Fractals are typically not self-similar

[@ryanmarita-davis3339]

"In some ways.. fractal geometry is a rebellion against calculus." That's just a beautiful statement.

[@brandongroves4465]

There is no such thing as rebellion in math.

[@brandongroves4465]

Also I agree

[@matthewfarrell6822]

You know I think I like fractals

[@bitonic589]

@@brandongroves4465 I truly do enjoy dividing zero by zero.

[@andistansbury4366]

Yea! Screw calculus!

[@Nevermind445]

"This is math, everything is made up" Love this quote!!

[@nathanwagester6665]

read philosophy

[@HUGOGARCAO]

There’s an interesting question that is “would aliens understand math?” It boils down to “Is math a human concept or is is it something absolute, that would exist no matter the view point? Is math just some ground rules someone thougth of and then we noticed some interesting results from applying those rules?”

[@dataexpunged3914]

How to make every math teacher very angry and how to claim 42 as a trial answer to everything.

[@dataexpunged3914]

@@HUGOGARCAO it depends on what part of math you mean. 1+1=2 is a physical law and aliens will understand if the are intelligent. If something is a fractal is made up so the will only understand if you explain it

[@antoy384]

On politics/news/rockets/car videos, everyone is bringing up politics and news and facts and polls and new models and engine tricks in the comments, often with sources. On those videos, it’s hard to challenge/discuss the body of the argumentation. So we’re left with making comments on the form or philosophy. It’s great on one side, because it shows a very very deep knowledge is being offered to us, that’s why we wouldn’t be able to criticize/comment/reflect. But it’s sad because I don’t feel we’re competent enough to deserve the author :D

[@Tomyb15]

This channel is youtube gold.

[@fossilfighters101]

+

[@eldiagrama]

+

[@umamaheshwaranl8554]

3blue1brown:maths::pbsspacetime:physics

[@vampyricon7026]

+

[@xshortguy]

I even enjoy listening to the ad at the end of the video.

[@huhneat1076]

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

[@Dracomandriuthus]

And that's what dimensions are all about

[@AlexanderGieg]

Gabriel's horn is a 3D shape with the property of having finite volume but infinite area, so it involves a seeming paradox in that, were you to construct one physically, you could fill it with a finite amount of (idealized) paint, but you could never paint its internal wall as that'd require infinite amounts of paint. I think the principle here may be the same: a 3D object with a volume that, were you to try and disassemble it to reassemble into a perfect 2D shape consisting solely of area, would result in an infinite area -- after all, that's how many "layers" of 2D planes with thickness = 0 are contained in any volume with a thickness > 0.

[@eliasmochan]

@@AlexanderGieg Gabriel's horn has infinite area and _surrounds_ a finite volume, but the inside of the horn and the surface of the horn are different things. It is a very interesting object, but it's not a counterexample of the statement that the dimension of an object defines what thing you can measure and have a non-inifinite non-zero value.

[@bettercalldelta]

never thought of it this way, you blew my mind

[@upanshulakhani6221]

I think this is what is meant by k-volume when discussing determinants, where k is the dimension. For example, 1-volume is length, 2-volume is area, 3-volume is our regular volume, and so on. So a 2D object would only have a 2-volume and finding its 1-volume/3-volume wouldn't make sense (or give a determinant of 0; PS not very sure about this statement)

[@Shubham_pandey-nk1un]

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

[@Poyni]

I immediately dismissed this thought as complete tomfoolery

[@MrMegaMetroid]

Well you still need to differentiate spacial dimensions and fractal dimensions. Those are two completely unrelated concepts only sharing the same linguistic designation.

[@Shubham_pandey-nk1un]

@@MrMegaMetroid Yes, You are right

[@archangelofsorrow]

@@MrMegaMetroid so there are three versions of dimension

[@MrMegaMetroid]

@@archangelofsorrow there are multiple definition of the word dimension. Dimension can also mean size in some contexts. A fractal dimension has nothing to do with a spacial dimension, and a spacial dimension has nothing to do with the dimensions (size) of an object. Also, dimension can mean paralel world, which is also a completely unrelated linguistic concept that has nothing to do with any of the former. The word dimension has multiple definitions, and spacial as well as fractal dimensions are 2 definitions that are entirely unrelated to each other. That must not be confused with the difference between space and time dimension, which are different, conceptually, talk about the same field in physics though and can thus be categorized as the same linguistic umbrella

[@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.

[@ultimatedeatrix9149]

Damn that's a neat piece of fact on its application!

[@danyilbutsenko6339]

"Paintings"

[@shobacon8263]

How did they do that

[@CalebSalstrom]

@@danyilbutsenko6339 there’s always one of you

[@CarlNiemi]

@@CalebSalstrom Always one correct person?

[@grande1900]

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

[@TheAustronaut03]

the coastline of Switzerland is 0-dimensional

[@killerfishe5092]

@@TheAustronaut03 yes, that is technically true

[@TheAustronaut03]

wahoo

[@bobfs9855]

The coastline of Norway is far superior.

[@trueriver1950]

@@TheAustronaut03 Unless you include the coast of Lake Geneva, of course.

[@jeanmariegrangon]

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

[@profbbfab6211]

Our teacher's assistant did! And I thank him for that

[@KasabianFan44]

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

[@redoxred5588]

KasabianFan44 recursive acronym

[@eednb4257]

So his name is Benoit Benoit B. Mandelbrot Mandelbrot. Ah, but now it's Benoit Benoit Benoit B. Mandelbrot Mandelbrot Mandelbrot. You know where this is going.

