Fractals are typically not self-similar

Рет қаралды 3,953,626

Күн бұрын

An 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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Пікірлер: 4 700

@Nevermind445 6 жыл бұрын

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

@nathanwagester6665 4 жыл бұрын

read philosophy

@HUGOGARCAO 4 жыл бұрын

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 жыл бұрын

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

@dataexpunged3914 4 жыл бұрын

@@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 жыл бұрын

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

@ryanmarita-davis3339 Жыл бұрын

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

@brandongroves4465 Жыл бұрын

There is no such thing as rebellion in math.

@brandongroves4465 Жыл бұрын

Also I agree

@matthewfarrell6822 11 ай бұрын

You know I think I like fractals

@bitonic589 8 ай бұрын

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

@andistansbury4366 6 ай бұрын

Yea! Screw calculus!

@Shubham_pandey-nk1un 3 жыл бұрын

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 жыл бұрын

I immediately dismissed this thought as complete tomfoolery

@MrMegaMetroid 3 жыл бұрын

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 жыл бұрын

@@MrMegaMetroid Yes, You are right

@archangelofsorrow 3 жыл бұрын

@@MrMegaMetroid so there are three versions of dimension

@MrMegaMetroid 3 жыл бұрын

@@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 жыл бұрын

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 жыл бұрын

And that's what dimensions are all about

@AlexanderGieg 2 жыл бұрын

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 жыл бұрын

@@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 Жыл бұрын

never thought of it this way, you blew my mind

@upanshulakhani6221 2 ай бұрын

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 жыл бұрын

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

@TheAustronaut03 4 жыл бұрын

the coastline of Switzerland is 0-dimensional

@killerfishe5092 4 жыл бұрын

@@TheAustronaut03 yes, that is technically true

@TheAustronaut03 4 жыл бұрын

wahoo

@bobfs9855 4 жыл бұрын

The coastline of Norway is far superior.

@trueriver1950 4 жыл бұрын

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

@matheusnever1plasmaman477 4 жыл бұрын

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

@ireallyneedalife6979 4 жыл бұрын

*Ah yes, enslaved fractal*

@ntck 4 жыл бұрын

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

@GlitchedBlox 4 жыл бұрын

Ah yes, Enslaved object

@1994mrmysteryman 4 жыл бұрын

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

@thegoofyarchive8300 4 жыл бұрын

10:39 *n i c e*

@NoahSpurrier 2 жыл бұрын

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 2 жыл бұрын

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

@danyilbutsenko6339 2 жыл бұрын

"Paintings"

@shobacon8263 2 жыл бұрын

How did they do that

@CalebSalstrom Жыл бұрын

@@danyilbutsenko6339 there’s always one of you

@CarlNiemi Жыл бұрын

@@CalebSalstrom Always one correct person?

@jeanmariegrangon 3 жыл бұрын

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

@profbbfab6211 Жыл бұрын

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

@Tomyb15 7 жыл бұрын

This channel is youtube gold.

@fossilfighters101 7 жыл бұрын

@eldiagrama 7 жыл бұрын

@umamaheshwaranl8554 7 жыл бұрын

3blue1brown:maths::pbsspacetime:physics

@vampyricon7026 7 жыл бұрын

@xshortguy 7 жыл бұрын

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

@KasabianFan44 7 жыл бұрын

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

@redoxred5588 6 жыл бұрын

KasabianFan44 recursive acronym

@eednb4257 6 жыл бұрын

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 жыл бұрын

This is the most brilliant pun i have ever seen

@blunderbus2695 6 жыл бұрын

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 жыл бұрын

A “Mandelbrot” set...

@tokiWren 4 жыл бұрын

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

@otesunki 4 жыл бұрын

Is that profile pic... from nick?

