Fractals are typically not self-similar
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- čas přidán 4. 05. 2024
- An explanation of fractal dimension.
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One technical note: It's possible to have fractals with an integer dimension. The example to have in mind is some very rough curve, which just so happens to achieve roughness level exactly 2. Slightly rough might be around 1.1-dimension; quite rough could be 1.5; but a very rough curve could get up to 2.0 (or more). A classic example of this is the boundary of the Mandelbrot set. The Sierpinski pyramid also has dimension 2 (try computing it!).
The proper definition of a fractal, at least as Mandelbrot wrote it, is a shape whose "Hausdorff dimension" is greater than its "topological dimension". Hausdorff dimension is similar to the box-counting one I showed in this video, in some sense counting using balls instead of boxes, and it coincides with box-counting dimension in many cases. But it's more general, at the cost of being a bit harder to describe.
Topological dimension is something that's always an integer, wherein (loosely speaking) curve-ish things are 1-dimensional, surface-ish things are two-dimensional, etc. For example, a Koch Curve has topological dimension 1, and Hausdorff dimension 1.262. A rough surface might have topological dimension 2, but fractal dimension 2.3. And if a curve with topological dimension 1 has a Hausdorff dimension that happens to be exactly 2, or 3, or 4, etc., it would be considered a fractal, even though it's fractal dimension is an integer.
See Mandelbrot's book "The Fractal Geometry of Nature" for the full details and more examples.
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Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
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"In some ways.. fractal geometry is a rebellion against calculus."
That's just a beautiful statement.
There is no such thing as rebellion in math.
Also I agree
You know I think I like fractals
@@brandongroves4465 I truly do enjoy dividing zero by zero.
Yea! Screw calculus!
"This is math, everything is made up"
Love this quote!!
read philosophy
There’s an interesting question that is “would aliens understand math?”
It boils down to “Is math a human concept or is is it something absolute, that would exist no matter the view point? Is math just some ground rules someone thougth of and then we noticed some interesting results from applying those rules?”
How to make every math teacher very angry and how to claim 42 as a trial answer to everything.
@@HUGOGARCAO it depends on what part of math you mean.
1+1=2 is a physical law and aliens will understand if the are intelligent.
If something is a fractal is made up so the will only understand if you explain it
On politics/news/rockets/car videos, everyone is bringing up politics and news and facts and polls and new models and engine tricks in the comments, often with sources. On those videos, it’s hard to challenge/discuss the body of the argumentation. So we’re left with making comments on the form or philosophy.
It’s great on one side, because it shows a very very deep knowledge is being offered to us, that’s why we wouldn’t be able to criticize/comment/reflect. But it’s sad because I don’t feel we’re competent enough to deserve the author :D
One interesting fact about fractal measure is how it can be used to distinguish Jackson Pollock paintings from imitations. This technique achieved a 93% accuracy rate for distinguishing genuine Pollock paintings from forgeries.
Damn that's a neat piece of fact on its application!
"Paintings"
How did they do that
@@danyilbutsenko6339 there’s always one of you
@@CalebSalstrom Always one correct person?
When I was in class 12th and was discussing with my friends about dimensions something like the idea of 2.5 dimension strikes in my mind. I was searching whether they exist or not and then sometime like 6 months after I discovered this video.
And this video satisfied my Curiosity.
Thanks 3 Blue 1 Brown
I immediately dismissed this thought as complete tomfoolery
Well you still need to differentiate spacial dimensions and fractal dimensions. Those are two completely unrelated concepts only sharing the same linguistic designation.
@@MrMegaMetroid Yes, You are right
@@MrMegaMetroid so there are three versions of dimension
@@archangelofsorrow there are multiple definition of the word dimension. Dimension can also mean size in some contexts. A fractal dimension has nothing to do with a spacial dimension, and a spacial dimension has nothing to do with the dimensions (size) of an object. Also, dimension can mean paralel world, which is also a completely unrelated linguistic concept that has nothing to do with any of the former. The word dimension has multiple definitions, and spacial as well as fractal dimensions are 2 definitions that are entirely unrelated to each other.
That must not be confused with the difference between space and time dimension, which are different, conceptually, talk about the same field in physics though and can thus be categorized as the same linguistic umbrella
"Ah,Yes,The fractal here is made out of fractal"
*Ah yes, enslaved fractal*
*ah* *yes,* *enslaved* *infinity*
Ah yes, Enslaved object
@@tthung8668 Get at it. Unless you "ate" it.
10:39 *n i c e*
Ah, yes, my favorite fractal!