[@alexkeil3445]

This is the most brilliant pun i have ever seen

[@blunderbus2695]

Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit Benoit... (infinitely many benoits later) Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot...

[@eyemotif]

A “Mandelbrot” set...

[@karthikprabhu3173]

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

[@samwang6515]

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

[@dangerouspie0319]

And that sucks so much. I hated school, but now I get home from work and just learn about every topic out there. School is set up to kill spirits first and educate second and I'm never going to forgive our society for doing that to kids.

[@stratowhammy]

I agree. The principle is the same for teachers too. When you want to approach a topic or example that is amazing but also need to fully conceive of it's delivery in a short amount of time; this feels analogous to students' being introduced an idea in an artificially short and high-stakes time window, and being expected to fully incorporate it's implications. The thing that fires my pistons about math(s) textbooks is that often they will break topics into discrete chunks that don't naturally flow into one another... Not only does NO ONE learn complex subject matter that way, but the combination of the two results in almost no iterative thought process skills being built. Under this kind of pressure the brain floods and it's physically impossible to absorb the material in a meaningful way. It's a lost opportunity at every level: math becomes the enemy and the amazing skill of developing an iterative thought process is never explicitly or implicitly taught through curriculum.

[@JuanLeon-oe6xe]

@@stratowhammy "bUT tAHt mEiK iTs HuRdeR aNd S0 iT tAKes MoR3 E4oRT". These "people" seriously need to hyper-complicate even the most trivial stuff to feel "superior" because they did something "difficult". Now that you mention it, separating subjects into chunks has another effect, it makes students incapable of seeing the relationships between subjects, and when one approach fails, well we're fucked. "tH1Nk oUtS1D3 DaH v0X", yeah, when everything we know is the Box (and trust me, they force us into ONLY KNOWING THE BOX), that is more of a formal proof of the impossibility of dreams and the ret@....ness of hope.

[@mohammedjafer9265]

@@dangerouspie0319 education system is a play ground for control did you really expect the benefits to outweigh their agenda...

[@matheusnever1plasmaman477]

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

[@ireallyneedalife6979]

*Ah yes, enslaved fractal*

[@ntck]

*ah* *yes,* *enslaved* *infinity*

[@GlitchedBlox]

Ah yes, Enslaved object

[@1994mrmysteryman]

@@tthung8668 Get at it. Unless you "ate" it.

[@thegoofyarchive8300]

10:39 *n i c e*

[@tokiWren]

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

[@otesunki]

Is that profile pic... from nick?

[@StarGarnet03]

The almond bread set

[@therandomshow1265]

I can see it

[@The_Tiffster]

That's just the name of the person who discovered it....

[@Tower_Swagman]

yay deustechland (sorry germans if i mispronounced or mispelled it)

[@karthikprabhu3173]

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

[@vale3242]

I need it.

[@papasscooperiaworker3649]

LOL

[@mrkun5905]

​@@papasscooperiaworker3649 DUDE

[@1ucasvb]

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

[@3blue1brown]

Indeed! This clip should give you some clue: kzbin.info/www/bejne/iIaTqIaZfs9mbNEm40s

[@michaelwulber8229]

The dimension of the Koch Snowflake approaches 2 as the theta in the seed approaches 0, right? So, depending on the theta, the dimension ranges from 1 to 2. Can this process be generalized for other dimensions? The seed of the Koch Snowflake is based on a line, but if there is a seed based on a plane could we alter one (or maybe more than one) attribute of that seed to obtain a curve that ranges from 2 to the 3 dimensions? I guess my question is that, given a seed of n dimensions, can one always obtain a curve that ranges from n to n + 1 dimensions?

[@dev02ify]

It would be interesting to see an animation of a shape going from 1d to 3d

[@NathanTAK]

Yes, but it might take a while :P.

[@1Thor61storm8]

+Michael Wulber That reminds me to Topology. Maybe there is some sort of fractal dimension topology. Written from my honest ignorance. I'm no mathematician, just a fan

[@mitchg9017]

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

[@yaiirable]

had to switch on subtitles to hear it: it was actually 'calculus'

[@gold_5600]

A rebellion against capitalism, huh?

[@raddrew42]

What part?

[@dcterr1]

I'm glad my salary isn't a fractal!

[@gamermapper]

☭ ☭ ☭ ☭ ☭ ☭ ☭ ☭ ☭ ☭ Unbreakable 🚫⛏️ Union 🔗 of Freeborn 🗽 Republics 🐘 Great 💪 Russia 🇷🇺 has welded 🤝 forever ⌚ to stand 👍. Created 🏗️ in struggle 🌋 by will of the people 🙋, United 1⃣ and mighty 👍, our Soviet ☭ land 🤝! Sing 🥇 to the Motherland 😎 , home 🏡 of the free 🆓 , Bulwark 👍 of peoples 🌍 in brotherhood 👫 strong 💪 . O Party 🏛️ of Lenin ☭ , the strength 💪 of the people 👭,To Communism's ☭ triumph 🎉 lead us 👉 on! (To be continued...)

[@wigglygrass3066]

10:42 thought you were gonna say" in the back of our minds, we know to say nice"

[@tomimated1638]

Wait so is it called fractals because they are fractional

[@CoolColourBlack]

yass, so I also thought this towards the end of this video and I was like whoaaaa.

