Fractals are typically not self-similar
Переглядів 3,904,110
День томуAn explanation of fractal dimension.
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An equally valuable form of support is to simply share some of the videos.
Special thanks to these supporters: 3b1b.co/fractals-thanks
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Home page: www.3blue1brown.com/
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.
Music by Vince Rubinetti: / riemann-zeta-function
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
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@ryanmarita-davis3339 Рік тому
"In some ways.. fractal geometry is a rebellion against calculus." That's just a beautiful statement.
@brandongroves4465 11 місяців тому
There is no such thing as rebellion in math.
@brandongroves4465 11 місяців тому
Also I agree
@matthewfarrell6822 9 місяців тому
You know I think I like fractals
@bitonic589 6 місяців тому
@@brandongroves4465 I truly do enjoy dividing zero by zero.
@andistansbury4366 4 місяці тому
Yea! Screw calculus!
@matheusnever1plasmaman477 4 роки тому
"Ah,Yes,The fractal here is made out of fractal"
@ireallyneedalife6979 4 роки тому
*Ah yes, enslaved fractal*
@ntck 4 роки тому
*ah* *yes,* *enslaved* *infinity*
@GlitchedBlox 4 роки тому
Ah yes, Enslaved object
@1994mrmysteryman 4 роки тому
@@tthung8668 Get at it. Unless you "ate" it.
@thegoofyarchive8300 4 роки тому
10:39 *n i c e*
@Shubham_pandey-nk1un 3 роки тому
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
@Poyni 3 роки тому
I immediately dismissed this thought as complete tomfoolery
@MrMegaMetroid 3 роки тому
Well you still need to differentiate spacial dimensions and fractal dimensions. Those are two completely unrelated concepts only sharing the same linguistic designation.
@Shubham_pandey-nk1un 3 роки тому
@@MrMegaMetroid Yes, You are right
@archangelofsorrow 2 роки тому
@@MrMegaMetroid so there are three versions of dimension
@MrMegaMetroid 2 роки тому
@@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
@huhneat1076 3 роки тому
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
@Dracomandriuthus 2 роки тому
And that's what dimensions are all about
@AlexanderGieg 2 роки тому
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.
@eliasmochan 2 роки тому
@@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.
@bettercalldelta 9 місяців тому
never thought of it this way, you blew my mind
@upanshulakhani6221 21 день тому
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)
@Tomyb15 7 років тому
This channel is youtube gold.
@fossilfighters101 7 років тому
+
@eldiagrama 7 років тому
+
@umamaheshwaranl8554 7 років тому
3blue1brown:maths::pbsspacetime:physics
@vampyricon7026 7 років тому
+
@xshortguy 7 років тому
I even enjoy listening to the ad at the end of the video.
@grande1900 4 роки тому
Ah, yes, my favorite fractal! *The Coastline of Britain*
@TheAustronaut03 4 роки тому
the coastline of Switzerland is 0-dimensional
@killerfishe5092 4 роки тому
@@TheAustronaut03 yes, that is technically true
@TheAustronaut03 4 роки тому
wahoo
@bobfs9855 4 роки тому
The coastline of Norway is far superior.
@trueriver1950 4 роки тому
@@TheAustronaut03 Unless you include the coast of Lake Geneva, of course.
@jeanmariegrangon 3 роки тому
As a physics graduate, I wish that our teacher had shown us this video when he tried to teach us about fractal dimension.
@profbbfab6211 Рік тому
Our teacher's assistant did! And I thank him for that
@NoahSpurrier 2 роки тому
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.
@ultimatedeatrix9149 Рік тому
Damn that's a neat piece of fact on its application!
@danyilbutsenko6339 Рік тому
"Paintings"
@shobacon8263 Рік тому
How did they do that
@CalebSalstrom Рік тому
@@danyilbutsenko6339 there’s always one of you
@CarlNiemi Рік тому
@@CalebSalstrom Always one correct person?
@Nevermind445 6 років тому
"This is math, everything is made up" Love this quote!!
@nathanwagester6665 4 роки тому
read philosophy
@HUGOGARCAO 4 роки тому
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?”
@dataexpunged3914 4 роки тому
How to make every math teacher very angry and how to claim 42 as a trial answer to everything.
