Fractals are typically not self-similar

  • Published on Jan 27, 2017
  • An explanation of fractal dimension.
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    One technical note: It's possible to have fractals with an integer dimension. The example to have in mind is some *very* rough curve, which just so happens to achieve roughness level exactly 2. Slightly rough might be around 1.1-dimension; quite rough could be 1.5; but a very rough curve could get up to 2.0 (or more). A classic example of this is the boundary of the Mandelbrot set. The Sierpinski pyramid also has dimension 2 (try computing it!).
    The proper definition of a fractal, at least as Mandelbrot wrote it, is a shape whose "Hausdorff dimension" is greater than its "topological dimension". Hausdorff dimension is similar to the box-counting one I showed in this video, in some sense counting using balls instead of boxes, and it coincides with box-counting dimension in many cases. But it's more general, at the cost of being a bit harder to describe.
    Topological dimension is something that's always an integer, wherein (loosely speaking) curve-ish things are 1-dimensional, surface-ish things are two-dimensional, etc. For example, a Koch Curve has topological dimension 1, and Hausdorff dimension 1.262. A rough surface might have topological dimension 2, but fractal dimension 2.3. And if a curve with topological dimension 1 has a Hausdorff dimension that *happens* to be exactly 2, or 3, or 4, etc., it would be considered a fractal, even though it's fractal dimension is an integer.
    See Mandelbrot's book "The Fractal Geometry of Nature" for the full details and more examples.
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Comments • 3 032

  • Ryan McAdams
    Ryan McAdams Day ago

    what dimension is a Fibonacci flower spiral

  • mitch dorsman
    mitch dorsman Day ago

    10:42 nice

  • KumPupp
    KumPupp 2 days ago

    So, class, four shapes: a line, a square, a cube and The Sierpinski Triangle!

  • God
    God 3 days ago

    Bruh, i had a math test about logarithms last week, and i didn't fail because of this video

  • Sumit Modak
    Sumit Modak 4 days ago +2

    you must get a patent for this explaination and also for making the pi fractal.

  • amoura saliha
    amoura saliha 5 days ago

    thank you very much , i have a part of my Phd about fractal , nice expalnation , thank you again

  • LemonadeCat :3
    LemonadeCat :3 8 days ago

    Your name should actually be 4blue1brown

  • 1.
    1. 8 days ago

    Touching 69
    What are the chances

  • Sara Kim
    Sara Kim 11 days ago

    Could someone explain me y a line,square and cube aren’t fractals? What’s the definition of a fractal?

    • Pluto
      Pluto 9 days ago

      Sara Kim A fractal is a shape with a non-integer dimension. He says in the video.

  • TheEgglet
    TheEgglet 14 days ago +1

    what dimension is the mandelbrot set?

  • Will Cahill
    Will Cahill 14 days ago

    Can someone show me how the curve of scaling factor converges to 4, like the fancy maths way of saying it

  • Shruggz Da Str8-Faced Clown

    1.585 just happens to be 3-sqrt2.

  • Suschi
    Suschi 16 days ago

    But whats about dimensions bigger than 3? Like time or anything else?

  • otherlego
    otherlego 16 days ago


  • Glitched Blox
    Glitched Blox 17 days ago


  • Reazul Ranga
    Reazul Ranga 17 days ago


  • Prashant Dahiya
    Prashant Dahiya 17 days ago

    I think we are just playing

  • Conor Clements
    Conor Clements 17 days ago

    10:35 nice

  • Myrmidon
    Myrmidon 22 days ago

    15:33 When you ask an engineer the value of pi.

  • Qaim Alvi
    Qaim Alvi 22 days ago

    hello people commenting here. Please be my friend. I hv trouble discussing any of this with my current friends and feel intellectually unfulfilled :(

  • Gogole PI Mukherjee
    Gogole PI Mukherjee 23 days ago

    Again a awesome work. wonderful .

  • sarahannkathrin
    sarahannkathrin 23 days ago

    This helped me so much!!! I study Geology and I couldn’t wrap my head around the concept of self similarity and dimensions 🙃

  • Noah Abraham
    Noah Abraham 24 days ago

    You be reported for touching boxes....

  • Arjab De
    Arjab De 26 days ago

    But tell me how to calculate or get scaling factor?

