Fractals are typically not self-similar

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An explanation of fractal dimension.
Help fund future projects: / 3blue1brown
An equally valuable form of support is to simply share some of the videos.
Special thanks to these supporters: 3b1b.co/fractals-thanks
And by Affirm: www.affirm.com/careers
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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Опубликовано:

19 май 2024

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Комментарии : 4,7 тыс.

@matheusnever1plasmaman477 4 года назад

"Ah,Yes,The fractal here is made out of fractal"

@ireallyneedalife6979 4 года назад

*Ah yes, enslaved fractal*

@ntck 4 года назад

*ah* *yes,* *enslaved* *infinity*

@GlitchedBlox 4 года назад

Ah yes, Enslaved object

@1994mrmysteryman 4 года назад

@@tthung8668 Get at it. Unless you "ate" it.

@thegoofyarchive8300 4 года назад

10:39 *n i c e*

@ryanmarita-davis3339 Год назад

"In some ways.. fractal geometry is a rebellion against calculus." That's just a beautiful statement.

@brandongroves4465 Год назад

There is no such thing as rebellion in math.

@brandongroves4465 Год назад

Also I agree

@matthewfarrell6822 9 месяцев назад

You know I think I like fractals

@bitonic589 7 месяцев назад

@@brandongroves4465 I truly do enjoy dividing zero by zero.

@andistansbury4366 4 месяца назад

Yea! Screw calculus!

@tokiWren 3 года назад

Fun fact: "Mandelbrot" translates to "Almond bread" from German.

@otesunki 3 года назад

Is that profile pic... from nick?

@StarGarnet03 3 года назад

The almond bread set

@therandomshow1265 3 года назад

I can see it

@The_Tiffster 3 года назад

That's just the name of the person who discovered it....

@Tower_Swagman 3 года назад

yay deustechland (sorry germans if i mispronounced or mispelled it)

@huhneat1076 3 года назад

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 года назад

And that's what dimensions are all about

@AlexanderGieg 2 года назад

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 года назад

@@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 10 месяцев назад

never thought of it this way, you blew my mind

@upanshulakhani6221 Месяц назад

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)

@grande1900 4 года назад

Ah, yes, my favorite fractal! *The Coastline of Britain*

@TheAustronaut03 4 года назад

the coastline of Switzerland is 0-dimensional

@killerfishe5092 4 года назад

@@TheAustronaut03 yes, that is technically true

@TheAustronaut03 4 года назад

wahoo

@bobfs9855 4 года назад

The coastline of Norway is far superior.

@trueriver1950 4 года назад

@@TheAustronaut03 Unless you include the coast of Lake Geneva, of course.

@Nevermind445 6 лет назад

"This is math, everything is made up" Love this quote!!

@nathanwagester6665 4 года назад

read philosophy

@HUGOGARCAO 4 года назад

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 года назад

How to make every math teacher very angry and how to claim 42 as a trial answer to everything.

@dataexpunged3914 4 года назад

@@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 года назад

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

@Shubham_pandey-nk1un 3 года назад

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 года назад

I immediately dismissed this thought as complete tomfoolery

@MrMegaMetroid 3 года назад

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 года назад

@@MrMegaMetroid Yes, You are right

@archangelofsorrow 2 года назад

@@MrMegaMetroid so there are three versions of dimension

@MrMegaMetroid 2 года назад

@@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

@NoahSpurrier 2 года назад

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 Год назад

Damn that's a neat piece of fact on its application!

@danyilbutsenko6339 Год назад

"Paintings"

@shobacon8263 Год назад

How did they do that

@CalebSalstrom Год назад

@@danyilbutsenko6339 there’s always one of you

@CarlNiemi Год назад

@@CalebSalstrom Always one correct person?

@Tomyb15 7 лет назад

This channel is youtube gold.

@fossilfighters101 7 лет назад

@eldiagrama 7 лет назад

@umamaheshwaranl8554 7 лет назад

3blue1brown:maths::pbsspacetime:physics

@vampyricon7026 7 лет назад

@xshortguy 7 лет назад

I even enjoy listening to the ad at the end of the video.

@KasabianFan44 7 лет назад

What does the B stand for in Benoit B. Mandelbrot? Benoit B. Mandelbrot.

@redoxred5588 6 лет назад

KasabianFan44 recursive acronym

@eednb4257 6 лет назад

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 лет назад

This is the most brilliant pun i have ever seen

@blunderbus2695 6 лет назад

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 лет назад

A “Mandelbrot” set...