@dataexpunged3914 4 роки тому
@@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
@antoy384 4 роки тому
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
@karthikprabhu3173 4 роки тому
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
@samwang6515 3 роки тому
Please create a merch line with "THIS IS MATH, EVERYTHING IS MADE UP"
@dangerouspie0319 3 роки тому
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.
@stratowhammy Рік тому
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.
@JuanLeon-oe6xe Рік тому
@@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.
@mohammedjafer9265 Рік тому
@@dangerouspie0319 education system is a play ground for control did you really expect the benefits to outweigh their agenda...
@tokiWren 3 роки тому
Fun fact: "Mandelbrot" translates to "Almond bread" from German.
@otesunki 3 роки тому
Is that profile pic... from nick?
@StarGarnet03 3 роки тому
The almond bread set
@therandomshow1265 3 роки тому
I can see it
@The_Tiffster 3 роки тому
That's just the name of the person who discovered it....
@Tower_Swagman 3 роки тому
yay deustechland (sorry germans if i mispronounced or mispelled it)
@lijacom 4 роки тому
Car salesman: (slaps sandpaper ) "This bad boy has a really high fractional dimention"
@dangerouspie0319 3 роки тому
But does it have a high frictional dimension?
@Green24152 3 роки тому
@@dangerouspie0319 Yes.
@fetterkeks2796 2 роки тому
(slaps Mandelbrot set): "This fractal can fit so many fractals in it"
@mrkun5905 2 місяці тому
Nahhh
@karthikprabhu3173 4 роки тому
Please create a merch line with "THIS IS MATH, EVERYTHING IS MADE UP"
@vale3242 3 роки тому
I need it.
@papasscooperiaworker3649 3 роки тому
LOL
@mrkun5905 2 місяці тому
@@papasscooperiaworker3649 DUDE
@KasabianFan44 7 років тому
What does the B stand for in Benoit B. Mandelbrot? Benoit B. Mandelbrot.
@redoxred5588 6 років тому
KasabianFan44 recursive acronym
@eednb4257 6 років тому
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.
@alexkeil3445 6 років тому
This is the most brilliant pun i have ever seen
@blunderbus2695 6 років тому
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...
@eyemotif 6 років тому
A “Mandelbrot” set...
@wigglygrass3066 2 роки тому
10:42 thought you were gonna say" in the back of our minds, we know to say nice"
@illusionist1872 3 роки тому
10:34 When you said "In the back of our minds," I thought you were going to make fun of how immature everyone is
@mitchg9017 5 років тому
when i first heard it, i thought he said "In some ways, fractal geometry is a rebellion against capitalism"
@yaiirable 4 роки тому
had to switch on subtitles to hear it: it was actually 'calculus'
@gold_5600 4 роки тому
A rebellion against capitalism, huh?
@raddrew42 4 роки тому
What part?
@dcterr1 4 роки тому
I'm glad my salary isn't a fractal!
@gamermapper 4 роки тому
☭ ☭ ☭ ☭ ☭ ☭ ☭ ☭ ☭ ☭ Unbreakable 🚫⛏️ Union 🔗 of Freeborn 🗽 Republics 🐘 Great 💪 Russia 🇷🇺 has welded 🤝 forever ⌚ to stand 👍. Created 🏗️ in struggle 🌋 by will of the people 🙋, 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...)
@George4943 7 років тому
Your videos remind me why I was a math major until I wrote my first program in 1962.
@TheRedstoneTaco 7 років тому
roasted.
@bigbox8992 7 років тому
What language did you use?
@George4943 7 років тому
FORGO (a version of FORTRAN)
@ricardo.mazeto 7 років тому
I'd like to hear you explain why or just talk more about that. I'm C programmer.
@bigbox8992 7 років тому
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?
@JustMoseyinAround 3 роки тому
10:48 *"In the back of our min-... Nice"*
@ConnoisseurOfExistence Рік тому
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?
@raphdm3776 Рік тому
you can't represent a fractal with a dimension over 3 in real world
@lakshya5946 Рік тому
@@raphdm3776 This is math Everything is and can be made up
@raphdm3776 Рік тому
@@lakshya5946 I know but we will never see what it looks like
@lakshya5946 Рік тому
@@raphdm3776 we can actually
@raphdm3776 Рік тому
@@lakshya5946 I don't know about you but I only live in 3 dimensions
@vishwas425 7 років тому
Waiting for a calculus series
@Ramzuiv 7 років тому
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
@Euquila 7 років тому
Ya I still watch the linear algebra series from time to time just because it's so good (especially the animations!)
