for mathematics but wants to improve. What do you suggest? Where to begin? I know the question is very vague - like extremely vague for mathematics is vast, but if could still guide me, in anyway, it would be very much appreciated. :)
+Nikhil Sasi Rajan I would start by ensuring that your fundamentals are sound. It's much too easy to excitedly learn about what is interesting, with not enough time spent on exercises and foundations. With some admitted bias, I'd recommend Khan Academy for that.
I am also a big Art of Problem Solving fan. They are aptly named, and provide a seemingly endless source of interesting problems that help to gain different perspectives on math.
And as general advice, always be ruthlessly honest with yourself about whether or not you understand something, and be patient enough to insist on a deep understanding before you move on.
+1 for AoPS!
Do you have an AoPS account?
Which software do you use?
He writes some of his own stuff github.com/3b1b/manim
Stop straining your eyes trying to see the tiny details
relax
Let it happen
Enjoy the smooth colors
The Hilbert curve is a continuous fractal space-filling curve first described by the German mathematician David Hilbert in 1891, as a variant of the space-filling Peano curves discovered by Giuseppe Peano in 1890. Because it is space-filling, its Hausdorff dimension is 2
Well thanks, now I've got jizz everywhere... D:
Everybody: wow this is really impressive and satisfying!
Me: Y o u T u b e G a m i n g l o g o ?
Same lol
Why is this so satisfying
I vaguely remember most of these. Vihart taught me good
What would a pi-dimensional fractal be?
TheRedstoneTaco
I don't know man
Because it would be greater than 3d
So I wonder how would we perceive it
Try tau dimensional first.
@@kulak8548 But tau is larger than 6..
We can't even begin to comprehend.
Better start with e
estava ansioso já
What a gorgeous, FlowSnake!
The razar team making the perfect logo
+3Blue1Brown didn't know "charm" and "porn" were synonyms ;D
*"porn"*
@3Blue1Brown: What happen if the angle tends to 180 degree at last ?
+Bhagirath Solanki We get the most boring fractal of all: A line.
xso.
its area is boring, but its 'node' spacing may be interesting
I don't know what this is but I love it.
It’s just.... so satisfying...
0:05 Slimes in Minecraft be like:
1:19 zelda players where you at
im pretty sure everyone knows what the triforce is
Flow snake was introduced to me in the terminals of jurassic park on snes, fun to watch.
2:09 that looks like an angel attack lmao
im not even paying attention
im just dancing to the music
I just woke up in the midlle of the night to tu-ru-ru-tu-tuu in my ears.
I can’t image there are so many beautiful curve in the world'
this is way too satisfying
i need this color palette why is it so good
I love your videos. Too much videos like "how to take the derivative of a function" on youtube, lol. This is really interesting stuffs, more like this! :)
The sierpinsky triangle shows up in pascal's triangle, it's a fractal and it can be made by the simplest of hilbert curves, it's my favorite shape from now on.
great animations.
IDK about ANY of this math stuff but boy all this looks real cool
Hey guy......stupid people like me want to understand this too. Your videos are fantastic.........do more explanation without rushing through or assuming everyone has had the same mathematical training. I know it's slumming, but cut us mortals a break. Keep the videos coming.
the previous video goes into more detail. If you've watched that already, then I'm sorry, but I find it weird that you comment this on this video(which is more of a visual show-off than anything else)
Excellent presentation, clear and elegant.
The scale plays the role of a second dimension as in holography.
It’s really cool when he explained it, now I kinda get it
The other tactic for making space-filling curves is to start with the shape you want to be filled, define a start and end point for the curve on the border of the shape, define a method for dividing it into segments (self-similar is easiest because a seed can be used, but not actually necessary), then defining an order and end points for the individuals (defining a partial space-filling curve by connecting), and finally defining a recursive segmentation for every segment. As long as the limit of iterated segmentation hits every point in the original shape, then the limit of the partial spacefilling curves will converge to a spacefilling curve This way you can make really funky space filling curves, like defining one which can fill a circle (divide into seven internal circles and a bunch of roulux triangle-esque shapes which can given a space filling curve in a similar manner to actual triangles by using arbitrary curve segments instead of line segments, the partitions will just get super distorted).
This gave you a space filling sub.
These would make excellent card backs for playing cards!!!
idk what i just watched but i enjoyed it
I have noi idea but it satisfies me
Actually went through the trouble of working out the formula for the koch curve's fractal index at various angles.
*SPOILER: FORMULA IS BELOW*
f(x)= log(2+2*cos(x/2))/log(2)
this should take forever to make. Thank you so much! I watched it, it was very interesting. My little brother liked watching it too, and he HATES math.
I'm so glad you're still making videos!!
this is a entire vibe tbh
.... ..... thats the best name I have ever heard of anything
Lol, it was maths that made me quit environmental geosciences at uni after just 8 weeks, but here i am again, watching fractal porn, numberphile and other mathematical stuff on youtube. but i'm happy being able to appreciate the beauty of maths and other sciences, instead of simply memorizing stuff like a fucking computer to proof some wannabe prof or teacher you're so fucking good at it, for a few weeks at least, whilst not giving a flying fuck about the subject anyway. Love this system. Guess I'm not the only one, eh?