@StarGarnet03 3 жыл бұрын

The almond bread set

@therandomshow1265 3 жыл бұрын

I can see it

@The_Tiffster 3 жыл бұрын

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

@Tower_Swagman 3 жыл бұрын

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

@illusionist1872 3 жыл бұрын

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

@karthikprabhu3173 4 жыл бұрын

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 жыл бұрын

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

@dangerouspie0319 3 жыл бұрын

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 жыл бұрын

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 2 жыл бұрын

@@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 Жыл бұрын

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

@karthikprabhu3173 4 жыл бұрын

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

@vale3242 3 жыл бұрын

I need it.

@papasscooperiaworker3649 3 жыл бұрын

LOL

@mrkun5905 4 ай бұрын

@@papasscooperiaworker3649 DUDE

@wigglygrass3066 3 жыл бұрын

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

@ConnoisseurOfExistence Жыл бұрын

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 Жыл бұрын

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

@lakshya5946 Жыл бұрын

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

@raphdm3776 Жыл бұрын

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

@lakshya5946 Жыл бұрын

@@raphdm3776 we can actually

@raphdm3776 Жыл бұрын

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

@mitchg9017 5 жыл бұрын

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

@yaiirable 4 жыл бұрын

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

@gold_5600 4 жыл бұрын

A rebellion against capitalism, huh?

@raddrew42 4 жыл бұрын

What part?

@dcterr1 4 жыл бұрын

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

@gamermapper 4 жыл бұрын

☭ ☭ ☭ ☭ ☭ ☭ ☭ ☭ ☭ ☭ 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...)

@lijacom 4 жыл бұрын

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

@dangerouspie0319 3 жыл бұрын

But does it have a high frictional dimension?

@Green24152 3 жыл бұрын

@@dangerouspie0319 Yes.

@fetterkeks2796 2 жыл бұрын

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

@mrkun5905 4 ай бұрын

Nahhh

@shantonudutta9726 2 жыл бұрын

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!

@JustMoseyinAround 3 жыл бұрын

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

@mediocreicerinkparodies1099 5 жыл бұрын

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

@hebpennington 5 жыл бұрын

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

@ferencgazdag1406 4 жыл бұрын

The 4th should have been a hypercube.

@squibble311 4 жыл бұрын

and a tesseract

@sketin 3 жыл бұрын

THE ARISTOCRATS!

@zmoot 3 жыл бұрын

And 4d cube.

@krisoluich9119 5 жыл бұрын

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

@ArchHeretic1 5 жыл бұрын

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

@Ayy_la 5 жыл бұрын

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

@ericjohnson1811 5 жыл бұрын

Pretty cool, huh? : )

@thekoldrex 5 жыл бұрын

@@Ayy_la you thin thas beaa...

@Xishnik94 5 жыл бұрын

@@thekoldrex z

@hasanirtija8996 3 жыл бұрын

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

@actuallyasriel 3 жыл бұрын

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.

@tomimated1638 5 жыл бұрын

Wait so is it called fractals because they are fractional

@CoolColourBlack 5 жыл бұрын

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

@fqidz 5 жыл бұрын

yes

@thekoldrex 5 жыл бұрын

@@fqidz YES

@dp271 5 жыл бұрын

well duh

@Julian-tf8nj 5 жыл бұрын

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". '

@alek6362 4 жыл бұрын

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 жыл бұрын

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

@GoogleAccount-pi9ct 4 жыл бұрын

Lol

@wisdom6458 4 жыл бұрын

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 жыл бұрын

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 жыл бұрын

😂

@shreeyamittal1771 3 жыл бұрын

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

@minhsieuquay8164 Жыл бұрын

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 Жыл бұрын

why did no one see this comment ._.

@NhuKhiet-cr8lh Жыл бұрын

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

@ThanhNguyen-ni4vw Жыл бұрын

That’s great!

@HuyenVu-di3et Жыл бұрын

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

@minhsieuquay8164 Жыл бұрын

@@NhuKhiet-cr8lh oh yea

@1ucasvb 7 жыл бұрын

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

@3blue1brown 7 жыл бұрын

Indeed! This clip should give you some clue: kzfaq.info/get/bejne/iLtgqraUrtCXZqM.htmlm40s

@michaelwulber8229 7 жыл бұрын

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 жыл бұрын

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

@NathanTAK 7 жыл бұрын

Yes, but it might take a while :P.