[@fqidz]

yes

[@thekoldrex]

@@fqidz YES

[@dp271]

well duh

[@Julian-tf8nj]

Not always fractional! Check out the video description: 'The proper definition of a fractal, at least as Mandelbrot wrote it, is a shape whose "Hausdorff dimension" is greater than its "topological dimension". '

[@karstenroelofs9216]

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!

[@tygodankers6526]

jup, had the same thought

[@Dhanush-zj7mf]

Right.....

[@alek6362]

In primary school i used to sometimes doodle and once i drew that Sierpinski triangle thinking that i just invented a new design/shape Ok so I completely forgot about this comment but please stop arguing in the reply section your points are so stupid lmao

[@fabiopilnik827]

I independently invented certain finite difference methods but I was too old to think they were new.

[@GoogleAccount-pi9ct]

Lol

[@wisdom6458]

lol i remember when i was 8 and discovered that 1 + 3 + 5 + ... + 2n+1 = (n+1)^2, i couldn't prove at the time, and i just thought i had discovered something lol Good Times, when everything was way more simple :)

[@sebastianjost]

I'm wanting something doesn't have to mean that you are the first to invent it. Coming up with something new on your own is what counts, not how many people have done the same before you. Many people can solve a Rubik's cube but the proportion of people who have figured out a solution themselves is really small.

[@dipendraphuyal8914]

😂

[@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.

[@RyuichiNoGekido]

10:40 Boxes touched: 69 Noice I’m sorry, I’ll start learning now.

[@Bass_Guy]

Nice

[@nadari9162]

I was praying to see this in the comments. Thanks

[@hypertriangle6531]

Nice

[@luigivercotti6410]

lucky next guy will put the 69th like here

[@nnmk01]

69th liker here *N O I C E*

[@samshrivastava2655]

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

[@illusionist1872]

10:34 When you said "In the back of our minds," I thought you were going to make fun of how immature everyone is

[@axz0nice]

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.

[@axz0nice]

note: I learned integrals

[@wiltherdelacuesta8175]

This voice and speaking speed is perfect to undertand complex topics...Good job!!!

[@M-F-H]

can't totally agree, it makes me sleepy... and/or in fear of being hypnotized...

[@George4943]

Your videos remind me why I was a math major until I wrote my first program in 1962.

[@TheRedstoneTaco]

roasted.

[@bigbox8992]

What language did you use?

[@George4943]

FORGO (a version of FORTRAN)

[@ricardo.mazeto]

I'd like to hear you explain why or just talk more about that. I'm C programmer.

[@bigbox8992]

George Steele Ultra rare language. Nothing on the internet about it. Can you please talk to us about that time and what did you do with your coder skills?

[@trueriver1950]

19:10 ... a numerical way to represent the fact that it's way more jaggedy ... Lovely expression of the idea of roughness

[@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?

[@raphdm3776]

you can't represent a fractal with a dimension over 3 in real world

[@lakshya5946]

@@raphdm3776 This is math Everything is and can be made up

[@raphdm3776]

@@lakshya5946 I know but we will never see what it looks like

[@lakshya5946]

@@raphdm3776 we can actually

[@raphdm3776]

@@lakshya5946 I don't know about you but I only live in 3 dimensions

[@Binyamin.Tsadik]

This channel is a KZbin 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.

[@arongil]

It is true: he is a gem. Hopefully he can continue to gain more viewers.

[@SkyWKing]

Most intellectual channels are intentionally toned down to accommodate the learning capability of the general public. It is not saying the general public is stupid but that most people don't try to learn anything (even in college), but only gather information from these videos. 3B1B's channel is for those who genuinely want to learn.

[@Isilduhh]

I hope not that he gets more viewers necessarily (often comment section gets meme'd and/or becomes unanswerable by owner, style changes to fit viewership), but that a greater percentage of existing viewers donate!

[@vampyricon7026]

I feel like this is for math what PBS Spacetime is for physics.

[@Binyamin.Tsadik]

Vampyricon I like PBS Spacetime too

[@Minecraftster148790]

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

[@ilikemitchhedberg]

thx m8. Was trying to remember that one.

[@maxbuskirk5302]

Benoit Benoit Benoit Benoit Benoit . . . Benoit Benoit Benoit B Mandelbrot Mandelbrot Mandelbrot . . . Mandelbrot Mandelbrot Mandelbrot Mandelbrot Mandelbrot.

[@pranavgarg9075]

Balls

[@thomassynths]

It stands for "Bacon".

[@ntofed]

@Minecraftster148790: Then his name is 1.226-dimensional. The string "Benoit B Mandelbrot" has a length of 19. The string "Benoit Benoit B Mandelbrot Mandelbrot" has a length of 37. So, log(37)/log(19) == 1.226.

[@yulongqiu]

this is the first time I understand the fractional dimensions. thank you.

[@shantonudutta9726]

As always, you made us understand a very abstract concept in an intuitive and logical way. The concept of non-integer dimension which did not make sense some minutes ago makes so much sense now! Thank you very much and keep making such beautiful content!

[@sogidochnet9304]

A level of breaking down complicated matter into understandable chunks which is rarely seen on YT and one might have thought, wasn’t even possible - but obviously it is. Thank you!

[@mediocreicerinkparodies1099]

"A line, a square, a cube..." "And a Sierpinski triangle"

[@hebpennington]

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

[@ferencgazdag1406]

The 4th should have been a hypercube.

[@squibble311]

and a tesseract

[@sketin]

THE ARISTOCRATS!

[@zmoot]

And 4d cube.

