# 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 • 2 739

• bazoo513 14 hours ago

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

• jamma246 Day ago

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

• Mayo naise Day ago

10:37
Nice

• Patriot513 Day ago

I’m high school I never thought I would ever find math fascinating, great video

• Edric Ndirangu 3 days ago

Tri triangle= tri force

• Singh Naveen 4 days ago

Hey grant
Big fan
Which software do you use?

• NoriMori 4 days ago

13:19 You're missing a box near Cardiff! And one near Glenluce!

• NoriMori 4 days ago

At 5:11, it clicked! The dimension of a self-similar shape is the logarithm of the mass scaling factor to the base of the scaling factor! It makes so much sense!

• William Callender 5 days ago

10:37 Nice.

• Tommye K 5 days ago +1

M Y. M I N D. I S. F U C K I N G. B. L. O. W. N.

• Dont understand all. But is amazin

• 나노별 6 days ago

This is just amazing... Thank you.

• What a terrible channel! I'm an applied mathematician, and I assure you that the similarity coefficient is not the same as the dimension. These are different concepts, and one does not depend on the other.

• Какой ужасный канал! Я прикладной математик, и уверяю вас, что коэффициент подобия это не тоже самое, что размерность. Это разные понятия, и одно не зависит от другого.

• devin devine 7 days ago

In between dimensions.
Like with flatland--if we imagine flatland being a piece of paper, that paper has some depth--it's partially extended into 3 dimensions. Meanwhile, here in the 3-d, sometimes it rains frogs. Unexplained phenomena occur because we are partially extended into the 4-d.
The math, showing us that a shape can have a fractional dimension, seems like a clue.
No?

• TheJ4RyD 7 days ago

This seems so pointless and stupid

• Astatine 8 days ago

10:37
NICE

• Anders Termansen 8 days ago

OP needs more sex.

• BGMC 16 8 days ago

#2*1.5=3

• vebrun 8 days ago

10:37 "69"
N I C E

• Dario Vurchio 9 days ago

One of the most interesting videos on USclip *EVER*

• Olivier Nusbaumer 9 days ago

If I scale a "normal" triangle by 1/2, its mass is also scaled by 1/3... so what is so special about Sierpinski fractal triangle ??

• Francisco Martínez 10 days ago

i love this video! such a great and clear explanation about fractals...thanks!!!

• Hank S 12 days ago

Cantor set has a dimension less than 1. D=(ln 2)/(ln 3) approximately 0.631

• Konstantin Kuzminykh 13 days ago

so that also means usual shapes are fractals with integer dimensions=) 15:10 in a more deep thinking we like 1,2,3 numbers so we like lines squares cubes more then map of england))) thats very natural

• Smuutti 13 days ago

11:00 2*2 is approximately 3.39?

• Ami Schaefer 14 days ago +1

Anybody how knows how to calculate the Dimension by oneself? I mean with a Programm like he used or sth. like that?🤔🙏🏼

• [수리 TV] Eagle Jr. 16 days ago

6:51 Is D log2(3) so it isn't a rational number?
Edit: wait, nvm, he said it.

• Lucas Romano 16 days ago

At 2:52 when he says that math is made up, does that sentence means that math is invented by humans? Becuase some people claim that we just found it, it already existed. I just want to know the opinion of the guy who makes this videos

• Bash Gamer 18 days ago

So you are Pi right?

• cjsturgis100 18 days ago

What about a 2.45 dimensional object

• When will it end 21 day ago +1

Who is acc here to learn

• AaronThe Proj 21 day ago +2

I literally exploded in my mind

My brother still considers the coastline of Brian is 2 dimensional

• Jimmy from Philly 22 days ago

Ok so I'm tripping my face off and this isn't helping,lol

• Abdul Haseeb 24 days ago +15

Me at 2:00
Either I dont know what a dimension is or he is gone crazy

• AAALE 24 days ago

21:25 Now calcolate the dimension.

• Saurabh Basu 26 days ago

can you do a video on Green's functions

• mint CHILL 26 days ago

Which music is used in background?

• brett knoss 26 days ago +1

Would an example of changing dimensions be, a flat plot of land as 2 dimensional, on a 3 dimensional globe?

• Icestrike411 Cubing 27 days ago +1

• Kev Alan 27 days ago +3

Would you consider making videos without the background music? There is a flaw in my brain that makes me about 90% less intelligent whenever there is a background track. :-P

• Discombobulated 28 days ago

Hey you have my eye as your icon :D

• todessushi 28 days ago

When it comes to the real world if you go to even higher scaling factors you will end up being in the quantum world where nothing is defined exact anymore.

