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 • 2 739

  • bazoo513
    bazoo513 14 hours ago

    A *very* clear explanation of fractional dimensionality, the best I have seen.

  • jamma246
    jamma246 Day ago

    Do you know what the B. stands for in Benoit B. Mandelbrot?
    It's Benoit B. Mandelbrot.

  • Mayo naise
    Mayo naise Day ago


  • Patriot513
    Patriot513 Day ago

    I’m high school I never thought I would ever find math fascinating, great video

  • Edric Ndirangu
    Edric Ndirangu 3 days ago

    Tri triangle= tri force

  • Singh Naveen
    Singh Naveen 4 days ago

    Hey grant
    Big fan
    Which software do you use?

  • NoriMori
    NoriMori 4 days ago

    13:19 You're missing a box near Cardiff! And one near Glenluce!

  • NoriMori
    NoriMori 4 days ago

    At 5:11, it clicked! The dimension of a self-similar shape is the logarithm of the mass scaling factor to the base of the scaling factor! It makes so much sense!

  • William Callender
    William Callender 5 days ago

    10:37 Nice.

  • Tommye K
    Tommye K 5 days ago +1

    M Y. M I N D. I S. F U C K I N G. B. L. O. W. N.

  • JLR O.o. Fuck yeah from Mexico.

    Dont understand all. But is amazin

  • 나노별
    나노별 6 days ago

    This is just amazing... Thank you.

  • Дмитрий Брилько

    What a terrible channel! I'm an applied mathematician, and I assure you that the similarity coefficient is not the same as the dimension. These are different concepts, and one does not depend on the other.

  • Дмитрий Брилько

    Какой ужасный канал! Я прикладной математик, и уверяю вас, что коэффициент подобия это не тоже самое, что размерность. Это разные понятия, и одно не зависит от другого.

  • devin devine
    devin devine 7 days ago

    In between dimensions.
    Like with flatland--if we imagine flatland being a piece of paper, that paper has some depth--it's partially extended into 3 dimensions. Meanwhile, here in the 3-d, sometimes it rains frogs. Unexplained phenomena occur because we are partially extended into the 4-d.
    The math, showing us that a shape can have a fractional dimension, seems like a clue.

  • TheJ4RyD
    TheJ4RyD 7 days ago

    This seems so pointless and stupid

  • Astatine
    Astatine 8 days ago


  • Anders Termansen
    Anders Termansen 8 days ago

    OP needs more sex.

  • BGMC 16
    BGMC 16 8 days ago


  • vebrun
    vebrun 8 days ago

    10:37 "69"
    N I C E

  • Dario Vurchio
    Dario Vurchio 9 days ago

    One of the most interesting videos on USclip *EVER*

  • Olivier Nusbaumer
    Olivier Nusbaumer 9 days ago

    If I scale a "normal" triangle by 1/2, its mass is also scaled by 1/3... so what is so special about Sierpinski fractal triangle ??

  • Francisco Martínez
    Francisco Martínez 10 days ago

    i love this video! such a great and clear explanation about fractals...thanks!!!

  • Hank S
    Hank S 12 days ago

    Cantor set has a dimension less than 1. D=(ln 2)/(ln 3) approximately 0.631

  • Konstantin Kuzminykh
    Konstantin Kuzminykh 13 days ago

    so that also means usual shapes are fractals with integer dimensions=) 15:10 in a more deep thinking we like 1,2,3 numbers so we like lines squares cubes more then map of england))) thats very natural

  • Smuutti
    Smuutti 13 days ago

    11:00 2*2 is approximately 3.39?

  • Ami Schaefer
    Ami Schaefer 14 days ago +1

    Anybody how knows how to calculate the Dimension by oneself? I mean with a Programm like he used or sth. like that?🤔🙏🏼

  • [수리 TV] Eagle Jr.
    [수리 TV] Eagle Jr. 16 days ago

    6:51 Is D log2(3) so it isn't a rational number?
    Edit: wait, nvm, he said it.

  • Lucas Romano
    Lucas Romano 16 days ago

    At 2:52 when he says that math is made up, does that sentence means that math is invented by humans? Becuase some people claim that we just found it, it already existed. I just want to know the opinion of the guy who makes this videos

  • Bash Gamer
    Bash Gamer 18 days ago

    So you are Pi right?