*The Coastline of Britain*
the coastline of Switzerland is 0-dimensional
@@TheAustronaut03 yes, that is technically true
wahoo
The coastline of Norway is far superior.
@@TheAustronaut03
Unless you include the coast of Lake Geneva, of course.
7:32 I just realized that the parallel of this is that the total "length" of an entire square's area is infinite and the total "volume" of its area is 0, but "area" is the only metric that will have a non-0, finite amount to measure it by
And that's what dimensions are all about
Gabriel's horn is a 3D shape with the property of having finite volume but infinite area, so it involves a seeming paradox in that, were you to construct one physically, you could fill it with a finite amount of (idealized) paint, but you could never paint its internal wall as that'd require infinite amounts of paint. I think the principle here may be the same: a 3D object with a volume that, were you to try and disassemble it to reassemble into a perfect 2D shape consisting solely of area, would result in an infinite area -- after all, that's how many "layers" of 2D planes with thickness = 0 are contained in any volume with a thickness > 0.
@@AlexanderGieg Gabriel's horn has infinite area and _surrounds_ a finite volume, but the inside of the horn and the surface of the horn are different things. It is a very interesting object, but it's not a counterexample of the statement that the dimension of an object defines what thing you can measure and have a non-inifinite non-zero value.
never thought of it this way, you blew my mind
I think this is what is meant by k-volume when discussing determinants, where k is the dimension. For example, 1-volume is length, 2-volume is area, 3-volume is our regular volume, and so on. So a 2D object would only have a 2-volume and finding its 1-volume/3-volume wouldn't make sense (or give a determinant of 0; PS not very sure about this statement)
As a physics graduate, I wish that our teacher had shown us this video when he tried to teach us about fractal dimension.
Our teacher's assistant did! And I thank him for that
This channel is youtube gold.
+
+
3blue1brown:maths::pbsspacetime:physics
+
I even enjoy listening to the ad at the end of the video.
What does the B stand for in Benoit B. Mandelbrot?
Benoit B. Mandelbrot.
KasabianFan44 recursive acronym
So his name is Benoit Benoit B. Mandelbrot Mandelbrot. Ah, but now it's Benoit Benoit Benoit B. Mandelbrot Mandelbrot Mandelbrot. You know where this is going.
This is the most brilliant pun i have ever seen
Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit... (infinitely many benoits later) Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot...
A “Mandelbrot” set...
Fun fact: "Mandelbrot" translates to "Almond bread" from German.
Is that profile pic...
from nick?
The almond bread set
I can see it
That's just the name of the person who discovered it....
yay deustechland (sorry germans if i mispronounced or mispelled it)
Car salesman: (slaps sandpaper ) "This bad boy has a really high fractional dimention"
But does it have a high frictional dimension?
@@dangerouspie0319 Yes.
(slaps Mandelbrot set): "This fractal can fit so many fractals in it"
Nahhh
Please create a merch line with
"THIS IS MATH, EVERYTHING IS MADE UP"
I need it.
LOL
@@papasscooperiaworker3649 DUDE
When you learn about a topic before you are taught in school, you see the topic as your friend and your ally rather than a nightmare how ever hard it is especially if you learnt it from 3 blue 1 brown
Please create a merch line with
"THIS IS MATH, EVERYTHING IS MADE UP"
And that sucks so much. I hated school, but now I get home from work and just learn about every topic out there.
School is set up to kill spirits first and educate second and I'm never going to forgive our society for doing that to kids.
I agree. The principle is the same for teachers too. When you want to approach a topic or example that is amazing but also need to fully conceive of it's delivery in a short amount of time; this feels analogous to students' being introduced an idea in an artificially short and high-stakes time window, and being expected to fully incorporate it's implications.
The thing that fires my pistons about math(s) textbooks is that often they will break topics into discrete chunks that don't naturally flow into one another... Not only does NO ONE learn complex subject matter that way, but the combination of the two results in almost no iterative thought process skills being built.
Under this kind of pressure the brain floods and it's physically impossible to absorb the material in a meaningful way. It's a lost opportunity at every level: math becomes the enemy and the amazing skill of developing an iterative thought process is never explicitly or implicitly taught through curriculum.
@@stratowhammy "bUT tAHt mEiK iTs HuRdeR aNd S0 iT tAKes MoR3 E4oRT". These "people" seriously need to hyper-complicate even the most trivial stuff to feel "superior" because they did something "difficult".
Now that you mention it, separating subjects into chunks has another effect, it makes students incapable of seeing the relationships between subjects, and when one approach fails, well we're fucked.