[@jaygreenwood422]

Don’t mind me, I’m just procrastinating

[@overlordcringe2715]

And I'm popping

[@overlordcringe2715]

Pope

[@overlordcringe2715]

Avacoda

[@h-hhh]

@@overlordcringe2715 h

[@indiumlove]

Never gonna give you up, Never gonna let you down, Never gonna run around and desert you. Never gonna make you cry, Never gonna say goodbye, Never gonna tell a lie and hurt you.

[@spacekid9680]

1:03 I like that you used my home island as an example. I see that shape and instantly think "home"

[@AwesomeCreeperBD]

I remember last year I tried to create a proof for the area of Sierpinski’s triangle, and showing it to my math teacher. Now I’m watching this video and realizing I didn’t come up with anything new lol. Amazing video though, keep up the amazing work :)

[@AshrZ]

Don't underplay your achievements! What you did is still incredible and shows that you're a wonderful mathematician. Keep it up!

[@lijacom]

Car salesman: (slaps sandpaper ) "This bad boy has a really high fractional dimention"

[@dangerouspie0319]

But does it have a high frictional dimension?

[@Green24152]

@@dangerouspie0319 Yes.

[@fetterkeks2796]

(slaps Mandelbrot set): "This fractal can fit so many fractals in it"

[@mrkun5905]

Nahhh

[@krisoluich9119]

I was living in a coma until I found this channel.

[@ArchHeretic1]

you think thats bad? I was living in a comma,

[@Ayy_la]

@@ArchHeretic1 you think that's bad? I was having a srtkoe

[@ericjohnson1811]

Pretty cool, huh? : )

[@thekoldrex]

@@Ayy_la you thin thas beaa...

[@Magnesius]

@@thekoldrex z

[@tetra6354]

Another way to show that when a disk gets scaled down by 1/2, it's area gets scaled down by 1/4: The area of the original circle = πr² Since the circle gets scaled down by 1/2, it's area also gets scaled down by 1/2. => The area of the new circle = π(r/2)²=π*r²/4

[@tetra6354]

why did no one see this comment ._.

[@NhuKhiet-cr8lh]

Do you mean the area of the new circle = π(r/2)²=π*r²/4 = 1/4 old area?

[@ThanhNguyen-ni4vw]

That’s great!

[@HuyenVu-di3et]

It seems that the author wants to use visualization to demonstrate fractals, he doesn't want to use mathematical formulas

[@tetra6354]

@@NhuKhiet-cr8lh oh yea

[@pikminfan6778]

Yo dawg, I heard you like Triforces.

[@haslan4885]

triforce-ception

[@purrplaysLE]

So I put 9 triforces In a stupid WiiU game

[@goldsrcorsource2551]

so i made it and infiniteforce

[@skoto8219]

Haven't seen that meme in a while lol

[@killerfishe5092]

so i put a triforce in your triforce so you can use a triforce with your triforce

[@seppobastian]

This guy knows his stuff. And makes it very interesting :)

[@IanZainea1990]

1:14 ... I heard "Fractals are a rebellion against capitalists." ... I was intrigued but knew I had to have heard wrong! haha.

[@nevanmasterson46]

Workers, unite! you have nothing to lose but your derivatives!

[@jackdesy2127]

@@nevanmasterson46 its integral that we seize the means of production

[@Davidelombardi18]

Funny fact. Marx was a excellent mathematician that independentely to cauchy and weierstrass realized the way for give solids foundation to differential calculus even if marx didn't have the same knowledge of the two mathematicians. It was also the first economist to use math in massive way in economy.

[@segmentsAndCurves]

@@Davidelombardi18 *He was also

[@monika.alt197]

Lmao

[@JustMoseyinAround]

10:48 *"In the back of our min-... Nice"*

[@itaiefrat3286]

Serious math question: In physics the idea of dimension is usually expressed as the number of degrees of freedom needed to describe the movement of a particle. Is there a sense in which a particle moving in a fractal has a non integer number of degrees of freedom?

[@ori5021]

This is exactly what bothers me

[@fossilfighters101]

+

[@zairaner1489]

This only really works for vectorspaces (and in a weaker version for manifolds)

[@SkyWKing]

Is there any motion at all along a fractal trajectory? The distance between any pair of points on that trajectory is infinite, so it seems that motion cannot be defined.

[@nmarbletoe8210]

How about an idea of motion through the iterations. Like, take a line and iterate it to look like one side of the Koch snowflake. Start at one end, Iterate once, then take one step along the line to where it bends (at 1/3 the length). Then iterate, and take a step, etc. You'd go 1/3, then 1/9, then 1/27 length of the original line.... With this idea of motion I guess there are two degrees of freedom of motion, since you could also go backwards (except for the first step). If we imagine that instead of staying on the line, you could also jump to nearby parts of the snowflake, there might be jump distances that would give an average degrees of freedom that is not an integer. Maybe that could be done more simply, what if a particle sits on a random spot on a y shape and can move along lines to intersections or end points. If it's on an ends it has one choice of motion. If it's in the middle it has 3 choices. So on average it would have 6/4 degrees of freedom?

[@vishwas425]

Waiting for a calculus series

[@Ramzuiv]

Vishwas Dubey The calculus is coming... if you support him on Patreon you'll get access to early drafts of a few of the videos

[@Euquila]

Ya I still watch the linear algebra series from time to time just because it's so good (especially the animations!)

[@justinward3679]

I need a topology series.