  • cjsturgis100
    cjsturgis100 18 days ago

    What about a 2.45 dimensional object

  • When will it end
    When will it end 21 day ago +1

    Who is acc here to learn

  • AaronThe Proj
    AaronThe Proj 21 day ago +2

    I literally exploded in my mind

    My brother still considers the coastline of Brian is 2 dimensional

  • Jimmy from Philly
    Jimmy from Philly 22 days ago

    Ok so I'm tripping my face off and this isn't helping,lol

  • Abdul Haseeb
    Abdul Haseeb 24 days ago +15

    Me at 2:00
    Either I dont know what a dimension is or he is gone crazy

    AAALE 24 days ago

    21:25 Now calcolate the dimension.

  • Saurabh Basu
    Saurabh Basu 26 days ago

    can you do a video on Green's functions

  • mint CHILL
    mint CHILL 26 days ago

    Which music is used in background?

  • brett knoss
    brett knoss 26 days ago +1

    Would an example of changing dimensions be, a flat plot of land as 2 dimensional, on a 3 dimensional globe?

  • Icestrike411 Cubing
    Icestrike411 Cubing 27 days ago +1

    almond bread

  • Kev Alan
    Kev Alan 27 days ago +3

    Would you consider making videos without the background music? There is a flaw in my brain that makes me about 90% less intelligent whenever there is a background track. :-P

  • Discombobulated
    Discombobulated 28 days ago

    Hey you have my eye as your icon :D

  • todessushi
    todessushi 28 days ago

    When it comes to the real world if you go to even higher scaling factors you will end up being in the quantum world where nothing is defined exact anymore.

  • TheRubyGames
    TheRubyGames 28 days ago

    0:49 no! THATS A TRIFORCE

  • Daniel Goetz
    Daniel Goetz 29 days ago

    This makes so much sense now! I put A LOT of incorrect information of my 7th grade math fair 2 years ago

  • Gaspard Berthelier
    Gaspard Berthelier Month ago

    but doesn't dividing by 2 the lenghts of a "normal" triangular shape (3 triangles put together with a triangular void in the middle) also give a division of area by 3 ? Would that mean that its dimension is also log(2,3) ?

    • Gaspard Berthelier
      Gaspard Berthelier 29 days ago

      @phiefer3 oh that's right thanks

    • phiefer3
      phiefer3 29 days ago

      No, because you are not scaling that shape down when you do that (you don't end up with a smaller version of the original, you just end up with a triangle with no void in the middle). Your example would be more like the circle example, when you scale it down by a factor of 2, you would end up with a smaller set of 3 triangles with a void between them, which if you did the math you'd find that it has a 'measure/mass' equal to 1/4 of the original (again, just like the circle example). So it has a dimension of 2.

      The thing to note is that just like the circle example, your 'tri-force' shape is not self-similar the way a line, square, cube or sierpinski triangle are. A tri-force is not made up of smaller copies of itself, it's made up of regular triangles.

  • Devilitionist
    Devilitionist Month ago

    van Koch was van coke when he make that snowflacke

  • Maurice A
    Maurice A Month ago +1

    WOW! At 3:47, I just noticed something! 😲😍
    line: 1-D --> 2^1 self-similar parts = 2
    plane: 2-D --> 2^2 self simliar parts = 4
    cube: 3-D --> 2^3 self-similar parts = 8
    sierpinski: 1.5849-D --> 2^1.5849 self-similar parts = 3

  • NyOS Gomboc
    NyOS Gomboc Month ago +2

    Sierpinski tetrahedron's Hausdorff dimension is 2, but, it's a fractal. So your integer vs. non-integer dimension is flawed.

  • Cheydinal
    Cheydinal Month ago +2

    What if the dimension is an imaginary number?

    • Harold
      Harold 28 days ago

      *FBI OPEN UP*

  • Cheydinal
    Cheydinal Month ago +1

    I don't like grids. They're coarse and irritating, and they inaccurately measure the area of two-dimensional objects everywhere

    • Vid Jančar
      Vid Jančar Month ago

      Especially when you have a circle on a grid. Annoying as hell.