"tH1Nk oUtS1D3 DaH v0X", yeah, when everything we know is the Box (and trust me, they force us into ONLY KNOWING THE BOX), that is more of a formal proof of the impossibility of dreams and the ret@....ness of hope.
@@dangerouspie0319 education system is a play ground for control did you really expect the benefits to outweigh their agenda...
10:34
When you said "In the back of our minds," I thought you were going to make fun of how immature everyone is
Another way to show that when a disk gets scaled down by 1/2, it's area gets scaled down by 1/4:
The area of the original circle = πr²
Since the circle gets scaled down by 1/2, it's area also gets scaled down by 1/2.
=> The area of the new circle = π(r/2)²=π*r²/4
why did no one see this comment ._.
Do you mean the area of the new circle = π(r/2)²=π*r²/4 = 1/4 old area?
That’s great!
It seems that the author wants to use visualization to demonstrate fractals, he doesn't want to use mathematical formulas
@@NhuKhiet-cr8lh oh yea
when i first heard it, i thought he said "In some ways, fractal geometry is a rebellion against capitalism"
had to switch on subtitles to hear it: it was actually 'calculus'
A rebellion against capitalism, huh?
What part?
I'm glad my salary isn't a fractal!
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United 1⃣ and mighty 👍, our Soviet ☭ land 🤝!
Sing 🥇 to the Motherland 😎 , home 🏡 of the free 🆓 ,
Bulwark 👍 of peoples 🌍 in brotherhood 👫 strong 💪 .
O Party 🏛️ of Lenin ☭ , the strength 💪 of the people 👭,To Communism's ☭ triumph 🎉 lead us 👉 on!
(To be continued...)
"A line, a square, a cube..."
"And a Sierpinski triangle"
"A line, a square, a cube, wand a Sierpinski triangle walk into a bar."
The 4th should have been a hypercube.
and a tesseract
THE ARISTOCRATS!
And 4d cube.
10:42 thought you were gonna say" in the back of our minds, we know to say nice"
Blows my mind... I wonder how it will look like some (somewhat regularly shaped) fractal with dimension equal to pi, or e ... Can a dimension be a negative number? How about complex?
you can't represent a fractal with a dimension over 3 in real world
@@raphdm3776 This is math Everything is and can be made up
@@lakshya5946 I know but we will never see what it looks like
@@raphdm3776 we can actually
@@lakshya5946 I don't know about you but I only live in 3 dimensions
17:00 I kind of remember this having a correlation with String Theory where there are these supposed "hidden dimensions" that would justify the 11 something dimensions required in order for the maths to check out. Very interesting to see how certain disciplines cross over!
jup, had the same thought
Right.....
In primary school i used to sometimes doodle and once i drew that Sierpinski triangle thinking that i just invented a new design/shape
Ok so I completely forgot about this comment but please stop arguing in the reply section your points are so stupid lmao
I independently invented certain finite difference methods but I was too old to think they were new.
Lol
lol i remember when i was 8 and discovered that 1 + 3 + 5 + ... + 2n+1 = (n+1)^2, i couldn't prove at the time, and i just thought i had discovered something lol Good Times, when everything was way more simple :)
I'm wanting something doesn't have to mean that you are the first to invent it.
Coming up with something new on your own is what counts, not how many people have done the same before you.
Many people can solve a Rubik's cube but the proportion of people who have figured out a solution themselves is really small.
😂
10:30 - Nice
Nice
10:48 *"In the back of our min-... Nice"*
Your videos remind me why I was a math major until I wrote my first program in 1962.
roasted.
What language did you use?
FORGO (a version of FORTRAN)
I'd like to hear you explain why or just talk more about that. I'm C programmer.
George Steele Ultra rare language. Nothing on the internet about it. Can you please talk to us about that time and what did you do with your coder skills?
This channel is a CZcams anomaly. It is the best intellectual channel on youtube with a fraction of the viewers from all of the other ones (VSauce, Numberphile, Veritasium, MinutePhysics... etc)
Higher quality videos, better explanations, better animations with a fraction of the subscribers.
If you scale it up it will touch more boxes than the inverse of the other channels scaled down.
It is true: he is a gem. Hopefully he can continue to gain more viewers.
Most intellectual channels are intentionally toned down to accommodate the learning capability of the general public. It is not saying the general public is stupid but that most people don't try to learn anything (even in college), but only gather information from these videos. 3B1B's channel is for those who genuinely want to learn.
I hope not that he gets more viewers necessarily (often comment section gets meme'd and/or becomes unanswerable by owner, style changes to fit viewership), but that a greater percentage of existing viewers donate!
