Fractals are typically not self-similar

  • Published: 27 January 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 • 2 817

  • whiteguywithacousticguitar

    I don't like fractals. I don't really like math either so I'm not sure why I'm watching this. Fractals freak me the fuck out.

    • shady shane
      shady shane  3 days back

      Norway with its "lovely crinkly edges".

      • Michael Fernando
        Michael Fernando  3 days back

        Teach me how to program please. Grant you make the best videos on YouTube. Knowing how difficult it is to write some of this stuff, your visual approach to all of this is as beautiful of the notation of math itself. Spectacular work!

        • Пикула Силоватељ

          10:37 Nice

          • timwins31
            timwins31  3 days back

            How do I say this. . .so based on this view, can anyone else kinda conceptualize how - maybe one of the reasons humans seem so damn inharmonious and always at odds with nature is that our consciousness is perceiving the world from a non-integer dimensional perspective. Line not from the 2nd OR 3rd or the 4th dimension specifically, but more like the 3.3th dimension - or any other non-integer dimension? And maybe the reason we see so many geometric patterns and shapes when we use psychedelics and feel that connected and harmonized feeling is because it either lowers our consciousness into the next integer dimension below or the next one above where we typically observe/perceive the cosmos? Maybe psychedelics are just making the roughness from the complexity of the cosmos just a little smoother which makes perception FAR less demanding of brain power, as in needing to run far fewer calculations and use far fewer algorithms to make sense of everything. . .

            Does any of that make even a little sense. I have a feeling there's a better way to propose the idea, but its late and I'm also very dumb.

            • Matthew Boyd
              Matthew Boyd  4 days back

              I noticed that I felt more natural when I changed my growing patterns from addition to a multiplication of phi.

              It's simple to add up 3 5 8 13 and so on, more than it is to just double. Or at least in my opinion.

              Or I just use the ratio 62% of original or +62% of the original.
              .1 .15 .25 .40 .6 1 1.6 2.6 4 6.2 10
              It repeats every 5 cycles

              • picake
                picake  4 days back

                What about 3D fractals?

                • Sam Shrivastava
                  Sam Shrivastava  6 days back

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

                  • Matheusnever1 Plasma Man

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

                    • GeWind Gaming
                      GeWind Gaming  6 days back

                      10:40 nice

                      • lone wolf
                        lone wolf  6 days back

                        space-time < geometricality < reality
                        space-time is a geometrical realization

                        • Hans Dampf
                          Hans Dampf  6 days back

                          thought about infinite spaces within finite space shortly before my brain push the reset button ^^

                          • uM3
                            uM3  6 days back


                            • Jeffrey Terleckyj
                              Jeffrey Terleckyj  6 days back

                              I always see dimensions as 1 dimension is just length, 2 dimension as length width, 3 dimension is length width hight, and 4 dimension is length width high and what ever the next thing that we can't understand is. This power thing seems to be just a coincidence. Like. It shouldn't be called dimensional, it should be called something else. Cause fractals can't be real, since in real life we have a smallest unit. So we can always measure somethings area or Volume no matter how complex.

                              • crazygame crafter
                                crazygame crafter  6 days back

                                10:40 nice

                                • Andras Libal
                                  Andras Libal  7 days back

                                  Physically you zoom into atoms so in the rigorous mathematical sense nothing in nature is a fractal, but you pointed out the more sensible approach. Thanks for the video, great way of presenting things as always.

                                  • Ryuu_OwO
                                    Ryuu_OwO  7 days back

                                    Thx for the brainfuck!

                                    • Steven Van Hulle
                                      Steven Van Hulle  7 days back

                                      What's with the 1970s-porn-movie-background-music?

                                      (Not I ever watched a 1970s porn movie, but of course you knew that.)

                                      • Micheal Rogers
                                        Micheal Rogers  7 days back

                                        The fact that you had to add that means you definitely watched a 70s porn movie.

                                    • naaviv
                                      naaviv  1 weeks back

                                      Isnt britain coast 2 dimensional? Its is finite and you can get to scales that are small enouth to see stright lines.

                                      • Hello sir
                                        Hello sir  1 weeks back

                                        10:35 nice

                                        • Razor M
                                          Razor M  1 weeks back

                                          just plugged 2^1.585, it equals 3 lol

                                          • Sprunkers
                                            Sprunkers  1 weeks back

                                            cmon matematicians we all know these are 2dimensional x any y. You're just counting boxes and pretending you found new dimensions

                                            The number seems more like base growth multiplier or seed for procedural generation

                                            • Bowling Box
                                              Bowling Box  1 weeks back

                                              My dum self legit thought that was Westeros not Britain. Jeez....

                                              • Tejas Kundapur
                                                Tejas Kundapur  1 weeks back

                                                Anybody know any examples for 2.5 dimensional object/shape?

