Fractals are typically not self-similar

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• Published: 27 January 2017
• An explanation of fractal dimension.
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.

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• 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  3 days back

Norway with its "lovely crinkly edges".

• 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!

• 10:37 Nice

• 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  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  4 days back

• Sam Shrivastava  6 days back

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"

• GeWind Gaming  6 days back

10:40 nice

• lone wolf  6 days back

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

• Hans Dampf  6 days back

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

• uM3  6 days back

.....ok

• 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  6 days back

10:40 nice

• 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  7 days back

Thx for the brainfuck!

• 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  7 days back

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

• 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  1 weeks back

10:35 nice

• Razor M  1 weeks back

just plugged 2^1.585, it equals 3 lol

• 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  1 weeks back

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

• Tejas Kundapur  1 weeks back

Anybody know any examples for 2.5 dimensional object/shape?

• jursamaj  5 days back

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

• Clare Kuehn  1 weeks back

Wow! Thank you. 😉😍

• Enrique Piatti  1 weeks back

• Pablo Repetto  1 weeks back

"Mathematicians are clearly making stuff up"

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

• B-ichi the tsundere  1 weeks back

10:36 nice

• Danial Ahmed  1 weeks back

Its really seems supernatural

• Doctor Medkit  2 weeks back

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

• sans the skeleton  2 weeks back

10:38 NICE !

• Axel Andru  2 weeks back

So you research/work 18 and sleep 6 ?

• jake warren  2 weeks back

Boxes Touching: Nice

• 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  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  1 weeks back

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

• 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  2 weeks back

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

• 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  6 days back

• FohenLeuer  1 weeks back

π?

• Sum Arbor  2 weeks back

"A New Science" - Stephen Wolfram.

• typhoonfire8  2 weeks back

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

• 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  3 weeks back

“How many boxes have you touched?”

• 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  4 weeks back

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

• PerfectArmonic  4 weeks back

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

• 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  4 weeks back

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

• jamma246  4 weeks back

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

It's Benoit B. Mandelbrot.

• Mayo naise  4 weeks back

10:37

Nice