• TheRubyGames 28 days ago

0:49 no! THATS A TRIFORCE

• Daniel Goetz 29 days ago

This makes so much sense now! I put A LOT of incorrect information of my 7th grade math fair 2 years ago

• Gaspard Berthelier Month ago

but doesn't dividing by 2 the lenghts of a "normal" triangular shape (3 triangles put together with a triangular void in the middle) also give a division of area by 3 ? Would that mean that its dimension is also log(2,3) ?

• Gaspard Berthelier 29 days ago

@phiefer3 oh that's right thanks

• phiefer3 29 days ago

No, because you are not scaling that shape down when you do that (you don't end up with a smaller version of the original, you just end up with a triangle with no void in the middle). Your example would be more like the circle example, when you scale it down by a factor of 2, you would end up with a smaller set of 3 triangles with a void between them, which if you did the math you'd find that it has a 'measure/mass' equal to 1/4 of the original (again, just like the circle example). So it has a dimension of 2.

The thing to note is that just like the circle example, your 'tri-force' shape is not self-similar the way a line, square, cube or sierpinski triangle are. A tri-force is not made up of smaller copies of itself, it's made up of regular triangles.

• Devilitionist Month ago

van Koch was van coke when he make that snowflacke

• Maurice A Month ago +1

WOW! At 3:47, I just noticed something! 😲😍
line: 1-D --> 2^1 self-similar parts = 2
plane: 2-D --> 2^2 self simliar parts = 4
cube: 3-D --> 2^3 self-similar parts = 8
sierpinski: 1.5849-D --> 2^1.5849 self-similar parts = 3
DID ANYONE ELSE NOTICE THIS???😍😍😍

• NyOS Gomboc Month ago +2

Sierpinski tetrahedron's Hausdorff dimension is 2, but, it's a fractal. So your integer vs. non-integer dimension is flawed.

• Cheydinal Month ago +2

What if the dimension is an imaginary number?

• Harold 28 days ago

*FBI OPEN UP*

• Cheydinal Month ago +1

I don't like grids. They're coarse and irritating, and they inaccurately measure the area of two-dimensional objects everywhere

• Vid Jančar Month ago

Especially when you have a circle on a grid. Annoying as hell.

• Glossolalia Online Month ago

• god I hate patreon

• Super Derpy Doge Month ago

Nigga that’s the triforce

• Adrian Morgan Month ago

Only thing I'm going to argue with is _"It seems to be the core differentiator between objects that arise naturally and those that are just man made."_ And I'm going to argue by quoting the epilogue to Fractal Vision by Dick Oliver (1992). _"Why does Mother Nature's work show a different geometry than our own? Aren't we nature, too? Indeed, the difference lies not so much in whose hand does the work, but in the swiftness of that hand. Linearity is the geometry of motion, of cutting of separation. The faster the saw blade, the smoother the cut. Nature, too, has her arcs and lines: the speeding orbits of the planets, the streaming trail of a light ray, the zip of a bee to the hive. Fractal geometry, on the other hand, grows from stillness, from layer upon layer of repeated joining. When humans slow down, we create lacework, Persian rugs, Baroque furniture, and Gothic palaces. That richly woven artifacts have all but disappeared from our culture is above all a reflection of its velocity. Who has time to weave?"_

• 凄く合点がいった。分かりやすかった！

• Brendan McCabe Month ago +2

The B in Benoit B. Mandelbrot stands for Benoit B. Mandelbrot

• Jesse Ramirez Month ago

But this concept would not make sense in many instances. One instance being the fact that the Siepinski Triangle I am seeing right now, does not have any mass. You see, light doesn’t have any mass. Would it be just to label everything with mass? Nope.

• Ash Weber-Campbell Month ago +1

"this is math, everything is made up" look out, you might trigger someone

• BoostAddict Month ago

Well yes, but actually no

• EvenStephen Month ago +1

10:35 nice.

• BigManDan 543_2 Month ago +1

This channel is USclip gold

• Eliora Ben-Gurion Month ago

Also how about an object which has diverging measures of boxes it touches at different levels, meaning a slope cannot be determined?

• MultiQuanzaAVQ Month ago

that's the triforce, sir

• I hate Drew Saddic Month ago

Why do you feel the need to mess with my entire concept of the universe so much, and so repeatedly.
Seriously speaking though, these videos are incredible. You’re the smartest person I know of. Keep it up

• Minh Tran Month ago

So to recap: fractal dimensions is synonymous with "fractional dimensions", an approach to make sense of 'real-valued dimensions' (e.g., 1.5-dimensional objects) by thinking about dimensions in terms of how scaling affects that object.