@1Thor61storm8 7 жыл бұрын

+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

@RyuichiNoGekido 4 жыл бұрын

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

@Bass_Guy 4 жыл бұрын

Nice

@nadari9162 4 жыл бұрын

I was praying to see this in the comments. Thanks

@hypertriangle6531 4 жыл бұрын

Nice

@luigivercotti6410 4 жыл бұрын

lucky next guy will put the 69th like here

@nnmk01 4 жыл бұрын

69th liker here *N O I C E*

@YBY3 3 жыл бұрын

10:30 - Nice

@meisnameless Жыл бұрын

Nice

@Ambettertenu Ай бұрын

Nice

@Cruveit 4 ай бұрын

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 3 ай бұрын

note: I learned integrals

@George4943 7 жыл бұрын

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

@TheRedstoneTaco 7 жыл бұрын

roasted.

@bigbox8992 7 жыл бұрын

What language did you use?

@George4943 7 жыл бұрын

FORGO (a version of FORTRAN)

@ricardo.mazeto 7 жыл бұрын

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

@bigbox8992 7 жыл бұрын

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?

@Binyamin.Tsadik 7 жыл бұрын

This channel is a KZfaq 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 жыл бұрын

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

@SkyWKing 7 жыл бұрын

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 жыл бұрын

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 жыл бұрын

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

@Binyamin.Tsadik 7 жыл бұрын

Vampyricon I like PBS Spacetime too

@smokey04200420 3 жыл бұрын

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.

@spacekid9680 Жыл бұрын

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

@tallyhallsally 4 жыл бұрын

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

@overlordcringe2715 4 жыл бұрын

Fuck you

@bin6549 4 жыл бұрын

@@overlordcringe2715 Jeez, who hurt you?

@kingaha3657 4 жыл бұрын

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 жыл бұрын

@@overlordcringe2715 Your username makes sense

@real_nosferatu 4 жыл бұрын

Cube*

@jaygreenwood422 4 жыл бұрын

Don’t mind me, I’m just procrastinating

@overlordcringe2715 4 жыл бұрын

And I'm popping

@overlordcringe2715 4 жыл бұрын

Pope

@overlordcringe2715 4 жыл бұрын

Avacoda

@h-hhh 4 жыл бұрын

@@overlordcringe2715 h

@theaccounter 4 жыл бұрын

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.

@harrycarpio 2 жыл бұрын

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

@akarshshrivastava3719 3 жыл бұрын

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

@Minecraftster148790 7 жыл бұрын

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

@ilikemitchhedberg 7 жыл бұрын

thx m8. Was trying to remember that one.

@maxbuskirk5302 7 жыл бұрын

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

@pranavgarg9075 7 жыл бұрын

Balls

@thomassynths 7 жыл бұрын

It stands for "Bacon".

@ntofed 7 жыл бұрын

@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.

@karstenroelofs9216 4 жыл бұрын

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 жыл бұрын

jup, had the same thought

@Dhanush-zj7mf 3 жыл бұрын

Right.....

@squiji9750 Жыл бұрын

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...

@kaitsune2262 2 жыл бұрын

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.

@wiltherdelacuesta8175 5 жыл бұрын

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

@M-F-H 4 жыл бұрын

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

@samshrivastava2655 4 жыл бұрын

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

@lukewaite9144 3 жыл бұрын

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 ❤️

@edwardlulofs444 11 ай бұрын

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.

@vishwas425 7 жыл бұрын

Waiting for a calculus series

@Ramzuiv 7 жыл бұрын

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 жыл бұрын

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

@justinward3679 7 жыл бұрын

I need a topology series.