  • 10
    10 27 days ago

    Why is everyone a pi symbol

  • Sam Eldridge
    Sam Eldridge 27 days ago +1


  • nic k.
    nic k. 27 days ago +1

    kevin: ( ͡° ͜ʖ ͡°)

  • 1infinity8 1i8
    1infinity8 1i8 28 days ago

    15:09 I think his mic change

  • AwesomeCreeperBD
    AwesomeCreeperBD 29 days ago

    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 :)

  • ThatSpicyDragon
    ThatSpicyDragon Month ago

    Zips law 11:42

  • Ghost Buster
    Ghost Buster Month ago

    I need to learn more before I watch this channel as I barely understand most of it

  • Bloky555
    Bloky555 Month ago

    I'm just a high school student who isn't even that far into mathematics and whenever I watch and rewatch your videos, I don't always understand everything. However, each time I do watch them I understand a little bit more. I don't have to understand everything to know that this is all beautiful, but understanding this all will make it even more wondrous. Hopefully by the time I get out of college I'll be watching these things and will understand everything in its beauty. Basically,, yeah math is really scary and yeah I don't understand every facet of it, but one day, maybe 8 or 9 years later, I will know more. Because of that, I'll see it's far more intimdating than I thought 8 years ago, but it will be MY scary ass monster. I intend to tame this crap and use it to help develop some proof, formula, or whatever, and discover something just as insightful as this guy's videos.

  • Atoy 1234 ihrs
    Atoy 1234 ihrs Month ago

    10:35 nice

  • Lycanroc Dusk
    Lycanroc Dusk Month ago

    Well this is just annoying.

  • Diggi
    Diggi Month ago

    So, britain is a fractal?

  • Arunanshu Biswas
    Arunanshu Biswas Month ago

    @ 10:37 math concluded me towards 69!!

    Anyways, the Video and Grant, both are awesome!
    Keep up the Good Work!

  • Ian Zainea
    Ian Zainea Month ago +1

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

  • DaNkO Dangerpea
    DaNkO Dangerpea Month ago

    10:36 Nice

  • Skeleton Masher
    Skeleton Masher Month ago +1

    200 IQ... trying to find 1.5 dimensions, not the 4th dimension..

  • Forest Dewberry
    Forest Dewberry Month ago +1

    Well, the sierpinsky triangle is man-made and doesn't have an integer dimension.

  • Bhanu Rathore
    Bhanu Rathore Month ago

    Can you please explain this with higher dimensions as well, like how would a fractal be with dimension higher than 3.

  • Matthieu Besançon
    Matthieu Besançon Month ago

    I truly enjoy what you do. Not only the quality of the videos but especially the new perspectives that you bring!

  • Robert Pemberton
    Robert Pemberton Month ago

    Nope, fractals are 2D, all of them exist on an x-y plane

  • Menya Savut
    Menya Savut Month ago

    Where do you get this annoying background music from? Do you create your videos in an elevator?

  • Menya Savut
    Menya Savut Month ago

    Terribly annoying background piano. Need to apply Fourier transform to get rid of it.

  • Aoi Tanlores
    Aoi Tanlores Month ago

    as an asian i give this guy reezpecc cuz i can't understand the complex word

  • Jan Segre
    Jan Segre Month ago

    So, a dense dotted line is zero dimensional because if you make that "number of squares touched" plot it eventually reaches a point where there is one square per dot and the graph flattens to a horizontal line which has inclination 0. I think that the whole idea of a dimension being a function instead of a real number is much more interesting. It can probably model the real world physics more accurately because at a zoomed matter is just particles but zoomed out is more like 3 dimensions, and if we zoom out even more it may be 0-dimensional again if it's all contained or n-dimensional if there is more stuff where we can't observe but arranged in a different way.

  • Papa {YT}
    Papa {YT} Month ago

    Going back to videos older than 1 year to find memes...

    nice 10:36

  • Malizma
    Malizma Month ago

    10:35 number of boxes are nice

  • Daffa_FM
    Daffa_FM Month ago

    2d fractal: 1.xx dimensions
    3d fractal: 2.xx dimensions
    4d fractal: 3.xx dimensions

  • Blaise Vignon
    Blaise Vignon Month ago

    I believe his seminal example was the coast of Brittany, not great Britain :-)

  • Somebody very nice
    Somebody very nice Month ago +2

    Oh yes, it's all coming together.

  • John Baldwin
    John Baldwin Month ago

    Fractals 1/0

  • YT InzAniTix
    YT InzAniTix Month ago

    the start is satisfying asf

  • An_Annoying_Cat
    An_Annoying_Cat Month ago

    We really need to make up new terms to describe these concepts.

  • The pathetic one
    The pathetic one Month ago

    4:08 should've made the yellow ones brown. missed opportunity!

  • Ami Schaefer
    Ami Schaefer Month ago +1

    Which Things do you think are interesting in fractal geometry? I want to write in a project (20-30 pages) about a permanent topic of fractals, so it should be a “Hard“ topic. Please, Any ideas 🙏🏼🙏🏼

  • Danny Danny
    Danny Danny Month ago

    i love this video honestly! it gives so much perspective of the world around us

  • pricken2000
    pricken2000 Month ago +1

    I'm against this because dimensions aren't shapes they are axis

    • An_Annoying_Cat
      An_Annoying_Cat Month ago

      There are no good terms for these so this is what he’s using.

  • MLN Reid
    MLN Reid Month ago +1

    One of the first programs I had to write for class was creating a Mandelbrot Set!!