# Fractals are typically not self-similar

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- 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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Ryan McAdamsDay agowhat dimension is a Fibonacci flower spiral

mitch dorsmanDay ago10:42 nice

KumPupp2 days agoSo, class, four shapes: a line, a square, a cube and The Sierpinski Triangle!

God3 days agoBruh, i had a math test about logarithms last week, and i didn't fail because of this video

Sumit Modak4 days ago^{+2}you must get a patent for this explaination and also for making the pi fractal.

amoura saliha5 days agothank you very much , i have a part of my Phd about fractal , nice expalnation , thank you again

LemonadeCat :38 days agoYour name should actually be 4blue1brown

1.8 days ago10:40

Touching 69

What are the chances

Sara Kim11 days agoCould someone explain me y a line,square and cube aren’t fractals? What’s the definition of a fractal?

Pluto9 days agoSara Kim A fractal is a shape with a non-integer dimension. He says in the video.

TheEgglet14 days ago^{+1}what dimension is the mandelbrot set?

Will Cahill14 days agoCan someone show me how the curve of scaling factor converges to 4, like the fancy maths way of saying it

Shruggz Da Str8-Faced Clown15 days ago1.585 just happens to be 3-sqrt2.

Suschi16 days agoBut whats about dimensions bigger than 3? Like time or anything else?

otherlego16 days agotest

Glitched Blox17 days agowat

Reazul Ranga17 days agoMind-blowing..!!

Prashant Dahiya17 days agoI think we are just playing

Conor Clements17 days ago10:35 nice

Myrmidon22 days ago15:33 When you ask an engineer the value of pi.

Qaim Alvi22 days agohello people commenting here. Please be my friend. I hv trouble discussing any of this with my current friends and feel intellectually unfulfilled :(

Gogole PI Mukherjee23 days agoAgain a awesome work. wonderful .

sarahannkathrin23 days agoThis helped me so much!!! I study Geology and I couldn’t wrap my head around the concept of self similarity and dimensions 🙃

Noah Abraham24 days agoYou be reported for touching boxes....

Arjab De26 days agoBut tell me how to calculate or get scaling factor?

1027 days agoWhy is everyone a pi symbol

Sam Eldridge27 days ago^{+1}r/woahdude

nic k.27 days ago^{+1}10:35

everybody:

kevin: ( ͡° ͜ʖ ͡°)

1infinity8 1i828 days ago15:09 I think his mic change

AwesomeCreeperBD29 days agoI 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 :)

ThatSpicyDragonMonth agoZips law 11:42

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

Bloky555Month agoI'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 ihrsMonth ago10:35 nice

Lycanroc DuskMonth ago15:05

Well this is just annoying.

DiggiMonth agoSo, britain is a fractal?

Arunanshu BiswasMonth ago@ 10:37 math concluded me towards 69!!

Anyways, the Video and Grant, both are awesome!

Keep up the Good Work!

Ian ZaineaMonth 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 DangerpeaMonth ago10:36 Nice

Skeleton MasherMonth ago^{+1}200 IQ... trying to find 1.5 dimensions, not the 4th dimension..

Forest DewberryMonth ago^{+1}Well, the sierpinsky triangle is man-made and doesn't have an integer dimension.

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

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

Robert PembertonMonth agoNope, fractals are 2D, all of them exist on an x-y plane

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

Menya SavutMonth agoTerribly annoying background piano. Need to apply Fourier transform to get rid of it.

Aoi TanloresMonth agoas an asian i give this guy reezpecc cuz i can't understand the complex word

Jan SegreMonth agoSo, 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}Month agoGoing back to videos older than 1 year to find memes...

nice 10:36

MalizmaMonth ago10:35 number of boxes are nice

Daffa_FMMonth ago2d fractal: 1.xx dimensions

3d fractal: 2.xx dimensions

4d fractal: 3.xx dimensions

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

Somebody very niceMonth ago^{+2}10:38

Oh yes, it's all coming together.

John BaldwinMonth agoFractals 1/0

YT InzAniTixMonth agothe start is satisfying asf

An_Annoying_CatMonth agoWe really need to make up new terms to describe these concepts.

The pathetic oneMonth ago4:08 should've made the yellow ones brown. missed opportunity!

Ami SchaeferMonth 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 DannyMonth agoi love this video honestly! it gives so much perspective of the world around us

pricken2000Month ago^{+1}I'm against this because dimensions aren't shapes they are axis

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

MLN ReidMonth ago^{+1}One of the first programs I had to write for class was creating a Mandelbrot Set!!