[@treyshaffer]

There's a guy on KZbin that made a pretty good topology introductory video series called "What is a Manifold?". I think the channel name is XyXyXyX or something to that degree

[@rikenm]

He already has the Calculus series at Khanacademy.org. There's old one which is of sal khan and the new one is from the 3blue1Brown. I saw his Calc3 series and it was better put forward. I think he also did some of differential. here's the playlist: www.khanacademy.org/math/calculus-home/multivariable-calculus

[@christianthompson7915]

i never knew that i had such a love and interest in math until i found this channel. I also noticed that fractals are beautiful

[@hasanirtija8996]

This was better than a 2-hour graduate lecture. Thank YOU!

[@tallyhallsally]

3 dimensional: *shows square* Him: “like we live in” Me: ahh, yes. It’s Minecraft time

[@overlordcringe2715]

Fuck you

[@bin6549]

@@overlordcringe2715 Jeez, who hurt you?

[@kingaha3657]

Dude I had the best idea to incorporate these zoomed out fractal shapes as tower-bases from a bird's eye view. 100% gonna try it maybe on rust too

[@theshermantanker7043]

@@overlordcringe2715 Your username makes sense

[@real_nosferatu]

Cube*

[@tcarrotgaming1639]

If only math at school was this fun...

[@bttfish]

The school is needless

[@hugo3222]

@@bttfish Yeah, this is what people usually say when they watch a video like this. But it is an illusion. First of all, if you look more closely at the video, you notice that it basically is a random repetition of the same few animated pictures over and over again. Nothing wrong with this per se. Probably it was hard work to create them, so using them several times to make up a video is cost efficient. But it means that the "math lesson" included in the video also somehow circulates around a few (superficial) topics over and over again. This tends to make the viewers believe they understood and learned something, but did they really? For example, we saw the Sierpinski triangle so often, that we believe we know what it is. But do we? It is a set of points in the plane. After watching the video, could you tell which points? Given that two corners are at (0,0), (1,0), does the point (1/8,1/6) belong to the set? You should be able to answer this question if you claim to "know" what the Sierpinski triangle is, and you need to be able to tell if you want to reproduce anything shown in the video on our own computer. Same with the Koch curve. It's like watching an orchestra playing a symphony in the background (actually the same excerpts over and over again) and listening to a musician explaining the compostion, the instruments, and how the condictor is managing everything to make it sound well, and then saying: "If only my violin lessons where so much fun."

[@kalisticmodiani2613]

Imagine if you had to come up with proofs of these ideas like Mandelbrot did, or even come up with new ideas, you'll need to go into a lot of tedious details that this won't cover.

[@bttfish]

Hu Go at least most educations in most secondary schools destroy the interest of most students.

[@bttfish]

Kalistic Modiani At least this gives an interesting introduction.

[@Oroborus12]

3Blue1Brown, I don’t know if you read comments or not, but ever since seeing a 3D projection of a tesseract, I have long wondered what a 3D projection of a 4D Menger Sponge might look like. I can’t think of anything valuable it may have to teach, but maybe there are some other (self similar) fractals that might have interesting or insightful expressions in 3D projections of 4D space. I hope you find the idea as interesting as I do. Whether you decide to use the idea or not, thank you for content that expands the way I think.

[@ДмитроПрищепа-д3я]

For some projection it should look exactly like default Menger sponge. Other projections should look like two Menger sponges intersecting and merging with each other in some way. I'm not 100% sure about it though.

[@karolakkolo123]

In some projections it will look like a normal menger sponge. In the most extreme case where all dimensions are twisted by 45 degrees to the camera, the projection would have 3-dimensional 6-pointed stars as holes, instead of cubes. Anything in between, I can't imagine. Maybe I'll be able to program a 4d menger-sponge viewer. It would be an interesting project actually

[@karthikprabhu3173]

www.google.com/url?sa=t&source=web&rct=j&url=%23&ved=2ahUKEwiSgd79-tjnAhXnyjgGHcFrCEkQwqsBMAB6BAgHEAQ&usg=AOvVaw0VTJzWxrN8ZFOD4xbU2nov

[@shreeyamittal1771]

"I mean, this is math. Everything's made up." Duude, you just got yourself another subscriber.

[@YounesLayachi]

I've been following this channel for *roughly* 2 years now, awesome content as usual

[@YounesLayachi]

***** 1.1 dimensional

[@Gadget622]

10:37 is when I found out that I'm not mature enough to be learning this stuff

[@RealClassixX]

I have a masters in electrical engineering, and I muttered 'nice' to myself. Maturity is a lie.

[@hesiod_delta9209]

@@RealClassixX L I E S

[@kerosene_turtle4715]

Yep you are 12

[@moadot720]

@@kerosene_turtle4715 Again, LOL, FuriousLightning!!!!

[@demerion]

I know exactly what you are talking about without checking what happened at 10:37 first xD

[@willemvandebeek]

So if a tesseract is scaled down one half, the mass is scaled down 1/16th?

[@3blue1brown]

Yup!

[@willemvandebeek]

Cool, Im trying to imagine it, but having a hard time with it ^^

[@adymode]

I might say i found 'mass' a bit overloaded a term for this "measure" I'm thinking of words like: travel, visitation, flood, fill. I can spend ages searching thesaurus for names though. It is the first time ive heard of this calculation and this video does describe it wonderfuly so mass worked out fine to go on.

[@xxnotmuchxx]

A 3-color piece of a pocket cube (2x2x2) is 1/8 of the whole cube and a 4-color piece of a 2x2x2x2 is 1/16 of the whole hypercube (you can see it with Magic Cube 4D).