@jeanmariegrangon 3 года назад

As a physics graduate, I wish that our teacher had shown us this video when he tried to teach us about fractal dimension.

@profbbfab6211 Год назад

Our teacher's assistant did! And I thank him for that

@wigglygrass3066 2 года назад

10:42 thought you were gonna say" in the back of our minds, we know to say nice"

@karthikprabhu3173 4 года назад

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 года назад

Please create a merch line with "THIS IS MATH, EVERYTHING IS MADE UP"

@dangerouspie0319 3 года назад

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 2 года назад

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 Год назад

@@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 Год назад

@@dangerouspie0319 education system is a play ground for control did you really expect the benefits to outweigh their agenda...

@karthikprabhu3173 4 года назад

Please create a merch line with "THIS IS MATH, EVERYTHING IS MADE UP"

@vale3242 3 года назад

I need it.

@papasscooperiaworker3649 3 года назад

LOL

@mrkun5905 3 месяца назад

@@papasscooperiaworker3649 DUDE

@illusionist1872 3 года назад

10:34 When you said "In the back of our minds," I thought you were going to make fun of how immature everyone is

@ConnoisseurOfExistence Год назад

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 Год назад

you can't represent a fractal with a dimension over 3 in real world

@lakshya5946 Год назад

@@raphdm3776 This is math Everything is and can be made up

@raphdm3776 Год назад

@@lakshya5946 I know but we will never see what it looks like

@lakshya5946 Год назад

@@raphdm3776 we can actually

@raphdm3776 Год назад

@@lakshya5946 I don't know about you but I only live in 3 dimensions

@mitchg9017 5 лет назад

when i first heard it, i thought he said "In some ways, fractal geometry is a rebellion against capitalism"

@yaiirable 4 года назад

had to switch on subtitles to hear it: it was actually 'calculus'

@gold_5600 4 года назад

A rebellion against capitalism, huh?

@raddrew42 4 года назад

What part?

@dcterr1 4 года назад

I'm glad my salary isn't a fractal!

@gamermapper 4 года назад

☭ ☭ ☭ ☭ ☭ ☭ ☭ ☭ ☭ ☭ 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...)

@tomimated1638 5 лет назад

Wait so is it called fractals because they are fractional

@CoolColourBlack 5 лет назад

yass, so I also thought this towards the end of this video and I was like whoaaaa.

@fqidz 5 лет назад

yes

@thekoldrex 5 лет назад

@@fqidz YES

@dp271 5 лет назад

well duh

@Julian-tf8nj 5 лет назад

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". '

@minhsieuquay8164 Год назад

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 Год назад

why did no one see this comment ._.

@NhuKhiet-cr8lh Год назад

Do you mean the area of the new circle = π(r/2)²=π*r²/4 = 1/4 old area?

@ThanhNguyen-ni4vw Год назад

That’s great!

@HuyenVu-di3et Год назад

It seems that the author wants to use visualization to demonstrate fractals, he doesn't want to use mathematical formulas

@minhsieuquay8164 Год назад

@@NhuKhiet-cr8lh oh yea

@JustMoseyinAround 3 года назад

10:48 *"In the back of our min-... Nice"*

@krisoluich9119 5 лет назад

I was living in a coma until I found this channel.

@ArchHeretic1 5 лет назад

you think thats bad? I was living in a comma,

@Ayy_la 5 лет назад

@@ArchHeretic1 you think that's bad? I was having a srtkoe

@ericjohnson1811 5 лет назад

Pretty cool, huh? : )

@thekoldrex 5 лет назад

@@Ayy_la you thin thas beaa...

@Xishnik94 5 лет назад

@@thekoldrex z

@vishwas425 7 лет назад

Waiting for a calculus series

@Ramzuiv 7 лет назад

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 лет назад

Ya I still watch the linear algebra series from time to time just because it's so good (especially the animations!)

@justinward3679 7 лет назад

I need a topology series.

@treyshaffer 7 лет назад

There's a guy on RU-vid 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 лет назад

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

@YBY3 3 года назад

10:30 - Nice

@meisnameless Год назад

Nice

@Ambettertenu 12 дней назад

Nice

@actuallyasriel 2 года назад

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.

@Minecraftster148790 7 лет назад

What does the B in Benoit B Mandelbrot stand for? Benoit B Mandelbrot

@ilikemitchhedberg 7 лет назад

thx m8. Was trying to remember that one.