@justinward3679 7 років тому
I need a topology series.
@treyshaffer 7 років тому
There's a guy on UKposts 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
@rikenm 7 років тому
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
@BinyaminTsadikBenMalka 7 років тому
This channel is a UKposts 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.
@arongil 7 років тому
It is true: he is a gem. Hopefully he can continue to gain more viewers.
@SkyWKing 7 років тому
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.
@Isilduhh 7 років тому
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!
@vampyricon7026 7 років тому
I feel like this is for math what PBS Spacetime is for physics.
@BinyaminTsadikBenMalka 7 років тому
Vampyricon I like PBS Spacetime too
@shreeyamittal1771 3 роки тому
"I mean, this is math. Everything's made up." Duude, you just got yourself another subscriber.
@YBY3 2 роки тому
10:30 - Nice
@meisnameless Рік тому
Nice
@krisoluich9119 5 років тому
I was living in a coma until I found this channel.
@ArchHeretic1 5 років тому
you think thats bad? I was living in a comma,
@Ayy_la 5 років тому
@@ArchHeretic1 you think that's bad? I was having a srtkoe
@ericjohnson1811 5 років тому
Pretty cool, huh? : )
@thekoldrex 5 років тому
@@Ayy_la you thin thas beaa...
@Xishnik94 5 років тому
@@thekoldrex z
@Minecraftster148790 7 років тому
What does the B in Benoit B Mandelbrot stand for? Benoit B Mandelbrot
@ilikemitchhedberg 7 років тому
thx m8. Was trying to remember that one.
@maxbuskirk5302 7 років тому
Benoit Benoit Benoit Benoit Benoit . . . Benoit Benoit Benoit B Mandelbrot Mandelbrot Mandelbrot . . . Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot.
@pranavgarg9075 7 років тому
Balls
@thomassynths 7 років тому
It stands for "Bacon".
@ntofed 7 років тому
@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.
@minhsieuquay8164 Рік тому
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
@minhsieuquay8164 Рік тому
why did no one see this comment ._.
@NhuKhiet-cr8lh 11 місяців тому
Do you mean the area of the new circle = π(r/2)²=π*r²/4 = 1/4 old area?
@ThanhNguyen-ni4vw 11 місяців тому
That’s great!
@HuyenVu-di3et 11 місяців тому
It seems that the author wants to use visualization to demonstrate fractals, he doesn't want to use mathematical formulas
@minhsieuquay8164 11 місяців тому
@@NhuKhiet-cr8lh oh yea
@actuallyasriel 2 роки тому
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.
@mediocreicerinkparodies1099 5 років тому
"A line, a square, a cube..." "And a Sierpinski triangle"
@hebpennington 5 років тому
"A line, a square, a cube, wand a Sierpinski triangle walk into a bar."
@ferencgazdag1406 4 роки тому
The 4th should have been a hypercube.
@squibble311 4 роки тому
and a tesseract
@sketin 3 роки тому
THE ARISTOCRATS!
@zmoot 3 роки тому
And 4d cube.
@karstenroelofs9216 4 роки тому
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!
@tygodankers6526 4 роки тому
jup, had the same thought
@Dhanush-zj7mf 3 роки тому
Right.....
@hasanirtija8996 3 роки тому
This was better than a 2-hour graduate lecture. Thank YOU!
@Cruveit 2 місяці тому
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.
@Cruveit Місяць тому
note: I learned integrals
@1ucasvb 7 років тому
Given a random real dimension D, is there an easy way to find a fractal with that dimension?
@3blue1brown 7 років тому
Indeed! This clip should give you some clue: ukposts.info/have/v-deo/ioVhq4magKBkl6M.htmlm40s
@michaelwulber8229 7 років тому
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?
@dev02ify 7 років тому
It would be interesting to see an animation of a shape going from 1d to 3d
@NathanTAK 7 років тому
Yes, but it might take a while :P.
@1Thor61storm8 7 років тому
+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
@IanZainea1990 4 роки тому
1:14 ... I heard "Fractals are a rebellion against capitalists." ... I was intrigued but knew I had to have heard wrong! haha.
@nevanmasterson46 4 роки тому
Workers, unite! you have nothing to lose but your derivatives!