  • Glossolalia Online
    Glossolalia Online Month ago

  • Cipher-Tu Productions

    god I hate patreon

  • Super Derpy Doge
    Super Derpy Doge Month ago

    Nigga that’s the triforce

  • Adrian Morgan
    Adrian Morgan Month ago

    Only thing I'm going to argue with is _"It seems to be the core differentiator between objects that arise naturally and those that are just man made."_ And I'm going to argue by quoting the epilogue to Fractal Vision by Dick Oliver (1992). _"Why does Mother Nature's work show a different geometry than our own? Aren't we nature, too? Indeed, the difference lies not so much in whose hand does the work, but in the swiftness of that hand. Linearity is the geometry of motion, of cutting of separation. The faster the saw blade, the smoother the cut. Nature, too, has her arcs and lines: the speeding orbits of the planets, the streaming trail of a light ray, the zip of a bee to the hive. Fractal geometry, on the other hand, grows from stillness, from layer upon layer of repeated joining. When humans slow down, we create lacework, Persian rugs, Baroque furniture, and Gothic palaces. That richly woven artifacts have all but disappeared from our culture is above all a reflection of its velocity. Who has time to weave?"_

  • げんまいちゃ!


  • Brendan McCabe
    Brendan McCabe Month ago +2

    The B in Benoit B. Mandelbrot stands for Benoit B. Mandelbrot

  • Jesse Ramirez
    Jesse Ramirez Month ago

    But this concept would not make sense in many instances. One instance being the fact that the Siepinski Triangle I am seeing right now, does not have any mass. You see, light doesn’t have any mass. Would it be just to label everything with mass? Nope.

  • Ash Weber-Campbell
    Ash Weber-Campbell Month ago +1

    "this is math, everything is made up" look out, you might trigger someone

  • BoostAddict
    BoostAddict Month ago

    Well yes, but actually no

  • EvenStephen
    EvenStephen Month ago +1

    10:35 nice.

  • BigManDan 543_2
    BigManDan 543_2 Month ago +1

    This channel is USclip gold

  • Eliora Ben-Gurion
    Eliora Ben-Gurion Month ago

    Also how about an object which has diverging measures of boxes it touches at different levels, meaning a slope cannot be determined?

  • MultiQuanzaAVQ
    MultiQuanzaAVQ Month ago

    that's the triforce, sir

  • I hate Drew Saddic
    I hate Drew Saddic Month ago

    Why do you feel the need to mess with my entire concept of the universe so much, and so repeatedly.
    Seriously speaking though, these videos are incredible. You’re the smartest person I know of. Keep it up

  • Minh Tran
    Minh Tran Month ago

    So to recap: fractal dimensions is synonymous with "fractional dimensions", an approach to make sense of 'real-valued dimensions' (e.g., 1.5-dimensional objects) by thinking about dimensions in terms of how scaling affects that object.

  • Angel Alegria
    Angel Alegria Month ago

    odd1sout sent me...

    i think

  • Agent Stache
    Agent Stache Month ago +1

    What would you do in the case of something whose apparent dimension doesn’t stabilize? Like what if the tightly wound helix you showed wasn’t actually a line wound around an axis but a much tighter helix wound around an axis, and that helix was made from an even tighter helix and so on? Wouldn’t it oscillate between appearing one dimensional and two dimensional? If you can’t define something’s fractal dimension what does that mean for it? Would there be another dimension you could define? What if you instead defined the original helix as having a helical axis that’s almost a line and consistent with whatever the scale factor between each helix is? Could you exploit its self similarity to define a fractal dimension for it?

  • The Prophet
    The Prophet Month ago

    When I first found out about fractional dimensions I assumed it was a misnomer for a way to compare how coarse an item is (maybe for designing rovers or for sandpaper and such, but just more encompassing) but this actually makes a lot of sense.
    I don't know how unusual this is, but I've never been able to visualize complicated ideas using real-world examples, so I'm glad you don't shy away from more abstract/purely mathematical concepts

  • Corrado Campisano
    Corrado Campisano Month ago

    @13:38 well, the shape would come with its own resolution, which I'd consider the "max one" and play scale down, not up

  • Erick Marín
    Erick Marín Month ago +1

    What would happen if you had an object like the one described that changes from 1 to 2 dimensional the finer grid you use, but by definition, shows that behavior infinitely, changing from spiral to tube the closer you look at it?
    How would you resolve for its dimension?