@maxbuskirk5302 7 лет назад

Benoit Benoit Benoit Benoit Benoit . . . Benoit Benoit Benoit B Mandelbrot Mandelbrot Mandelbrot . . . Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot.

@pranavgarg9075 7 лет назад

Balls

@thomassynths 7 лет назад

It stands for "Bacon".

@ntofed 7 лет назад

@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.

@George4943 7 лет назад

Your videos remind me why I was a math major until I wrote my first program in 1962.

@TheRedstoneTaco 7 лет назад

roasted.

@bigbox8992 7 лет назад

What language did you use?

@George4943 7 лет назад

FORGO (a version of FORTRAN)

@ricardo.mazeto 7 лет назад

I'd like to hear you explain why or just talk more about that. I'm C programmer.

@bigbox8992 7 лет назад

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?

@shreeyamittal1771 3 года назад

"I mean, this is math. Everything's made up." Duude, you just got yourself another subscriber.

@shantonudutta9726 2 года назад

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!

@lijacom 4 года назад

Car salesman: (slaps sandpaper ) "This bad boy has a really high fractional dimention"

@dangerouspie0319 3 года назад

But does it have a high frictional dimension?

@Green24152 3 года назад

@@dangerouspie0319 Yes.

@fetterkeks2796 2 года назад

(slaps Mandelbrot set): "This fractal can fit so many fractals in it"

@mrkun5905 3 месяца назад

Nahhh

@1ucasvb 7 лет назад

Given a random real dimension D, is there an easy way to find a fractal with that dimension?

@3blue1brown 7 лет назад

Indeed! This clip should give you some clue: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-RU0wScIj36o.htmlm40s

@michaelwulber8229 7 лет назад

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 лет назад

It would be interesting to see an animation of a shape going from 1d to 3d

@NathanTAK 7 лет назад

Yes, but it might take a while :P.

@1Thor61storm8 7 лет назад

+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

@Cruveit 3 месяца назад

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 Месяц назад

note: I learned integrals

@hasanirtija8996 3 года назад

This was better than a 2-hour graduate lecture. Thank YOU!

@mediocreicerinkparodies1099 5 лет назад

"A line, a square, a cube..." "And a Sierpinski triangle"

@hebpennington 5 лет назад

"A line, a square, a cube, wand a Sierpinski triangle walk into a bar."

@ferencgazdag1406 4 года назад

The 4th should have been a hypercube.

@squibble311 4 года назад

and a tesseract

@sketin 3 года назад

THE ARISTOCRATS!

@zmoot 3 года назад

And 4d cube.

@jaygreenwood422 4 года назад

Don’t mind me, I’m just procrastinating

@overlordcringe2715 4 года назад

And I'm popping

@overlordcringe2715 4 года назад

Pope

@overlordcringe2715 4 года назад

Avacoda

@h-hhh 4 года назад

@@overlordcringe2715 h

@theaccounter 3 года назад

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.

@spacekid9680 Год назад

1:03 I like that you used my home island as an example. I see that shape and instantly think "home"

@akarshshrivastava3719 3 года назад

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

@karstenroelofs9216 4 года назад

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 года назад

jup, had the same thought

@Dhanush-zj7mf 3 года назад

Right.....

@BinyaminTsadikBenMalka 7 лет назад

This channel is a RU-vid 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 лет назад

It is true: he is a gem. Hopefully he can continue to gain more viewers.

@SkyWKing 7 лет назад

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 лет назад

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 лет назад

I feel like this is for math what PBS Spacetime is for physics.

@BinyaminTsadikBenMalka 7 лет назад

Vampyricon I like PBS Spacetime too

@squiji9750 Год назад

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...

@smokey04200420 3 года назад

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.

@alek6362 4 года назад

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 года назад

I independently invented certain finite difference methods but I was too old to think they were new.

@GoogleAccount-pi9ct 4 года назад

Lol

@wisdom6458 4 года назад

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 года назад

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 года назад

😂

@wiltherdelacuesta8175 5 лет назад

This voice and speaking speed is perfect to undertand complex topics...Good job!!!

@M-F-H 4 года назад

can't totally agree, it makes me sleepy... and/or in fear of being hypnotized...

@xDMrGarrison 3 года назад

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

@edwardlulofs444 9 месяцев назад

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.

@IanZainea1990 4 года назад

1:14 ... I heard "Fractals are a rebellion against capitalists." ... I was intrigued but knew I had to have heard wrong! haha.

@nevanmasterson46 4 года назад

Workers, unite! you have nothing to lose but your derivatives!