[@hisxmark]

Let's see, the determinant is the scaling factor of the "mass" under transformation, so... OW! ... I think I hurt my head... and if a particle and antiparticle separated in minkowski space but entangled because in lateral dimensions they are still the same particle/wave... OW! OW! ... and if the "mass" is fractal as you scale the transformation... OW! OW! OW! ... the closer I look the fuzzier it gets... OWOWOWOW!

[@YogiMcCaw]

What strikes me is that the closer you zoom in to a natural fractal (as opposed to an abstract fractal) is how the slope varies from the earlier (let's say "less zoomed in") slope. Thus, we are just arbitrarily choosing (say, from the coastline of Britain) the slope that we find the most useful AT OUR OWN SCALE OF EXISTENCE (sorry for the caps, but the youtube text editor won't let me make italics). So, in reality, the slope (and therefore the dimension of fractal-ness) actually keeps changing all the time. That's kind of mind-blowing if you meditate on it for a minute. Nature once again eludes our penchant for trying to quantify it.

[@pablorepetto2759]

"Mathematicians are clearly making stuff up" Well yeah... but no. It's complicated?

[@piggywink333boyfriend6]

Well yes, but not really

[@mlgproplayer2915]

Well yes, but actually no.

[@arthurthekyogre9155]

All words are made up, all letters are made up, all numbers are made up, every type of character you can think off is made up

[@Trix-Valrae]

@Maximal's Personal Profile Hmm... I don't particularly understand what you're approaching can you explain it in a sense of Deatil.

[@Trix-Valrae]

@Maximal's Personal Profile Yes, I can clearly see it. But THEY don't accept it because they think they are universal which is so stupid and misleading like GOD we humans created the existence of GOD to explain something that cannot be explained Questions That has no answer as of right now we call them MIRACLES but they're just a bunch of in-adequate MISUNDERSTANDINGS. But THEY just don't accept it not because it can change their perspective just because they BELIEVE it's not true and that's an OPINION not an answer. Which is why debating with these kinds of idiots are essentially WASTING time for yourself. I can agree with you That Math is just a rule that we humans made up to describe things and explain it more better but because it explains things PEOPLE think it's universal but it's NOT it!

[@grevel1376]

Yesterday I was living in 3D space. You have changed my life.

[@alexwang982]

DODO You still live in 3D space

[@piggywink333boyfriend6]

If you lived in 3D yesterday, and you posted this a year ago HOW DID YOU TIME TRAVEL

[@pe3akpe3et99]

@@piggywink333boyfriend6 you know he's living in 4 dimensions now, the fourth is time

[@piggywink333boyfriend6]

@@pe3akpe3et99 Thanks

[@nickshowman4606]

Imagine a Sierpinski Pyramid. It will break apart into 4 copies of itself, meaning a 1/2 length scale translates to a 1/4 mass scale. Since 1/4 = (1/2)^2, a Sierpinski Pyramid is 2-dimensional, yet a pyramid is 3-dimensional. ?yo what

[@valium97582]

In the same way that a Sierpinski triangle is represented as a 2-dimensional drawing (which is bigger than its own dimension), the Sierpinski pyramid you've come up with _is_ 2-dimensional: the 3-dimensional shape is just a representation of it.

[@uklu]

Read the description ;)

[@yesssint7243]

When I initially heard about fractals, I was told that a fractal is simply a figure/shape with infinite perimeter

[@minuspi8372]

Any object with dimension other than exactly 2 doesn't have a perimeter.

[@JJean64]

@@minuspi8372 yes it does. For example, the perimeter of a line is just the length of the line, and the perimeter of a cube is just the surface area of a cube

[@mirosawzotorowicz8042]

Fractal with pi dimension?

[@migram4190]

Oh what could happen?? 😲

[@theone2-three438]

I think it would be a theoretical circle

[@ham1533]

it will be something with infinite volume and 0 hyper-volume (i think, dont yell at me lol)

[@dusty6299]

maybe it will be a circle

[@dusty6299]

kzbin.info/www/bejne/kHmldZuDoKt1jbc

[@OrchidAlloy]

When you suggested programming a way to calculate the fractal dimensión, I was actually terrified.

[@mushyomens6885]

" I mean this is math. Everything's made up. " Now that I think about it. Well said ...

[@hassannabil9792]

This the best way of explaining non integer dimensions. Thanks for presenting it.

[@rileydragunas9112]

It's honestly crazy how videos like this manage to take someone like me, who wasn't even good enough to pass algebra 2 in high school, super interested in higher level math

[@cherylr.3262]

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 😞

[@Em.P14]

Kind of interresting, also one sentence i never thought i'd say: Luckily i speak enough math to be able to understand this! At this point, with all the shapes, concepts and formulas, math is less of a pure concept but more of a language, if not even a subculture on its own already (my english isn't the best so please excuse me if i can't fully express this thought understandably with the wirds i chose), also you got to learn how to speak math to fully understand it, likewise with another language, basic math is like the first few words, cat, dog, hello, thank you ..., enough to describe basic things, needs and so on and on. But then you get sentences with their rules, placement of words and you can describe things in nature, formulars in math as the counterpart. Graphs are just a way to write it down, (you speak a sentence, if you write it down you can see every piece of it from the beginning to its end and how it conected, like the graph of a forlular) also writing is also like a language of its own, funny isn't it ? Add sentences to sentences and weave them together and you get a speech (mayby not a good word at this place, i told ya, my english isn't good enough to give it enouch credit, but it should be good enouch for the gist) that can deliver a greater meaning that non cohering sentences on their own. Some people say: math is the language god has written the universe in. I say: math is just a translation of the universe xD Now i understand it, thx for that mental breakthrough. Oh and something that i found out abbout my comment if i start thinking of it, isn't it just the same as one of those shapes above ? Things consisting of itself ? Oh no then it would be just a that a word would consist out of words, WAIT THEY DO! and even if you take another approach, like not just words consisting out of smaller words and sentences out of smaller ones, also words consist out of letters that are just a fraction of the word itself ! Would be interresting to see what the dimention of that would be, or just the struggle of mathematicians trying to calculate it XD. Does this even make sence annymore or do i just get insane ? Or do i just start talking in another language ? Got to check out if vsauce got a video abbout the concept of language.