@treyshaffer 7 жыл бұрын

There's a guy on KZfaq 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 жыл бұрын

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

@IanZainea1990 4 жыл бұрын

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

@nevanmasterson46 4 жыл бұрын

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

@jackdesy2127 4 жыл бұрын

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

@DL-fb8jd 3 жыл бұрын

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 жыл бұрын

@@DL-fb8jd *He was also

@monika.alt197 2 жыл бұрын

Lmao

@deadchannel4619 3 жыл бұрын

**slaps roof of fractal** This bad boy can fit so much fractal in it.

@SmokeyDope 3 жыл бұрын

Other valid way to end the joke is "This bad boy can fit so much perimeter in it

@xDMrGarrison 3 жыл бұрын

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

@sogidochnet9304 4 жыл бұрын

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 жыл бұрын

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

@pauln1557 3 жыл бұрын

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

@willemvandebeek 7 жыл бұрын

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

@3blue1brown 7 жыл бұрын

Yup!

@willemvandebeek 7 жыл бұрын

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

@adymode 7 жыл бұрын

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 жыл бұрын

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 жыл бұрын

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!

@Gadget622 7 жыл бұрын

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

@RealClassixX 5 жыл бұрын

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

@hesiod_delta9209 5 жыл бұрын

@@RealClassixX L I E S

@kerosene_turtle4715 5 жыл бұрын

Yep you are 12

@moadot720 5 жыл бұрын

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

@demerion 5 жыл бұрын

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

@knightlypoleaxe2501 3 жыл бұрын

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.

@raghavherugu6934 4 жыл бұрын

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

@tcarrotgaming1639 7 жыл бұрын

If only math at school was this fun...

@bttfish 5 жыл бұрын

The school is needless

@hugo3222 5 жыл бұрын

@@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 жыл бұрын

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 жыл бұрын

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

@bttfish 5 жыл бұрын

Kalistic Modiani At least this gives an interesting introduction.

@pablorepetto2759 4 жыл бұрын

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

@piggywink333boyfriend6 4 жыл бұрын

Well yes, but not really

@mlgproplayer2915 4 жыл бұрын

Well yes, but actually no.

@arthurthekyogre9155 4 жыл бұрын

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 жыл бұрын

@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 жыл бұрын

@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!

@jrjr1313jrjr 3 жыл бұрын

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. :-)

@PronteCo 2 жыл бұрын

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

@AwesomeCreeperBD 4 жыл бұрын

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 жыл бұрын

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

@seppobastian 7 жыл бұрын

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

@back2d_lobby Жыл бұрын

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

@itsyaboigalaxium 2 жыл бұрын

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

@mushyomens6885 4 жыл бұрын

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

@nickshowman4606 4 жыл бұрын

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 жыл бұрын

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 жыл бұрын

Read the description ;)

@shawnyao7850 3 жыл бұрын

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

@CMDRunematti Жыл бұрын

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

@trueriver1950 4 жыл бұрын

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

@christianthompson7915 4 жыл бұрын

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

@nefarioustoast 3 жыл бұрын

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?

@cassiopeiasfire6457 Жыл бұрын

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

@federicovolpe3389 7 жыл бұрын

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

@YounesLayachi 7 жыл бұрын

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

@YounesLayachi 7 жыл бұрын

***** 1.1 dimensional

@richardray7976 3 жыл бұрын

Beautiful description. Well done!

@lilywater3683 2 жыл бұрын

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!

@pikminfan6778 5 жыл бұрын

Yo dawg, I heard you like Triforces.

@haslan4885 5 жыл бұрын

triforce-ception

@purrplaysLE 5 жыл бұрын

So I put 9 triforces In a stupid WiiU game

@goldsrcorsource2551 5 жыл бұрын

so i made it and infiniteforce

@skoto8219 5 жыл бұрын

Haven't seen that meme in a while lol

@killerfishe5092 4 жыл бұрын

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

@itaiefrat3286 7 жыл бұрын

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 жыл бұрын

This is exactly what bothers me

@fossilfighters101 7 жыл бұрын

@zairaner1489 7 жыл бұрын

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

@SkyWKing 7 жыл бұрын

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 жыл бұрын

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?