I feel like this is for math what PBS Spacetime is for physics.
Vampyricon I like PBS Spacetime too
As always, you made us understand a very abstract concept in an intuitive and logical way. The concept of non-integer dimension which did not make sense some minutes ago makes so much sense now! Thank you very much and keep making such beautiful content!
I dabble in 3D art, and so much of how 3D art is made relies on fractal geometry. You actually start to be able to point out Voronoi fractals in textures after a while. I feel like this gave me some better sense of how it all actually works, though. "Dimension" is a control on many Blender nodes, and now I know how it actually affects the output in something of an intuitive way.
10:40
Boxes touched: 69
Noice
I’m sorry, I’ll start learning now.
Nice
I was praying to see this in the comments. Thanks
Nice
lucky next guy will put the 69th like here
69th liker here
*N O I C E*
I was living in a coma until I found this channel.
you think thats bad? I was living in a comma,
@@ArchHeretic1 you think that's bad? I was having a srtkoe
Pretty cool, huh? : )
@@Ayy_la you thin thas beaa...
@@thekoldrex z
Hey, 3Blue1Brown. You're the sole reason I had a mathematical miracle after I got a 38% in 7th grade. You're the reason I saw the beauty in math, and I'm now studying extreme math, way above my level, for fun, not for school. I already know all the material needed for the exams now. Thanks for fueling the love of math in me.
note: I learned integrals
This was better than a 2-hour graduate lecture. Thank YOU!
What does the B in Benoit B Mandelbrot stand for? Benoit B Mandelbrot
thx m8. Was trying to remember that one.
Benoit Benoit Benoit Benoit Benoit . . . Benoit Benoit Benoit B Mandelbrot Mandelbrot Mandelbrot . . . Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot.
Balls
It stands for "Bacon".
@Minecraftster148790: Then his name is 1.226-dimensional. The string "Benoit B Mandelbrot" has a length of 19. The string "Benoit Benoit B Mandelbrot Mandelbrot" has a length of 37. So, log(37)/log(19) == 1.226.
Don’t mind me, I’m just procrastinating
And I'm popping
Pope
Avacoda
@@overlordcringe2715 h
Never gonna give you up, Never gonna let you down, Never gonna run around and desert you. Never gonna make you cry, Never gonna say goodbye, Never gonna tell a lie and hurt you.
"I mean, this is math. Everything's made up."
Duude, you just got yourself another subscriber.
1:03 I like that you used my home island as an example. I see that shape and instantly think "home"
Wait so is it called fractals because they are fractional
yass, so I also thought this towards the end of this video and I was like whoaaaa.
yes
@@fqidz YES
well duh
Not always fractional! Check out the video description: 'The proper definition of a fractal, at least as Mandelbrot wrote it, is a shape whose "Hausdorff dimension" is greater than its "topological dimension". '
Waiting for a calculus series
Vishwas Dubey The calculus is coming... if you support him on Patreon you'll get access to early drafts of a few of the videos
Ya I still watch the linear algebra series from time to time just because it's so good (especially the animations!)
I need a topology series.
There's a guy on CZcams that made a pretty good topology introductory video series called "What is a Manifold?". I think the channel name is XyXyXyX or something to that degree
He already has the Calculus series at Khanacademy.org. There's old one which is of sal khan and the new one is from the 3blue1Brown. I saw his Calc3 series and it was better put forward. I think he also did some of differential. here's the playlist: www.khanacademy.org/math/calculus-home/multivariable-calculus
I am doing DIP and this Video just helped me a lot to understand what to do in my assignment.
Thanks for such great content
This is so interesting. Thanks! I’m a student, and having things like this to supplement and make things more interesting and memorable is super helpful.
This voice and speaking speed is perfect to undertand complex topics...Good job!!!
can't totally agree, it makes me sleepy... and/or in fear of being hypnotized...
Given a random real dimension D, is there an easy way to find a fractal with that dimension?
Indeed! This clip should give you some clue: czcams.com/video/RU0wScIj36o/video.htmlm40s
The dimension of the Koch Snowflake approaches 2 as the theta in the seed approaches 0, right? So, depending on the theta, the dimension ranges from 1 to 2. Can this process be generalized for other dimensions? The seed of the Koch Snowflake is based on a line, but if there is a seed based on a plane could we alter one (or maybe more than one) attribute of that seed to obtain a curve that ranges from 2 to the 3 dimensions? I guess my question is that, given a seed of n dimensions, can one always obtain a curve that ranges from n to n + 1 dimensions?