[@Em.P14]

@Yu-Chen Chang i will surely have one as my longer comments rarely get anny attention normally and it is nice to see that 6 people got interested and one even wrote back (although i expected to find a less good will comment as i read the first few lines in my newsfeed, as i do write some lazy comments when im not in the mood for correcting all of my translation mistakes xD), normally people tend to skip long comments as those look too intimidating for most people when they arent properly organized with gaps etc. I hope you got a nice day too !

[@adir6094]

bRuH that's what i try to tell everybody! math is the language of the universe! I wrote it for my math document/paper that i have to do in 12th grade, i'm doing it on the Chaos Theory and fractals so that's why i'm here, lol. But math is definitely a language. Like, an easy example would be translating the english description of speed into speed = distance / time. This usually convinces them lol.

[@akarshshrivastava3719]

I am doing DIP and this Video just helped me a lot to understand what to do in my assignment. Thanks for such great content

[@akselai]

'A dimension is a dimension, you can't say it's a half!' - TJ """Hausdorff""" Yoshi

[@matthieudeloget8998]

10:38 Oh yes, it's all coming together.

[@dartmarbleracing1762]

Who doesnt like 69 doesnt like memes... And guess what? I dont like 69 And I go Crazy And Crazy again And Crazy YET again

[@asriel522]

nice

[@lilywater3683]

Wait. If you have a shape so that if you scale it down, it’s mass increases, does that mean it exists in negative spatial dimension! That would be so cool!

[@CMDRunematti]

I have watched this at least 3 times now. It seems I watch it yearly... I don't know why but this specific idea fascinates me

[@Math_oma]

Ah, but is everything _really_ made up in math?

[@Math_oma]

+Joshua Speckman But is saying that it's a good way to model reality a good reason to think it's made up? If anything it seems to suggest that it's not just made up.

[@oldcowbb]

mathematicians made up some rules and see how the logic works, that roughly how math works. Sometime it turns out to be useful to model reality. if one create math just for the sake of modeling reality i would define that as science instead.

[@JM-us3fr]

Math is more real than reality

[@АлександрБагмутов]

For a proper discussion, firstly we need to assume some strict definitions (which is really hard in this case at the very edge of our concepts). For example more strict version of a question: Given enough time, will every possible intellectual entity, studying the universe, develope the same mathematical models as ours, or it is possible to use completely different concepts to describe universe? (I`d say they certainly will arrive at the same destination, which speaks in favour of assumption of math-universe. I can try to argue for this..) I`m interested to see other ways to formalize this intuitively natural question!

[@duckymomo7935]

There are multiple approaches to math Currently, we use axiomatic set theory and ZF axiom of choice in most/mainline math (what you learn all your life up to undergrad). There is also category theory to describe math..

[@sarahannkathrin3204]

This helped me so much!!! I study Geology and I couldn’t wrap my head around the concept of self similarity and dimensions 🙃

[@orangerthings8234]

geology is the study of rocks

[@henryzhang3961]

@@orangerthings8234 rocks are self similar, or so ive heard

[@ryanpost13]

13:21 he missed a bit of coast line, towards the bottom left

[@mitalisharma440]

the most beautiful and comprehensive explanation ever

[@TheHadMatters]

Fun factoid: The German word for "measure" is "Maß" (Pronounced "mass" with a long vowel). Clearly, the English definition of "mass" acknowledges its universal meaningfulness. ["mass" in German would be "Masse", and yes, those words are also somewhat linked in German. It's all in a way related to "maza" from ancient Greek for "chunk".]

[@AgeofReason]

Laetitian Madhatter It all goes back further to Hebrew and Phoenician. Phonics. Phoenix. Phoenician. Salam, Salaam, Shalom. Etc. Eye Am. i am. the first glyph in the name Allah looks like an i, which represents man. God created man in his image. We are a fractal piece of God and his or it's consciousness, delusion, madness, happiness and unconsciousness.

[@AgeofReason]

German word for 1 is Ein. Hebrew glyph for one in Ayin, pronounced Ein. I. Eye. One eye. Pineal gland. Etc. these are simply jump off points to consider.

[@over900aayan9]

0:50 *Oh, I never knew they mentioned the Hyrule Triangle in this. Ok, let's go. I'm listening*

[@wonderland22]

The music has moments of Breath of the Wild shrines too.

[@TBillie]

Bio tech needs without my child 🎉

[@bigbox8992]

Why we see fractals when we are on lysergic acid? What is the relationship between biochemistry, human perception and fractal geometry?

[@tehlolzfactor]

There are actually some very complex mathematics descriptions of such fractals and shapes and their relationship to vision, especially regarding the usage of LSD and marijuana. If you look up the paper, "Geometric visual hallucinations, Euclidean symmetry and the functional architecture of the striate cortex" by Paul C. Bressloff, Jack D. Cowan, and others, published by the Royal Society. I would love to see 3Blue1Brown do some sort of video simplifying the concepts for us laypeople, though the topic as it is described in the original paper is still very interesting.