@jackdesy2127 4 роки тому
@@nevanmasterson46 its integral that we seize the means of production
@DL-fb8jd 3 роки тому
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.
@segmentsAndCurves 3 роки тому
@@DL-fb8jd *He was also
@monika.alt197 2 роки тому
Lmao
@shantonudutta9726 2 роки тому
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!
@spacekid9680 Рік тому
1:03 I like that you used my home island as an example. I see that shape and instantly think "home"
@jaygreenwood422 4 роки тому
Don’t mind me, I’m just procrastinating
@overlordcringe2715 4 роки тому
And I'm popping
@overlordcringe2715 4 роки тому
Pope
@overlordcringe2715 4 роки тому
Avacoda
@h-hhh 4 роки тому
@@overlordcringe2715 h
@theaccounter 3 роки тому
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.
@samshrivastava2655 4 роки тому
Read a book on fractals in 1992. Was fascinated. Understood it in 2019. What a great channel. UKposts rocks.
@yesssint7243 2 роки тому
When I initially heard about fractals, I was told that a fractal is simply a figure/shape with infinite perimeter
@minuspi8372 Рік тому
Any object with dimension other than exactly 2 doesn't have a perimeter.
@JJean64 Рік тому
@@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
@henk7747 Рік тому
@@JJean64 generalized perimeter :p
@elnico5623 2 роки тому
So....... is a sierpinski tetrahedron... 2d? Even tho it lives in space?
@Gadget622 7 років тому
10:37 is when I found out that I'm not mature enough to be learning this stuff
@RealClassixX 5 років тому
I have a masters in electrical engineering, and I muttered 'nice' to myself. Maturity is a lie.
@hesiod_delta9209 5 років тому
@@RealClassixX L I E S
@kerosene_turtle4715 5 років тому
Yep you are 12
@moadot720 5 років тому
@@kerosene_turtle4715 Again, LOL, FuriousLightning!!!!
@demerion 5 років тому
I know exactly what you are talking about without checking what happened at 10:37 first xD
@alek6362 4 роки тому
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
@fabiopilnik827 4 роки тому
I independently invented certain finite difference methods but I was too old to think they were new.
@GoogleAccount-pi9ct 4 роки тому
Lol
@wisdom6458 4 роки тому
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 :)
@sebastianjost 4 роки тому
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.
@dipendraphuyal8914 4 роки тому
😂
@kaitsune2262 2 роки тому
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.
@louisbrill891 10 місяців тому
I love fractal videos solely to see how low the creator can drive the bitrate
@tomimated1638 5 років тому
Wait so is it called fractals because they are fractional
@CoolColourBlack 5 років тому
yass, so I also thought this towards the end of this video and I was like whoaaaa.
@fqidz 5 років тому
yes
@thekoldrex 5 років тому
@@fqidz YES
@dp271 5 років тому
well duh
@Julian-tf8nj 5 років тому
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". '
@tallyhallsally 4 роки тому
3 dimensional: *shows square* Him: “like we live in” Me: ahh, yes. It’s Minecraft time
@overlordcringe2715 4 роки тому
Fuck you
@bin6549 4 роки тому
@@overlordcringe2715 Jeez, who hurt you?
@kingaha3657 4 роки тому
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
@theshermantanker7043 4 роки тому
@@overlordcringe2715 Your username makes sense
@real_nosferatu 4 роки тому
Cube*
@harrycarpio 2 роки тому
I love this video. Thank you so much for taking the trouble to make it and share it to all.
@akarshshrivastava3719 3 роки тому
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
@wiltherdelacuesta8175 5 років тому
This voice and speaking speed is perfect to undertand complex topics...Good job!!!
@M-F-H 4 роки тому
can't totally agree, it makes me sleepy... and/or in fear of being hypnotized...
@tcarrotgaming1639 7 років тому
If only math at school was this fun...
@bttfish 5 років тому
The school is needless
@hugo3222 5 років тому
@@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."
@kalisticmodiani2613 5 років тому
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.
@bttfish 5 років тому
Hu Go at least most educations in most secondary schools destroy the interest of most students.
@bttfish 5 років тому
Kalistic Modiani At least this gives an interesting introduction.