It would be interesting to see an animation of a shape going from 1d to 3d
Yes, but it might take a while :P.
+Michael Wulber That reminds me to Topology. Maybe there is some sort of fractal dimension topology. Written from my honest ignorance. I'm no mathematician, just a fan
Wow. Thank you for putting into words this property of the universe that I’ve been trying to describe. Suddenly, it makes so much more sense now that you also showed how to quantize it.
I love this video. Thank you so much for taking the trouble to make it and share it to all.
3 dimensional: *shows square*
Him: “like we live in”
Me: ahh, yes. It’s Minecraft time
Fuck you
@@overlordcringe2715 Jeez, who hurt you?
Dude I had the best idea to incorporate these zoomed out fractal shapes as tower-bases from a bird's eye view. 100% gonna try it maybe on rust too
@@overlordcringe2715 Your username makes sense
Cube*
Read a book on fractals in 1992. Was fascinated. Understood it in 2019. What a great channel. CZcams rocks.
11:25 - "While I was editing this". You have so much dedication...
I thought you had a team of editors since your videos are so well thought out. You are like the Bob Ross of math...
Once again you have given me a deeper understanding for the Mathematics I’m studying, thank you very much you are a treasure big love ❤️
A level of breaking down complicated matter into understandable
chunks which is rarely seen on YT and one might have thought, wasn’t even
possible - but obviously it is. Thank you!
1:14 ... I heard "Fractals are a rebellion against capitalists." ... I was intrigued but knew I had to have heard wrong! haha.
Workers, unite! you have nothing to lose but your derivatives!
@@nevanmasterson46 its integral that we seize the means of production
Funny fact. Marx was a excellent mathematician that independentely to cauchy and weierstrass realized the way for give solids foundation to differential calculus even if marx didn't have the same knowledge of the two mathematicians. It was also the first economist to use math in massive way in economy.
@@DL-fb8jd *He was also
Lmao
This is mindblowing... Redefining dimensionality to be a scaling factor raised to the power of the number of dimensions and then realizing that that scaling factor doesn't have to be an integer at all..
There are so many things in math where you take a new perspective on something and then use that to find things that make sense in that new perspective, but translated back to the old perspective can look extremely weird (like a shape that lives in a fractional number of dimensions), and that is really beautiful and exciting. When you make analogies in normal speech, it breaks down very very quickly and is just used to illustrate a point, but in math, these analogies can be so deep that they can branch out into completely new areas of math, or give deep and valid insights into things you already knew about.
And 3Blue1Brown allows us to appreciate all this beauty :D
Great video, thanks. I had to teach myself all of this in the 1980s when part of my dissertation was on fractals. I had the privilege of talking to Mandelbrot.
I remember last year I tried to create a proof for the area of Sierpinski’s triangle, and showing it to my math teacher. Now I’m watching this video and realizing I didn’t come up with anything new lol. Amazing video though, keep up the amazing work :)
Don't underplay your achievements! What you did is still incredible and shows that you're a wonderful mathematician. Keep it up!
19:10 ... a numerical way to represent the fact that it's way more jaggedy ...
Lovely expression of the idea of roughness
One of the best mathematic channel ever. Keep it up!
A nice, clearly presented, concise video, thanks for posting.
This guy knows his stuff. And makes it very interesting :)
10:37 is when I found out that I'm not mature enough to be learning this stuff
I have a masters in electrical engineering, and I muttered 'nice' to myself.
Maturity is a lie.
@@RealClassixX L I E S
Yep you are 12
@@kerosene_turtle4715 Again, LOL, FuriousLightning!!!!
I know exactly what you are talking about without checking what happened at 10:37 first xD
I had to change the device I was watching this on, realize descriptions weren't visible there either, then pull this video up on a third device, all just to read the "more accurate definition" you directly referenced. Please consider this in how you structure your videos (which I love)
I've wondered what is meant by fractional dimensions for a long time. I've seen many definitions, but none had really helped much, and it's always been a concept I'd left on the back burner to delve into if the need arose. However, I stumbled upon this video and, based on my general appreciation of 3Blue1Brown videos, checked it out. HUGE jump in my understanding of the concept. Thank you. :-)
So if a tesseract is scaled down one half, the mass is scaled down 1/16th?
Yup!
Cool, Im trying to imagine it, but having a hard time with it ^^
I might say i found 'mass' a bit overloaded a term for this "measure" I'm thinking of words like: travel, visitation, flood, fill. I can spend ages searching thesaurus for names though.