• Angel Alegria Month ago

odd1sout sent me...

i think

• Agent Stache Month ago +1

What would you do in the case of something whose apparent dimension doesn’t stabilize? Like what if the tightly wound helix you showed wasn’t actually a line wound around an axis but a much tighter helix wound around an axis, and that helix was made from an even tighter helix and so on? Wouldn’t it oscillate between appearing one dimensional and two dimensional? If you can’t define something’s fractal dimension what does that mean for it? Would there be another dimension you could define? What if you instead defined the original helix as having a helical axis that’s almost a line and consistent with whatever the scale factor between each helix is? Could you exploit its self similarity to define a fractal dimension for it?

• The Prophet Month ago

When I first found out about fractional dimensions I assumed it was a misnomer for a way to compare how coarse an item is (maybe for designing rovers or for sandpaper and such, but just more encompassing) but this actually makes a lot of sense.
I don't know how unusual this is, but I've never been able to visualize complicated ideas using real-world examples, so I'm glad you don't shy away from more abstract/purely mathematical concepts

• Corrado Campisano Month ago

@13:38 well, the shape would come with its own resolution, which I'd consider the "max one" and play scale down, not up

• Erick Marín Month ago +1

What would happen if you had an object like the one described that changes from 1 to 2 dimensional the finer grid you use, but by definition, shows that behavior infinitely, changing from spiral to tube the closer you look at it?
How would you resolve for its dimension?

• Cheryl Robishaw Month ago +11

My brain hurts quite seriously, but that’s my fault. Great job on explaining it, even tho my last brain cells can’t comprehend it still 😞

• Matthew Mo Zhao Month ago

That rough pi is in pain

• Yuval Month ago

I'd love to see a video or just have some info on the tools you use to create these stunning videos!
Great job as always (:

Edit: for other curious cookies - www.3blue1brown.com/faq#manim

• Martin Nolte Month ago

I am

convinced

• Richard Sammons Month ago

Is it just me that has a problem with him saying we made up meth. From my understanding we did not come up with math the universe did, we just stumbled upon it.

• The Earth is a fractal, a plane infinite

• Danny Month ago

1:10 wtf im crying and shidding and farding rn

• I wish I could understand what you mean but I am way to young and that you are talking in english makes it even harder because I am not a native speaker

• The Kitten Gamer Month ago

0:50 infinite triforces yay

• Rolling_Mellons Month ago

Amazing

• "A line, a square, a cube..."

"And a Sierpinski triangle"

• heather pennington Month ago

"A line, a square, a cube, wand a Sierpinski triangle walk into a bar."

• nice disk

• fractals be like x = x^x

• soulslicer 42 2 months ago

With this logic could you backtrack to create new fractals

• Immort47 2 months ago

So does that imply that if you had a fractal that is some curve with dimension 2, then that curve fills an area?

• Robert Watson 2 months ago

Hasn't this got something to do with Slartibartfast?

• Francis Kolárik Month ago +1

It's all the fiddly bits and fjords, isn't it. Much rough, very fractal.

• Biednymaniek 2 months ago

How much boxes Tuch an other Amplitude?

• Spice Master II 2 months ago

But, the boxes themselves are 2 dimensional. Does this mean the dimension is 1.5 or 1.2 compared to 2 dimensional "box"?

• Prénom Nom 2 months ago

1.46

• Prénom Nom 2 months ago

I knew nothing of this and it is great

• JonahDimes 2 months ago

You just got your 1,806,148th subscriber!

• Pizza Gaming 2 months ago +1

The Sierpinski triangle does not have mass?

• Vadim Khudyakov 2 months ago

What is the dimention of final pi-fractal?

• Socks With Sandals 2 months ago

19:25 Didn't you get the answers the wrong way round? Natural roughness is not self-similar at different scales.

• David Wilkie 2 months ago

"Zooming in in Calculus is smooth", ay? An idea I've never thought of because I learned Perspective Drawing at about the same time as the "mathemagically" transformative nature of The Calculus, expanding or contracting to functional relationships like series formulae that have no physical concept of size.

• Vawlpe 2 months ago

Quick question, can we reverse the process of getting that D value? As in, can you choose a random number and create a fractal from it using this fractal dimension concept?

• Streichel 2 months ago

I can smell smoke while watching this i might stop functioning soon

• paper2222 2 months ago +7

yeah but in terms
if you were to live in a fractal
what "is" 1.5 dimension?
like, do you see half a y axis?

• Nutik Wulf Month ago +1

paper2222 I think at this point it’s a different definition of dimension. It would be impossible for a 1.5D object irl, maybe even computationally. What would that .5 be?

• Rr Ii 2 months ago

Is there a self similar limit as to be restricted with geometric functions as their base shape ?