# Fractals are typically not self-similar

HTML-code

**Published: 27 January 2017**- An explanation of fractal dimension.

Home page: 3blue1brown.com/

Brought to you by you: 3b1b.co/fractals-thanks

And by Affirm: 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.

------------------

3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that).

If you are new to this channel and want to see more, a good place to start is this playlist: vietnamtour.mobi/user/playlist?list=PLZHQObOWTQDPHP40bzkb0TKLRPwQGAoC-

Various social media stuffs:

Twitter: twitter.com/3Blue1Brown

Facebook: facebook.com/3blue1brown/

Reddit: reddit.com/r/3Blue1Brown

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.

Norway with its "lovely crinkly edges".

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!

10:37 Nice

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.

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

What about 3D fractals?

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

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

10:40 nice

space-time < geometricality < reality

space-time is a geometrical realization

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

.....ok

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.

10:40 nice

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.

Thx for the brainfuck!

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

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

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

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

10:35 nice

just plugged 2^1.585, it equals 3 lol

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

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

Anybody know any examples for 2.5 dimensional object/shape?

https://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension

Wow! Thank you. 😉😍

Comine with this video to blow your mind: https://www.youtube.com/watch?v=qhbuKbxJsk8

"Mathematicians are clearly making stuff up"

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

10:36 nice

Its really seems supernatural

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

10:38 NICE !

So you research/work 18 and sleep 6 ?

Boxes Touching: Nice

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?

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?

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

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?

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

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. . .

Styop! jdfjsdf.. sorry mad typing

π?

"A New Science" - Stephen Wolfram.

3B1B: programmers tend to be fond of them

psychedelic users: am I a joke to you?

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)

“How many boxes have you touched?”

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

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?

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

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

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

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

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

It's Benoit B. Mandelbrot.

10:37

Nice