@deadchannel4619 3 роки тому
**slaps roof of fractal** This bad boy can fit so much fractal in it.
@HerbalEngineering 3 роки тому
Other valid way to end the joke is "This bad boy can fit so much perimeter in it
@smokey04200420 3 роки тому
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.
@seppobastian 7 років тому
This guy knows his stuff. And makes it very interesting :)
@AwesomeCreeperBD 4 роки тому
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 :)
@AshrZ Рік тому
Don't underplay your achievements! What you did is still incredible and shows that you're a wonderful mathematician. Keep it up!
@xDMrGarrison 3 роки тому
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
@squiji9750 Рік тому
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...
@yulongqiu 5 років тому
this is the first time I understand the fractional dimensions. thank you.
@pablorepetto2759 4 роки тому
"Mathematicians are clearly making stuff up" Well yeah... but no. It's complicated?
@piggywink333boyfriend6 3 роки тому
Well yes, but not really
@mlgproplayer2915 3 роки тому
Well yes, but actually no.
@arthurthekyogre9155 3 роки тому
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
@Trix-Valrae 3 роки тому
@Maximal's Personal Profile Hmm... I don't particularly understand what you're approaching can you explain it in a sense of Deatil.
@Trix-Valrae 3 роки тому
@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!
@cassiopeiasfire6457 Рік тому
Going from Vi Hart's Dragon Curve videos to this feels like growing up, I love it.
@esisimp123456 Рік тому
The relation between fractional dimension of an object and its artificialness was really amazing.
@sogidochnet9304 4 роки тому
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!
@grevel1376 5 років тому
Yesterday I was living in 3D space. You have changed my life.
@alexwang982 5 років тому
DODO You still live in 3D space
@piggywink333boyfriend6 3 роки тому
If you lived in 3D yesterday, and you posted this a year ago HOW DID YOU TIME TRAVEL
@pe3akpe3et99 3 роки тому
@@piggywink333boyfriend6 you know he's living in 4 dimensions now, the fourth is time
@piggywink333boyfriend6 3 роки тому
@@pe3akpe3et99 Thanks
@CMDRunematti 11 місяців тому
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
@knightlypoleaxe2501 3 роки тому
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.
@willemvandebeek 7 років тому
So if a tesseract is scaled down one half, the mass is scaled down 1/16th?
@3blue1brown 7 років тому
Yup!
@willemvandebeek 7 років тому
Cool, Im trying to imagine it, but having a hard time with it ^^
@adymode 7 років тому
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.
@xxnotmuchxx 7 років тому
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).
@hisxmark 7 років тому
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!
@mushyomens6885 4 роки тому
" I mean this is math. Everything's made up. " Now that I think about it. Well said ...
@edwardlulofs444 9 місяців тому
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.
@pauln1557 2 роки тому
A nice, clearly presented, concise video, thanks for posting.
@christianthompson7915 4 роки тому
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
@matthieudeloget8998 4 роки тому
10:38 Oh yes, it's all coming together.
@dartmarbleracing1762 3 роки тому
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
@asriel522 2 роки тому
nice
@lukewaite9144 3 роки тому
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 ❤️
@mastershooter64 3 роки тому
1:19 Mandelbrot saw this as "overly idealized" dauym mandelbrot was a savage dude!
@trueriver1950 4 роки тому
19:10 ... a numerical way to represent the fact that it's way more jaggedy ... Lovely expression of the idea of roughness
@nickshowman4606 4 роки тому
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
@valium97582 3 роки тому
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.
@uklu 2 роки тому
Read the description ;)
@ryanpost13 4 роки тому
13:21 he missed a bit of coast line, towards the bottom left
@cycrothelargeplanet 2 роки тому
2:39 A tesseract is 4D A penteract is 5D A hexeract is 6D A hepteract is 7D An octeract is 8D An enneract is 9D A dekeract is 10D
@user-wz9sc2vr9q 7 місяців тому
☠️
@OrchidAlloy 4 роки тому
When you suggested programming a way to calculate the fractal dimensión, I was actually terrified.
@itaiefrat3286 7 років тому
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?
@ori5021 7 років тому
This is exactly what bothers me
@fossilfighters101 7 років тому
+
@zairaner1489 7 років тому
This only really works for vectorspaces (and in a weaker version for manifolds)
@SkyWKing 7 років тому
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.
@nmarbletoe8210 7 років тому
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?