                                                • jursamaj
                                                  jursamaj  5 days back


                                              • Clare Kuehn
                                                Clare Kuehn  1 weeks back

                                                Wow! Thank you. 😉😍

                                                • Enrique Piatti
                                                  Enrique Piatti  1 weeks back

                                                  Comine with this video to blow your mind:

                                                  • Pablo Repetto
                                                    Pablo Repetto  1 weeks back

                                                    "Mathematicians are clearly making stuff up"

                                                    Well yeah... but no. It's complicated?

                                                    • B-ichi the tsundere
                                                      B-ichi the tsundere  1 weeks back

                                                      10:36 nice

                                                      • Danial Ahmed
                                                        Danial Ahmed  1 weeks back

                                                        Its really seems supernatural

                                                        • Doctor Medkit
                                                          Doctor Medkit  2 weeks back

                                                          I love your use of shape tweening on the text. Adds a little flourish

                                                          • sans the skeleton
                                                            sans the skeleton  2 weeks back

                                                            10:38 NICE !

                                                            • Axel Andru
                                                              Axel Andru  2 weeks back

                                                              So you research/work 18 and sleep 6 ?

                                                              • jake warren
                                                                jake warren  2 weeks back

                                                                Boxes Touching: Nice

                                                                • mamuburaa
                                                                  mamuburaa  2 weeks back

                                                                  You don't think the universe is self-similar? There are certainly a lot of self-similarities.
                                                                  So isn't it plausible that any aspect that isn't might simply be repeated on a scale unobserved by us?
                                                                  I mean the macro and microverse are strikingly similar, aren't they?

                                                                  • Peter2kx
                                                                    Peter2kx  2 weeks back

                                                                    What if we had fractional time dimensions? Instead of your 3 space 1 time dimension universe, what would a 0.5, or perhaps 1.5 time dimension even be?

                                                                    • 50 Pct Amused
                                                                      50 Pct Amused  1 weeks back

                                                                      Well let's first start with what a universe with two time dimensions would look like.

                                                                  • Explain the fun away
                                                                    Explain the fun away  2 weeks back

                                                                    Would it be conceptually on the right track to say that the calm ocean picture you show is 2.0-ish dimensional because it's close to a flat plane, while the choppy ocean is higher-dimensional because it's closer to a 3D surface?

                                                                    • Gopal Prabhu Iy
                                                                      Gopal Prabhu Iy  2 weeks back

                                                                      What about the fractal dimension of the universe. What if it's 3.141...

                                                                      • timwins31
                                                                        timwins31  3 days back

                                                                        What if its more like - our consciousness is embedded in the D=pi but our physical bodies exist in the integer dimension 3? Could definitely help explain why we are so disconnect and always viewed at inharmonious and at odds with the physical world we live in, always having to manipulate and modify and dominate it. . .

                                                                      • Nikhil Kelkar
                                                                        Nikhil Kelkar  6 days back

                                                                        Styop! jdfjsdf.. sorry mad typing

                                                                      • FohenLeuer
                                                                        FohenLeuer  1 weeks back


                                                                    • Sum Arbor
                                                                      Sum Arbor  2 weeks back

                                                                      "A New Science" - Stephen Wolfram.

                                                                      • typhoonfire8
                                                                        typhoonfire8  2 weeks back

                                                                        3B1B: programmers tend to be fond of them
                                                                        psychedelic users: am I a joke to you?

                                                                        • Meezi van Houtendijk
                                                                          Meezi van Houtendijk  3 weeks back

                                                                          He says that we life in 3D but that not completely true,
                                                                          For math we life in 3D (x y z) but for physics we life in 4D ( x y z Time)

                                                                          • wahhaj musthafa
                                                                            wahhaj musthafa  3 weeks back

                                                                            “How many boxes have you touched?”
                                                                            “How many slices of bread have you eaten in your life?”

                                                                            • Richard
                                                                              Richard  3 weeks back

                                                                              So do most rough items in nature actually maintain this ratio of boxes in a grid that they touch at different levels of zoom. It doesn't seem intuitive to me that just because the ratio of boxes covered at x2 to x4 and x8 to x16 are the same that x128 to x256 would also be the same?

                                                                              • Anthony
                                                                                Anthony  4 weeks back

                                                                                So when you take hallucinogens and you see fractals you're simply observing matter that isnt tangible?

                                                                                • PerfectArmonic
                                                                                  PerfectArmonic  4 weeks back

                                                                                  Please can you show us a fractal or a shape which has the “Fibonacci Number” dimension?

                                                                                  • Desperado Shao
                                                                                    Desperado Shao  4 weeks back

                                                                                    so that's why the universe is in an regular shape and if the whole universe is right 4 dimensional, it is manmade.

                                                                                    • bazoo513
                                                                                      bazoo513  4 weeks back

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

                                                                                      • jamma246
                                                                                        jamma246  4 weeks back

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

                                                                                        It's Benoit B. Mandelbrot.

                                                                                        • Mayo naise
                                                                                          Mayo naise  4 weeks back