It is the first time ive heard of this calculation and this video does describe it wonderfuly so mass worked out fine to go on.
A 3-color piece of a pocket cube (2x2x2) is 1/8 of the whole cube and a 4-color piece of a 2x2x2x2 is 1/16 of the whole hypercube (you can see it with Magic Cube 4D).
Let's see, the determinant is the scaling factor of the "mass" under transformation, so... OW! ... I think I hurt my head... and if a particle and antiparticle separated in minkowski space but entangled because in lateral dimensions they are still the same particle/wave... OW! OW! ... and if the "mass" is fractal as you scale the transformation... OW! OW! OW! ... the closer I look the fuzzier it gets... OWOWOWOW!
this is the first time I understand the fractional dimensions. thank you.
beautifully direct and to the point, great explanation and great visual examples
11:20 It's like circumscribing and inscribing shapes based on a set circle size and then measuring the area of each, the shape's area slowly approaches the area of the circle it is inscribed/circumscribed around.
If only math at school was this fun...
The school is needless
@@bttfish Yeah, this is what people usually say when they watch a video like this. But it is an illusion. First of all, if you look more closely at the video, you notice that it basically is a random repetition of the same few animated pictures over and over again. Nothing wrong with this per se. Probably it was hard work to create them, so using them several times to make up a video is cost efficient. But it means that the "math lesson" included in the video also somehow circulates around a few (superficial) topics over and over again. This tends to make the viewers believe they understood and learned something, but did they really?
For example, we saw the Sierpinski triangle so often, that we believe we know what it is. But do we? It is a set of points in the plane. After watching the video, could you tell which points? Given that two corners are at (0,0), (1,0), does the point (1/8,1/6) belong to the set? You should be able to answer this question if you claim to "know" what the Sierpinski triangle is, and you need to be able to tell if you want to reproduce anything shown in the video on our own computer. Same with the Koch curve.
It's like watching an orchestra playing a symphony in the background (actually the same excerpts over and over again) and listening to a musician explaining the compostion, the instruments, and how the condictor is managing everything to make it sound well, and then saying: "If only my violin lessons where so much fun."
Imagine if you had to come up with proofs of these ideas like Mandelbrot did, or even come up with new ideas, you'll need to go into a lot of tedious details that this won't cover.
Hu Go at least most educations in most secondary schools destroy the interest of most students.
Kalistic Modiani At least this gives an interesting introduction.
"Mathematicians are clearly making stuff up"
Well yeah... but no. It's complicated?
Well yes, but not really
Well yes, but actually no.
All words are made up, all letters are made up, all numbers are made up, every type of character you can think off is made up
@Maximal's Personal Profile Hmm... I don't particularly understand what you're approaching can you explain it in a sense of Deatil.
@Maximal's Personal Profile Yes, I can clearly see it. But THEY don't accept it because they think they are universal which is so stupid and misleading like GOD we humans created the existence of GOD to explain something that cannot be explained Questions That has no answer as of right now we call them MIRACLES but they're just a bunch of in-adequate MISUNDERSTANDINGS. But THEY just don't accept it not because it can change their perspective just because they BELIEVE it's not true and that's an OPINION not an answer. Which is why debating with these kinds of idiots are essentially WASTING time for yourself. I can agree with you That Math is just a rule that we humans made up to describe things and explain it more better but because it explains things PEOPLE think it's universal but it's NOT it!
Going from Vi Hart's Dragon Curve videos to this feels like growing up, I love it.
I have watched this at least 3 times now. It seems I watch it yearly... I don't know why but this specific idea fascinates me
Serious math question: In physics the idea of dimension is usually expressed as the number of degrees of freedom needed to describe the movement of a particle. Is there a sense in which a particle moving in a fractal has a non integer number of degrees of freedom?
This is exactly what bothers me
+
This only really works for vectorspaces (and in a weaker version for manifolds)
Is there any motion at all along a fractal trajectory? The distance between any pair of points on that trajectory is infinite, so it seems that motion cannot be defined.
How about an idea of motion through the iterations. Like, take a line and iterate it to look like one side of the Koch snowflake. Start at one end, Iterate once, then take one step along the line to where it bends (at 1/3 the length). Then iterate, and take a step, etc. You'd go 1/3, then 1/9, then 1/27 length of the original line....
With this idea of motion I guess there are two degrees of freedom of motion, since you could also go backwards (except for the first step).
If we imagine that instead of staying on the line, you could also jump to nearby parts of the snowflake, there might be jump distances that would give an average degrees of freedom that is not an integer.
