Fractals are typically not self-similar

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.

Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld

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Various social media stuffs:

[ryanmarita-davis]

"In some ways.. fractal geometry is a rebellion against calculus."

That's just a beautiful statement.

[Nevermind]

"This is math, everything is made up"


Love this quote!!

[Shubham_pandey-nkun]

When I was in class 12th and was discussing with my friends about dimensions something like the idea of 2.5 dimension strikes in my mind. I was searching whether they exist or not and then sometime like 6 months after I discovered this video.
And this video satisfied my Curiosity.
Thanks 3 Blue 1 Brown

[NoahSpurrier]

One interesting fact about fractal measure is how it can be used to distinguish Jackson Pollock paintings from imitations. This technique achieved a 93% accuracy rate for distinguishing genuine Pollock paintings from forgeries.

[huhneat]

7:32 I just realized that the parallel of this is that the total "length" of an entire square's area is infinite and the total "volume" of its area is 0, but "area" is the only metric that will have a non-0, finite amount to measure it by

[KasabianFan]

What does the B stand for in Benoit B. Mandelbrot?



Benoit B. Mandelbrot.

[tokiWren]

Fun fact: "Mandelbrot" translates to "Almond bread" from German.

[grande]

Ah, yes, my favorite fractal!
*The Coastline of Britain*

[matheusneverplasmaman]

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

[jeanmariegrangon]

As a physics graduate, I wish that our teacher had shown us this video when he tried to teach us about fractal dimension.

[axznice]

Hey, 3Blue1Brown. You're the sole reason I had a mathematical miracle after I got a 38% in 7th grade. You're the reason I saw the beauty in math, and I'm now studying extreme math, way above my level, for fun, not for school. I already know all the material needed for the exams now. Thanks for fueling the love of math in me.

[karthikprabhu]

When you learn about a topic before you are taught in school, you see the topic as your friend and your ally rather than a nightmare how ever hard it is especially if you learnt it from 3 blue 1 brown

[karthikprabhu]

Please create a merch line with
"THIS IS MATH, EVERYTHING IS MADE UP"

[mitchg]

when i first heard it, i thought he said "In some ways, fractal geometry is a rebellion against capitalism"

[actuallyasriel]

I dabble in 3D art, and so much of how 3D art is made relies on fractal geometry. You actually start to be able to point out Voronoi fractals in textures after a while. I feel like this gave me some better sense of how it all actually works, though. "Dimension" is a control on many Blender nodes, and now I know how it actually affects the output in something of an intuitive way.

[tomimated]

Wait so is it called fractals because they are fractional

[Binyamin.Tsadik]

This channel is a Youtube anomaly. It is the best intellectual channel on youtube with a fraction of the viewers from all of the other ones (VSauce, Numberphile, Veritasium, MinutePhysics... etc)

Higher quality videos, better explanations, better animations with a fraction of the subscribers.
If you scale it up it will touch more boxes than the inverse of the other channels scaled down.

[ucasvb]

Given a random real dimension D, is there an easy way to find a fractal with that dimension?

[ConnoisseurOfExistence]

Blows my mind... I wonder how it will look like some (somewhat regularly shaped) fractal with dimension equal to pi, or e ... Can a dimension be a negative number? How about complex?

[karstenroelofs]

17:00 I kind of remember this having a correlation with String Theory where there are these supposed "hidden dimensions" that would justify the 11 something dimensions required in order for the maths to check out. Very interesting to see how certain disciplines cross over!