# Fractals are typically not self-similar

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**Published on Jan 27, 2017**- 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.

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bazoo51314 hours agoA *very* clear explanation of fractional dimensionality, the best I have seen.

jamma246Day agoDo you know what the B. stands for in Benoit B. Mandelbrot?

It's Benoit B. Mandelbrot.

Mayo naiseDay ago10:37

Nice

Patriot513Day agoI’m high school I never thought I would ever find math fascinating, great video

Edric Ndirangu3 days agoTri triangle= tri force

Singh Naveen4 days agoHey grant

Big fan

Which software do you use?

NoriMori4 days ago13:19 You're missing a box near Cardiff! And one near Glenluce!

NoriMori4 days agoAt 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 Callender5 days ago10:37 Nice.

Tommye K5 days ago^{+1}M Y. M I N D. I S. F U C K I N G. B. L. O. W. N.

JLR O.o. Fuck yeah from Mexico.5 days agoDont understand all. But is amazin

나노별6 days agoThis is just amazing... Thank you.

Дмитрий Брилько7 days agoWhat 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.

Дмитрий Брилько7 days agoКакой 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 devine7 days agoIn 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?

TheJ4RyD7 days agoThis seems so pointless and stupid

Astatine8 days ago10:37

NICE

Anders Termansen8 days agoOP needs more sex.

BGMC 168 days ago#2*1.5=3

vebrun8 days ago10:37 "69"

N I C E

Dario Vurchio9 days agoOne of the most interesting videos on USclip *EVER*

Olivier Nusbaumer9 days agoIf 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 Martínez10 days agoi love this video! such a great and clear explanation about fractals...thanks!!!

Hank S12 days agoCantor set has a dimension less than 1. D=(ln 2)/(ln 3) approximately 0.631

Konstantin Kuzminykh13 days agoso 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

Smuutti13 days ago11:00 2*2 is approximately 3.39?

Ami Schaefer14 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 ago6:51 Is D log2(3) so it isn't a rational number?

Edit: wait, nvm, he said it.

Lucas Romano16 days agoAt 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 Gamer18 days agoSo you are Pi right?

cjsturgis10018 days agoWhat about a 2.45 dimensional object

When will it end21 day ago^{+1}Who is acc here to learn

AaronThe Proj21 day ago^{+2}I literally exploded in my mind

My brother still considers the coastline of Brian is 2 dimensional

Jimmy from Philly22 days agoOk so I'm tripping my face off and this isn't helping,lol

Abdul Haseeb24 days ago^{+15}Me at 2:00

Either I dont know what a dimension is or he is gone crazy

Martin Kocian4 days agoHaha

DragonOmnicron X4 days ago???

AAALE24 days ago21:25 Now calcolate the dimension.

Saurabh Basu26 days agocan you do a video on Green's functions

mint CHILL26 days agoWhich music is used in background?

brett knoss26 days ago^{+1}Would an example of changing dimensions be, a flat plot of land as 2 dimensional, on a 3 dimensional globe?

Icestrike411 Cubing27 days ago^{+1}almond bread

Kev Alan27 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

Discombobulated28 days agoHey you have my eye as your icon :D

todessushi28 days agoWhen 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.

TheRubyGames28 days ago0:49 no! THATS A TRIFORCE

Daniel Goetz29 days agoThis makes so much sense now! I put A LOT of incorrect information of my 7th grade math fair 2 years ago

Gaspard BerthelierMonth agobut 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 Berthelier29 days ago@phiefer3 oh that's right thanks

phiefer329 days agoNo, 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.

DevilitionistMonth agovan Koch was van coke when he make that snowflacke

Maurice AMonth 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 GombocMonth ago^{+2}Sierpinski tetrahedron's Hausdorff dimension is 2, but, it's a fractal. So your integer vs. non-integer dimension is flawed.

CheydinalMonth ago^{+2}What if the dimension is an imaginary number?

Harold28 days ago*FBI OPEN UP*

CheydinalMonth ago^{+1}I don't like grids. They're coarse and irritating, and they inaccurately measure the area of two-dimensional objects everywhere

Vid JančarMonth agoEspecially when you have a circle on a grid. Annoying as hell.

Glossolalia OnlineMonth agoCipher-Tu ProductionsMonth agogod I hate patreon

Super Derpy DogeMonth agoNigga that’s the triforce

Adrian MorganMonth agoOnly 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?"_

げんまいちゃ!Month ago凄く合点がいった。分かりやすかった！

Brendan McCabeMonth ago^{+2}The B in Benoit B. Mandelbrot stands for Benoit B. Mandelbrot

Jesse RamirezMonth agoBut 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-CampbellMonth ago^{+1}"this is math, everything is made up" look out, you might trigger someone

BoostAddictMonth agoWell yes, but actually no

EvenStephenMonth ago^{+1}10:35 nice.

BigManDan 543_2Month ago^{+1}This channel is USclip gold

Eliora Ben-GurionMonth agoAlso how about an object which has diverging measures of boxes it touches at different levels, meaning a slope cannot be determined?

MultiQuanzaAVQMonth agothat's the triforce, sir

I hate Drew SaddicMonth agoWhy 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 TranMonth agoSo 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 AlegriaMonth agoodd1sout sent me...

i think

Agent StacheMonth 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 ProphetMonth agoWhen 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 CampisanoMonth 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 MarínMonth 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 RobishawMonth 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 ZhaoMonth agoThat rough pi is in pain

YuvalMonth agoI'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 NolteMonth agoI am

convinced

Richard SammonsMonth agoIs 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.

Ωρξφπβ ζλψυσMonth agoThe Earth is a fractal, a plane infinite

DannyMonth ago1:10 wtf im crying and shidding and farding rn

Dimitrios FotopoulosMonth agoI 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 GamerMonth ago0:50 infinite triforces yay

Rolling_MellonsMonth agoAmazing

mediocre ice rink parodiesMonth ago^{+2}"A line, a square, a cube..."

"And a Sierpinski triangle"

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

Stupid Doll Changeling DollMonth agonice disk

Stupid Doll Changeling DollMonth agofractals be like x = x^x

soulslicer 422 months agoWith this logic could you backtrack to create new fractals

Immort472 months agoSo does that imply that if you had a fractal that is some curve with dimension 2, then that curve fills an area?

Robert Watson2 months agoHasn't this got something to do with Slartibartfast?

Francis KolárikMonth ago^{+1}It's all the fiddly bits and fjords, isn't it. Much rough, very fractal.

Biednymaniek2 months agoHow much boxes Tuch an other Amplitude?

Spice Master II2 months agoBut, the boxes themselves are 2 dimensional. Does this mean the dimension is 1.5 or 1.2 compared to 2 dimensional "box"?

Prénom Nom2 months ago1.46

Prénom Nom2 months agoI knew nothing of this and it is great

JonahDimes2 months agoYou just got your 1,806,148th subscriber!

Pizza Gaming2 months ago^{+1}The Sierpinski triangle does not have mass?

Vadim Khudyakov2 months agoWhat is the dimention of final pi-fractal?

Socks With Sandals2 months ago19:25 Didn't you get the answers the wrong way round? Natural roughness is not self-similar at different scales.

David Wilkie2 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.

Vawlpe2 months agoQuick 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?

Streichel2 months agoI can smell smoke while watching this i might stop functioning soon

paper22222 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 WulfMonth 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 Ii2 months agoIs there a self similar limit as to be restricted with geometric functions as their base shape ?