@jessieattrey5901 4 жыл бұрын

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

@elizabetelauce1116 3 жыл бұрын

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!🦾🧠

@eggsoup6296 4 жыл бұрын

My IQ is increasing exponentially because of this video.

@AnindyaMahajan 4 жыл бұрын

1^x = 1 for all x jk lmao

@kartikaalst7354 4 жыл бұрын

@@AnindyaMahajan not if x is a complex number it isn't

@kartikaalst7354 4 жыл бұрын

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 жыл бұрын

@@georgehajnal2723 I think he means crystallized intelligence.

@moodleblitz 4 жыл бұрын

And I'm not so sure about mine anymore...

@bigbox8992 7 жыл бұрын

Why we see fractals when we are on lysergic acid? What is the relationship between biochemistry, human perception and fractal geometry?

@tehlolzfactor 7 жыл бұрын

There are actually some very complex mathematics descriptions of such fractals and shapes and their relationship to vision, especially regarding the usage of LSD and marijuana. If you look up the paper, "Geometric visual hallucinations, Euclidean symmetry and the functional architecture of the striate cortex" by Paul C. Bressloff, Jack D. Cowan, and others, published by the Royal Society. I would love to see 3Blue1Brown do some sort of video simplifying the concepts for us laypeople, though the topic as it is described in the original paper is still very interesting.

@primarysecondaryxd 7 жыл бұрын

I think it's really funny that someone can basically ask "Why do I see weird big shapes when I'm high", and people can mathematically and scientifically answer that question without joking about it.

@hhaavvvvii 7 жыл бұрын

It's questions like these that we actually want to answer. They provide real insight into our world.

@ClaytonLivsey 7 жыл бұрын

My take: because our visual apparatuses are fractals as well.

@daggawagga 7 жыл бұрын

Damn. I never would have guessed there was a specific word for the geometric shapes that appeared when I pressed my eyelids when I was a kid (wikipedia: "form constant").

@moreydumo5925 2 жыл бұрын

love it!!! great job

@kumarmanglam2362 4 жыл бұрын

It’s such a nice and detailed video.❤️i just loved it.this video is too visionary.

@mirosawzotorowicz8042 5 жыл бұрын

Fractal with pi dimension?

@migram4190 5 жыл бұрын

Oh what could happen?? 😲

@theone2-three438 5 жыл бұрын

I think it would be a theoretical circle

@ham1533 5 жыл бұрын

it will be something with infinite volume and 0 hyper-volume (i think, dont yell at me lol)

@dusty6299 5 жыл бұрын

maybe it will be a circle

@dusty6299 5 жыл бұрын

kzfaq.info/get/bejne/kK5yd8t-0Kymh4k.html

@dudeman3981 7 жыл бұрын

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

@ryanpost13 4 жыл бұрын

13:21 he missed a bit of coast line, towards the bottom left

@zolvaring9503 2 жыл бұрын

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)

@grevel1376 5 жыл бұрын

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

@alexwang982 5 жыл бұрын

DODO You still live in 3D space

@piggywink333boyfriend6 4 жыл бұрын

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

@pe3akpe3et99 3 жыл бұрын

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

@piggywink333boyfriend6 3 жыл бұрын

@@pe3akpe3et99 Thanks

@OrchidAlloy 4 жыл бұрын

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

@guardian-X 3 жыл бұрын

Awesome episode. I learned something today!

@danwillits7954 Жыл бұрын

I never knew how one got the dimension of a fractal until this video. Thank you.

@Oroborus12 5 жыл бұрын

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 жыл бұрын

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 жыл бұрын

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 жыл бұрын

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

@rileydragunas9112 4 жыл бұрын

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

@esisimp123456 Жыл бұрын

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

@noahglimcher5445 11 ай бұрын

I always love to watch your videos, and find them the perfect mix of fun and educational. I did just notice the tiniest mistake though: at 14:13 while display the coastline of Britain a lot of the parts of the line are not yellowed out. Did that affect the count?