# Fractals are typically not self-similar

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• Published on Jan 27, 2017
• An explanation of fractal dimension.
Brought to you by you: 3b1b.co/fractals-thanks
And by Affirm: www.affirm.com/
Music by Vince Rubinetti: soundcloud.com/vincerubinetti/riemann-zeta-function
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 Day ago

what dimension is a Fibonacci flower spiral

• mitch dorsman Day ago

10:42 nice

• KumPupp 2 days ago

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

• 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 4 days ago +2

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

• 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 8 days ago

Your name should actually be 4blue1brown

• 1. 8 days ago

10:40
Touching 69
What are the chances

• 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 9 days ago

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

• TheEgglet 14 days ago +1

what dimension is the mandelbrot set?

• 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

• 1.585 just happens to be 3-sqrt2.

• Suschi 16 days ago

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

• otherlego 16 days ago

test

• Glitched Blox 17 days ago

wat

• Reazul Ranga 17 days ago

Mind-blowing..!!

• Prashant Dahiya 17 days ago

I think we are just playing

• Conor Clements 17 days ago

10:35 nice

• Myrmidon 22 days ago

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

• 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 23 days ago

Again a awesome work. wonderful .

• 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 24 days ago

You be reported for touching boxes....

• Arjab De 26 days ago

But tell me how to calculate or get scaling factor?

• 10 27 days ago

Why is everyone a pi symbol

• Sam Eldridge 27 days ago +1

r/woahdude

• nic k. 27 days ago +1

10:35
everybody:
kevin: ( ͡° ͜ʖ ͡°)

• 1infinity8 1i8 28 days ago

15:09 I think his mic change

• 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 Month ago

Zips law 11:42

• Ghost Buster Month ago

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

• 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 Month ago

10:35 nice

• Lycanroc Dusk Month ago

15:05
Well this is just annoying.

• Diggi Month ago

So, britain is a fractal?

• 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 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 Month ago

10:36 Nice

• Skeleton Masher Month ago +1

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

• Forest Dewberry Month ago +1

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

• 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 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 Month ago

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

• Menya Savut Month ago

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

• Menya Savut Month ago

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

• Aoi Tanlores Month ago

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

• 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} Month ago

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

nice 10:36

• Malizma Month ago

10:35 number of boxes are nice

• Daffa_FM Month ago

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

• Blaise Vignon Month ago

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

• Somebody very nice Month ago +2

10:38
Oh yes, it's all coming together.

• John Baldwin Month ago

Fractals 1/0

• YT InzAniTix Month ago

the start is satisfying asf

• An_Annoying_Cat Month ago

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

• The pathetic one Month ago

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

• 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 Month ago

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

• pricken2000 Month ago +1

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

• An_Annoying_Cat Month ago

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

• MLN Reid Month ago +1

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