  • Cheryl Robishaw
    Cheryl Robishaw Month ago +11

    My brain hurts quite seriously, but that’s my fault. Great job on explaining it, even tho my last brain cells can’t comprehend it still 😞

  • Matthew Mo Zhao
    Matthew Mo Zhao Month ago

    That rough pi is in pain

  • Yuval
    Yuval Month ago

    I'd love to see a video or just have some info on the tools you use to create these stunning videos!
    Great job as always (:

    Edit: for other curious cookies -

  • Martin Nolte
    Martin Nolte Month ago

    I am


  • Richard Sammons
    Richard Sammons Month ago

    Is it just me that has a problem with him saying we made up meth. From my understanding we did not come up with math the universe did, we just stumbled upon it.

  • Ωρξφπβ ζλψυσ

    The Earth is a fractal, a plane infinite

  • Danny
    Danny Month ago

    1:10 wtf im crying and shidding and farding rn

  • Dimitrios Fotopoulos

    I wish I could understand what you mean but I am way to young and that you are talking in english makes it even harder because I am not a native speaker

  • The Kitten Gamer
    The Kitten Gamer Month ago

    0:50 infinite triforces yay

  • Rolling_Mellons
    Rolling_Mellons Month ago


  • mediocre ice rink parodies

    "A line, a square, a cube..."

    "And a Sierpinski triangle"

    • heather pennington
      heather pennington Month ago

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

  • Stupid Doll Changeling Doll

    nice disk

  • Stupid Doll Changeling Doll

    fractals be like x = x^x

  • soulslicer 42
    soulslicer 42 2 months ago

    With this logic could you backtrack to create new fractals

  • Immort47
    Immort47 2 months ago

    So does that imply that if you had a fractal that is some curve with dimension 2, then that curve fills an area?

  • Robert Watson
    Robert Watson 2 months ago

    Hasn't this got something to do with Slartibartfast?

    • Francis Kolárik
      Francis Kolárik Month ago +1

      It's all the fiddly bits and fjords, isn't it. Much rough, very fractal.

  • Biednymaniek
    Biednymaniek 2 months ago

    How much boxes Tuch an other Amplitude?

  • Spice Master II
    Spice Master II 2 months ago

    But, the boxes themselves are 2 dimensional. Does this mean the dimension is 1.5 or 1.2 compared to 2 dimensional "box"?

  • Prénom Nom
    Prénom Nom 2 months ago


  • Prénom Nom
    Prénom Nom 2 months ago

    I knew nothing of this and it is great

  • JonahDimes
    JonahDimes 2 months ago

    You just got your 1,806,148th subscriber!

  • Pizza Gaming
    Pizza Gaming 2 months ago +1

    The Sierpinski triangle does not have mass?

  • Vadim Khudyakov
    Vadim Khudyakov 2 months ago

    What is the dimention of final pi-fractal?

  • Socks With Sandals
    Socks With Sandals 2 months ago

    19:25 Didn't you get the answers the wrong way round? Natural roughness is not self-similar at different scales.

  • David Wilkie
    David Wilkie 2 months ago

    "Zooming in in Calculus is smooth", ay? An idea I've never thought of because I learned Perspective Drawing at about the same time as the "mathemagically" transformative nature of The Calculus, expanding or contracting to functional relationships like series formulae that have no physical concept of size.

  • Vawlpe
    Vawlpe 2 months ago

    Quick question, can we reverse the process of getting that D value? As in, can you choose a random number and create a fractal from it using this fractal dimension concept?

  • Streichel
    Streichel 2 months ago

    I can smell smoke while watching this i might stop functioning soon

  • paper2222
    paper2222 2 months ago +7

    yeah but in terms
    if you were to live in a fractal
    what "is" 1.5 dimension?
    like, do you see half a y axis?

    • Nutik Wulf
      Nutik Wulf Month ago +1

      paper2222 I think at this point it’s a different definition of dimension. It would be impossible for a 1.5D object irl, maybe even computationally. What would that .5 be?

  • Rr Ii
    Rr Ii 2 months ago

    Is there a self similar limit as to be restricted with geometric functions as their base shape ?