@jackdesy2127 4 года назад

@@nevanmasterson46 its integral that we seize the means of production

@DL-fb8jd 3 года назад

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 года назад

@@DL-fb8jd *He was also

@monika.alt197 2 года назад

Lmao

@samshrivastava2655 4 года назад

Read a book on fractals in 1992. Was fascinated. Understood it in 2019. What a great channel. RU-vid rocks.

@kaitsune2262 2 года назад

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.

@lukewaite9144 3 года назад

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 ❤️

@willemvandebeek 7 лет назад

So if a tesseract is scaled down one half, the mass is scaled down 1/16th?

@3blue1brown 7 лет назад

Yup!

@willemvandebeek 7 лет назад

Cool, Im trying to imagine it, but having a hard time with it ^^

@adymode 7 лет назад

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 лет назад

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 лет назад

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!

@tcarrotgaming1639 7 лет назад

If only math at school was this fun...

@bttfish 5 лет назад

The school is needless

@hugo3222 5 лет назад

@@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 лет назад

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 лет назад

Hu Go at least most educations in most secondary schools destroy the interest of most students.

@bttfish 5 лет назад

Kalistic Modiani At least this gives an interesting introduction.

@harrycarpio 2 года назад

I love this video. Thank you so much for taking the trouble to make it and share it to all.

@pauln1557 3 года назад

A nice, clearly presented, concise video, thanks for posting.

@Gadget622 7 лет назад

10:37 is when I found out that I'm not mature enough to be learning this stuff

@RealClassixX 5 лет назад

I have a masters in electrical engineering, and I muttered 'nice' to myself. Maturity is a lie.

@hesiod_delta9209 5 лет назад

@@RealClassixX L I E S

@kerosene_turtle4715 5 лет назад

Yep you are 12

@moadot720 5 лет назад

@@kerosene_turtle4715 Again, LOL, FuriousLightning!!!!

@demerion 5 лет назад

I know exactly what you are talking about without checking what happened at 10:37 first xD

@RyuichiNoGekido 4 года назад

10:40 Boxes touched: 69 Noice I’m sorry, I’ll start learning now.

@Bass_Guy 4 года назад

Nice

@nadari9162 4 года назад

I was praying to see this in the comments. Thanks

@hypertriangle6531 4 года назад

Nice

@luigivercotti6410 4 года назад

lucky next guy will put the 69th like here

@nnmk01 4 года назад

69th liker here *N O I C E*

@jrjr1313jrjr 3 года назад

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. :-)

@raghavherugu6934 4 года назад

One of the best mathematic channel ever. Keep it up!

@AwesomeCreeperBD 4 года назад

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 2 года назад

Don't underplay your achievements! What you did is still incredible and shows that you're a wonderful mathematician. Keep it up!

@pikminfan6778 5 лет назад

Yo dawg, I heard you like Triforces.

@haslan4885 5 лет назад

triforce-ception

@purrplaysLE 5 лет назад

So I put 9 triforces In a stupid WiiU game

@goldsrcorsource2551 5 лет назад

so i made it and infiniteforce

@skoto8219 5 лет назад

Haven't seen that meme in a while lol

@killerfishe5092 4 года назад

so i put a triforce in your triforce so you can use a triforce with your triforce

@PronteCo 2 года назад

beautifully direct and to the point, great explanation and great visual examples

@pablorepetto2759 4 года назад

"Mathematicians are clearly making stuff up" Well yeah... but no. It's complicated?

@piggywink333boyfriend6 3 года назад

Well yes, but not really

@mlgproplayer2915 3 года назад

Well yes, but actually no.

@arthurthekyogre9155 3 года назад

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 года назад

@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 года назад

@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!

@tallyhallsally 4 года назад

3 dimensional: *shows square* Him: “like we live in” Me: ahh, yes. It’s Minecraft time

@overlordcringe2715 4 года назад

Fuck you

@bin6549 4 года назад

@@overlordcringe2715 Jeez, who hurt you?

@kingaha3657 4 года назад

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 года назад

@@overlordcringe2715 Your username makes sense

@real_nosferatu 4 года назад

Cube*

@itsyaboigalaxium 2 года назад

The most mind-capturing video I've ever watched. Amazing!

@cassiopeiasfire6457 Год назад

Going from Vi Hart's Dragon Curve videos to this feels like growing up, I love it.

@sogidochnet9304 4 года назад

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!