Maybe that could be done more simply, what if a particle sits on a random spot on a y shape and can move along lines to intersections or end points. If it's on an ends it has one choice of motion. If it's in the middle it has 3 choices. So on average it would have 6/4 degrees of freedom?
i never knew that i had such a love and interest in math until i found this channel. I also noticed that fractals are beautiful
The relation between fractional dimension of an object and its artificialness was really amazing.
Incredible presentation. I wish I had have these resources when I was at school
Yesterday I was living in 3D space. You have changed my life.
DODO
You still live in 3D space
If you lived in 3D yesterday, and you posted this a year ago
HOW DID YOU TIME TRAVEL
@@piggywink333boyfriend6 you know he's living in 4 dimensions now, the fourth is time
@@pe3akpe3et99 Thanks
I've been following this channel for *roughly* 2 years now, awesome content as usual
***** 1.1 dimensional
The most mind-capturing video I've ever watched.
Amazing!
Imagine a Sierpinski Pyramid. It will break apart into 4 copies of itself, meaning a 1/2 length scale translates to a 1/4 mass scale. Since 1/4 = (1/2)^2, a Sierpinski Pyramid is 2-dimensional, yet a pyramid is 3-dimensional. ?yo what
In the same way that a Sierpinski triangle is represented as a 2-dimensional drawing (which is bigger than its own dimension), the Sierpinski pyramid you've come up with _is_ 2-dimensional: the 3-dimensional shape is just a representation of it.
Read the description ;)
Thank you! You've explained clearly a difficult concept! I'm happy I discovered the world of fractal. Subscribed!
Wow, might digest this many times, before my small brain will connect everything together, but thank you for giving education in such intricate and beautiful things!🦾🧠
Awesome and inspiring! It opens a new dimension to my mind.
It's honestly crazy how videos like this manage to take someone like me, who wasn't even good enough to pass algebra 2 in high school, super interested in higher level math
10:38
Oh yes, it's all coming together.
Who doesnt like 69 doesnt like memes...
And guess what? I dont like 69
And I go Crazy
And Crazy again
And Crazy YET again
nice
thank you 3b1b, you just made my life so much better!
I know this video is old but I’m majoring in bio and got into a class in biomathematics with a professor specialized in fractals in nature. His name is Pedro Miramontes and has lots of papers towards this subject that forever fascinates me. He recently talked about the genome sequence and how it relates to fractals. Basically u can form a siepinski triangle in R3 with binary and some XOR with genomes
Yo dawg, I heard you like Triforces.
triforce-ception
So I put 9 triforces
In a stupid WiiU game
so i made it and infiniteforce
Haven't seen that meme in a while lol
so i put a triforce in your triforce so you can use a triforce with your triforce
" I mean this is math. Everything's made up. "
Now that I think about it. Well said ...
Did you just give an intuitive way to understand Haussdorf dimension and then a fairly working definition, all under 20 minutes? Teach me pliss, Senpai.
Also, I'm really interested in Lie algebras and representation theory, and I'm mentioning this because your videos explaining the exponential map are really my favourite. That really presents it in a way that easily generalizes to the exponential map from a Lie algebra to a corresponding Lie group.
You rox, and so does your team and other contributors ♥️.
19:29 This is a nice way of generalising what I was thinking when I saw the monolith for the first time in 2001: A space odyssey; it was too smooth to be something natural.
3Blue1Brown, I don’t know if you read comments or not, but ever since seeing a 3D projection of a tesseract, I have long wondered what a 3D projection of a 4D Menger Sponge might look like. I can’t think of anything valuable it may have to teach, but maybe there are some other (self similar) fractals that might have interesting or insightful expressions in 3D projections of 4D space. I hope you find the idea as interesting as I do. Whether you decide to use the idea or not, thank you for content that expands the way I think.
For some projection it should look exactly like default Menger sponge. Other projections should look like two Menger sponges intersecting and merging with each other in some way. I'm not 100% sure about it though.
In some projections it will look like a normal menger sponge. In the most extreme case where all dimensions are twisted by 45 degrees to the camera, the projection would have 3-dimensional 6-pointed stars as holes, instead of cubes. Anything in between, I can't imagine.
Maybe I'll be able to program a 4d menger-sponge viewer. It would be an interesting project actually
www.google.com/url?sa=t&source=web&rct=j&url=%23&ved=2ahUKEwiSgd79-tjnAhXnyjgGHcFrCEkQwqsBMAB6BAgHEAQ&usg=AOvVaw0VTJzWxrN8ZFOD4xbU2nov
I would love a video on the Fourier series and transformation. Your animations would make it look so beautiful and intuitive!