@back2d_lobby Рік тому
Incredible presentation. I wish I had have these resources when I was at school
@helphelphelphelpmehelphelphelp 8 днів тому
Thank you, I need to write an assignment about fractals WITH NO PRIOR INTRODUCTION TO THEM, and this video was a huge help!
@rileydragunas9112 4 роки тому
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
@RyuichiNoGekido 4 роки тому
10:40 Boxes touched: 69 Noice I’m sorry, I’ll start learning now.
@Bass_Guy 4 роки тому
Nice
@nadari9162 4 роки тому
I was praying to see this in the comments. Thanks
@hypertriangle6531 4 роки тому
Nice
@luigivercotti6410 4 роки тому
lucky next guy will put the 69th like here
@nnmk01 4 роки тому
69th liker here *N O I C E*
@jrjr1313jrjr 2 роки тому
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. :-)
@crazierchimp 3 роки тому
"This is math, everything's made up." has to be one of my favorite quotes.
@YounesLayachi 7 років тому
I've been following this channel for *roughly* 2 years now, awesome content as usual
@YounesLayachi 7 років тому
***** 1.1 dimensional
@pikminfan6778 5 років тому
Yo dawg, I heard you like Triforces.
@haslan4885 5 років тому
triforce-ception
@purrplaysLE 5 років тому
So I put 9 triforces In a stupid WiiU game
@goldsrcorsource2551 5 років тому
so i made it and infiniteforce
@skoto8219 5 років тому
Haven't seen that meme in a while lol
@killerfishe5092 4 роки тому
so i put a triforce in your triforce so you can use a triforce with your triforce
@raghavherugu6934 4 роки тому
One of the best mathematic channel ever. Keep it up!
@PronteCo 2 роки тому
beautifully direct and to the point, great explanation and great visual examples
@dudeman3981 7 років тому
I would love a video on the Fourier series and transformation. Your animations would make it look so beautiful and intuitive!
@Em.P14 4 роки тому
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.
@Em.P14 4 роки тому
@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 !
@adir6094 4 роки тому
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.
@lilywater3683 2 роки тому
Wait. If you have a shape so that if you scale it down, it’s mass increases, does that mean it exists in negative spatial dimension! That would be so cool!
@artyommoxid6233 3 роки тому
The golden rule of maths: Everything is made up. The question is - is it useful?
@mohammedjafer9265 Рік тому
Barely any is really useful yea the basics to create project's but the word's of display is senseless in terms of reality cuz we do not know the reality of everything
@eggsoup6296 4 роки тому
My IQ is increasing exponentially because of this video.
@AnindyaMahajan 4 роки тому
1^x = 1 for all x jk lmao
@kartikaalst7354 4 роки тому
@@AnindyaMahajan not if x is a complex number it isn't
@kartikaalst7354 4 роки тому
You aren't looking at the full picture, 1 is also expressed as exp(i 2 Pi) = 1 Exp((i 2 Pi) *x) = only 1 if x is a whole real number
@brendarojas5613 4 роки тому
@@georgehajnal2723 I think he means crystallized intelligence.
@moodleblitz 3 роки тому
And I'm not so sure about mine anymore...
@federicovolpe3389 6 років тому
Thank you! You've explained clearly a difficult concept! I'm happy I discovered the world of fractal. Subscribed!
@mitalisharma440 3 роки тому
the most beautiful and comprehensive explanation ever
@zolvaring9503 2 роки тому
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)
@akselai 4 роки тому
'A dimension is a dimension, you can't say it's a half!' - TJ """Hausdorff""" Yoshi
@Oroborus12 5 років тому
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.
@user-xh9pu2wj6b 5 років тому
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.
@karolakkolo123 5 років тому
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
@karthikprabhu3173 4 роки тому
www.google.com/url?sa=t&source=web&rct=j&url=%23&ved=2ahUKEwiSgd79-tjnAhXnyjgGHcFrCEkQwqsBMAB6BAgHEAQ&usg=AOvVaw0VTJzWxrN8ZFOD4xbU2nov
@nefarioustoast 2 роки тому
17:12 so at the point where you'd be zooming in atomically, would the limit of the slope just be 1? so the points at the very very far right would form a y=x graph? but we still only consider the slope=1.21 part of the graph given the context that we don't inspect coastlines atomically, but rather from space/birds eye height?
@lunarknightess7056 7 місяців тому
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
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