@yulongqiu 5 лет назад

this is the first time I understand the fractional dimensions. thank you.

@CMDRunematti 11 месяцев назад

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

@yesssint7243 2 года назад

When I initially heard about fractals, I was told that a fractal is simply a figure/shape with infinite perimeter

@minuspi8372 2 года назад

Any object with dimension other than exactly 2 doesn't have a perimeter.

@JJean64 Год назад

@@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 Год назад

@@JJean64 generalized perimeter :p

@seppobastian 7 лет назад

This guy knows his stuff. And makes it very interesting :)

@trueriver1950 4 года назад

19:10 ... a numerical way to represent the fact that it's way more jaggedy ... Lovely expression of the idea of roughness

@zolvaring9503 2 года назад

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)

@back2d_lobby Год назад

Incredible presentation. I wish I had have these resources when I was at school

@itaiefrat3286 7 лет назад

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 лет назад

This is exactly what bothers me

@fossilfighters101 7 лет назад

@zairaner1489 7 лет назад

This only really works for vectorspaces (and in a weaker version for manifolds)

@SkyWKing 7 лет назад

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 лет назад

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?

@grevel1376 5 лет назад

Yesterday I was living in 3D space. You have changed my life.

@alexwang982 5 лет назад

DODO You still live in 3D space

@piggywink333boyfriend6 3 года назад

If you lived in 3D yesterday, and you posted this a year ago HOW DID YOU TIME TRAVEL

@pe3akpe3et99 3 года назад

@@piggywink333boyfriend6 you know he's living in 4 dimensions now, the fourth is time

@piggywink333boyfriend6 3 года назад

@@pe3akpe3et99 Thanks

@knightlypoleaxe2501 3 года назад

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.

@elizabetelauce1116 3 года назад

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!🦾🧠

@christianthompson7915 4 года назад

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

@OrchidAlloy 4 года назад

When you suggested programming a way to calculate the fractal dimensión, I was actually terrified.

@lilywater3683 2 года назад

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!

@esisimp123456 Год назад

The relation between fractional dimension of an object and its artificialness was really amazing.

@YounesLayachi 7 лет назад

I've been following this channel for *roughly* 2 years now, awesome content as usual

@YounesLayachi 7 лет назад

***** 1.1 dimensional

@nickshowman4606 4 года назад

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 года назад

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 года назад

Read the description ;)

@louisbrill891 11 месяцев назад

I love fractal videos solely to see how low the creator can drive the bitrate

@lunarknightess7056 8 месяцев назад

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

@Oroborus12 5 лет назад

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 лет назад

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 лет назад

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 года назад

www.google.com/url?sa=t&source=web&rct=j&url=%23&ved=2ahUKEwiSgd79-tjnAhXnyjgGHcFrCEkQwqsBMAB6BAgHEAQ&usg=AOvVaw0VTJzWxrN8ZFOD4xbU2nov

@matthieudeloget8998 4 года назад

10:38 Oh yes, it's all coming together.

@dartmarbleracing1762 3 года назад

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 года назад

nice

@shawnyao7850 3 года назад

thank you 3b1b, you just made my life so much better!

@urmi-e-azar1776 2 года назад

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 ♥️.

@mushyomens6885 4 года назад

" I mean this is math. Everything's made up. " Now that I think about it. Well said ...

@federicovolpe3389 6 лет назад

Thank you! You've explained clearly a difficult concept! I'm happy I discovered the world of fractal. Subscribed!

@icemaster523 3 года назад

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.

@mastershooter64 3 года назад

1:19 Mandelbrot saw this as "overly idealized" dauym mandelbrot was a savage dude!

@MetalSilvan 7 лет назад

That Markus Persson is one of your Patrons? Notch?

@DocteurZeuhl 7 лет назад

It's "that" Markus Persson indeed :)

@duckymomo7935 7 лет назад

who is Markus Persson?

@MetalSilvan 7 лет назад

Mi Les Notch! The creator of minecraft!

@alxjones 7 лет назад

Yes its confirmed that Notch supports 3b1b

@suwinkhamchaiwong8382 6 лет назад

notch

@rileydragunas9112 4 года назад

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

@jessieattrey5901 3 года назад

Awesome and inspiring! It opens a new dimension to my mind.

@richardray7976 3 года назад

Beautiful description. Well done!

@dudeman3981 7 лет назад

I would love a video on the Fourier series and transformation. Your animations would make it look so beautiful and intuitive!

@Em.P14 4 года назад

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 года назад

@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 года назад

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.