When I initially heard about fractals, I was told that a fractal is simply a figure/shape with infinite perimeter
Any object with dimension other than exactly 2 doesn't have a perimeter.
@@minuspi8372 yes it does.
For example, the perimeter of a line is just the length of the line, and the perimeter of a cube is just the surface area of a cube
@@JJean64 generalized perimeter :p
the most beautiful and comprehensive explanation ever
This makes so much sense now! I put A LOT of incorrect information of my 7th grade math fair 2 years ago
Kind of interresting, also one sentence i never thought i'd say:
Luckily i speak enough math to be able to understand this! At this point, with all the shapes, concepts and formulas, math is less of a pure concept but more of a language, if not even a subculture on its own already (my english isn't the best so please excuse me if i can't fully express this thought understandably with the wirds i chose), also you got to learn how to speak math to fully understand it, likewise with another language, basic math is like the first few words, cat, dog, hello, thank you ..., enough to describe basic things, needs and so on and on. But then you get sentences with their rules, placement of words and you can describe things in nature, formulars in math as the counterpart. Graphs are just a way to write it down, (you speak a sentence, if you write it down you can see every piece of it from the beginning to its end and how it conected, like the graph of a forlular) also writing is also like a language of its own, funny isn't it ? Add sentences to sentences and weave them together and you get a speech (mayby not a good word at this place, i told ya, my english isn't good enough to give it enouch credit, but it should be good enouch for the gist) that can deliver a greater meaning that non cohering sentences on their own.
Some people say: math is the language god has written the universe in.
I say: math is just a translation of the universe xD
Now i understand it, thx for that mental breakthrough.
Oh and something that i found out abbout my comment if i start thinking of it, isn't it just the same as one of those shapes above ? Things consisting of itself ? Oh no then it would be just a that a word would consist out of words, WAIT THEY DO! and even if you take another approach, like not just words consisting out of smaller words and sentences out of smaller ones, also words consist out of letters that are just a fraction of the word itself ! Would be interresting to see what the dimention of that would be, or just the struggle of mathematicians trying to calculate it XD.
Does this even make sence annymore or do i just get insane ? Or do i just start talking in another language ?
Got to check out if vsauce got a video abbout the concept of language.
@Yu-Chen Chang i will surely have one as my longer comments rarely get anny attention normally and it is nice to see that 6 people got interested and one even wrote back (although i expected to find a less good will comment as i read the first few lines in my newsfeed, as i do write some lazy comments when im not in the mood for correcting all of my translation mistakes xD), normally people tend to skip long comments as those look too intimidating for most people when they arent properly organized with gaps etc.
I hope you got a nice day too !
bRuH that's what i try to tell everybody! math is the language of the universe! I wrote it for my math document/paper that i have to do in 12th grade, i'm doing it on the Chaos Theory and fractals so that's why i'm here, lol. But math is definitely a language. Like, an easy example would be translating the english description of speed into speed = distance / time. This usually convinces them lol.
It’s such a nice and detailed video.❤️i just loved it.this video is too visionary.
Beautiful description. Well done!
That Markus Persson is one of your Patrons? Notch?
It's "that" Markus Persson indeed :)
who is Markus Persson?
Mi Les Notch! The creator of minecraft!
Yes its confirmed that Notch supports 3b1b
notch
Why we see fractals when we are on lysergic acid? What is the relationship between biochemistry, human perception and fractal geometry?
There are actually some very complex mathematics descriptions of such fractals and shapes and their relationship to vision, especially regarding the usage of LSD and marijuana. If you look up the paper, "Geometric visual hallucinations, Euclidean symmetry and the functional architecture of the striate cortex" by Paul C. Bressloff, Jack D. Cowan, and others, published by the Royal Society. I would love to see 3Blue1Brown do some sort of video simplifying the concepts for us laypeople, though the topic as it is described in the original paper is still very interesting.
I think it's really funny that someone can basically ask "Why do I see weird big shapes when I'm high", and people can mathematically and scientifically answer that question without joking about it.
It's questions like these that we actually want to answer. They provide real insight into our world.
My take: because our visual apparatuses are fractals as well.
Damn. I never would have guessed there was a specific word for the geometric shapes that appeared when I pressed my eyelids when I was a kid (wikipedia: "form constant").
I never knew how one got the dimension of a fractal until this video. Thank you.
1:19 Mandelbrot saw this as "overly idealized"
dauym mandelbrot was a savage dude!
When you suggested programming a way to calculate the fractal dimensión, I was actually terrified.