[@primarysecondaryxd]

I think it's really funny that someone can basically ask "Why do I see weird big shapes when I'm high", and people can mathematically and scientifically answer that question without joking about it.

[@hhaavvvvii]

It's questions like these that we actually want to answer. They provide real insight into our world.

[@ClaytonLivsey]

My take: because our visual apparatuses are fractals as well.

[@daggawagga]

Damn. I never would have guessed there was a specific word for the geometric shapes that appeared when I pressed my eyelids when I was a kid (wikipedia: "form constant").

[@DerIchBinDa]

7 years late but, wow, I did not understand Fractals the first time I read about them, and neither years later in Wikipedia. Thank you for your exceptional explanation and animation it made now finally "click"!

[@BudCharlesUnderVlogs]

Can we describe any other features of these shapes (e.g. the UK coastline) beside their fractal dimension? Like is there a way to mathematically show the UK coastline is different to some other arbitrary 1.21 dimensional shape? And can we apply this to the real world to help us better approximate these shapes in regular 2 or 3 dimensional space, like to create a better area or volume formula for them?

[@thepatheticone4457]

4:08 should've made the yellow ones brown. missed opportunity!

[@JM-lh8rl]

Did anyone else notice that no cubes were colored yellow at 4:10?

[@tandyy8634]

THATS RACIST

[@cupcakeflareon324]

Yes

[@TaigiTWeseFormosanDiplomat]

wt

[@louisbrill891]

I love fractal videos solely to see how low the creator can drive the bitrate

[@cycrothelargeplanet]

2:39 A tesseract is 4D A penteract is 5D A hexeract is 6D A hepteract is 7D An octeract is 8D An enneract is 9D A dekeract is 10D

[@SajjadKhan2-m5b]

☠️

[@MetalSilvan]

That Markus Persson is one of your Patrons? Notch?

[@DocteurZeuhl]

It's "that" Markus Persson indeed :)

[@duckymomo7935]

who is Markus Persson?

[@MetalSilvan]

Mi Les Notch! The creator of minecraft!

[@alxjones]

Yes its confirmed that Notch supports 3b1b

[@suwinkhamchaiwong8382]

notch

[@sherumann]

It's really interesting. Could a coil shape like that have some kind of "alternating" dimension if the "line" that forms the coil turns out to be a coil in itself, and so on?

[@qwerty11111122]

sherumann i think that that's what string theory hangs onto, that extra dimensions could be wound up and bound into small spaces

[@WritingMyOwnElegy]

All i can imagine from that is how dna compacts itself around protiens

[@timh.6872]

This is how Nash did his embedding theorem, no?

[@edwardlulofs444]

Great video, thanks. I had to teach myself all of this in the 1980s when part of my dissertation was on fractals. I had the privilege of talking to Mandelbrot.

[@nefarioustoast]

17:12 so at the point where you'd be zooming in atomically, would the limit of the slope just be 1? so the points at the very very far right would form a y=x graph? but we still only consider the slope=1.21 part of the graph given the context that we don't inspect coastlines atomically, but rather from space/birds eye height?

[@OmnipotentEntity]

in your spiral example where the fractional Dimension fluctuated between 1 and 2 would it be theoretically possible to construct a shape that would continue to oscillate between 1 and 2 by making these spirals ever more tightly coiled? And if so what would be fractional dimension of such an object be because it doesn't seem to approach a limit?

[@OmnipotentEntity]

"Spirals ever more tightly coiled," meaning the spiral itself is made of a spiral (which is made by a spiral, etc.)

[@fossilfighters101]

+

[@MrKohlenstoff]

It probably is possible rather easily using standard replacement rules. Possibly somewhat like this: You start with a straight line, and then always apply these two rules in order: 1) every line is cut into, say, 50 sublines of equal lengths 2) every (sub)line is replaced by a (curvy) line, which moves from its start to its end point in a spirally way, with the spiral's radius being directly proportional to the line's length (and a relatively big factor, like >10 I guess) At least I think that would work out for an arbitrary number of steps. You could even merge the two rules into one. If I'm not mistaken this would cause the desired effect, and, but I'm just guessing here, the dimension would probably be considered to be undefined when working with the numerical measurement. There still might be a dimensionality that can be derived in an analytic way by taking the rules into account though.

[@Jesin00]

I believe the dimension of such a recursive spiral would probably depend on the radius and spacing of the spiral at each level, though I'm not sure exactly what the correspondence would be or how to prove it. I wonder if there's a way to construct a sequence of parameters that would give a well-defined curve whose dimension would fail to converge...

The only confusing part about this is that the word "dimension" is used to refer to different things, with different definitions. This is perfectly exemplified in 15:57 when Grant casually says: "In 3D, by the way..." :D

[@redeniousmusic7551]

they don't refer to different things. the tube IS a 3d object. measuring it in a more detailed way doesn't remove it from its existence in euclidian space... unless you are a materialist mathematician to whom numbers mean nothing except for arbitrary social constructs invented by rich european men for the purpose of subjugating others because qualia is merely billiard ball atoms bumping against each other in a conveniently too-complicated way for anyone to consider suggesting that objective reality exists

[@zonesquestiloveunderworld]

It's one thing that always irritated me about mathematics, and why I hated it so much as a child: it uses common words in ways that often don't relate to the original usage. Come up with new words! It just seems needlessly confusing to the layman, much like law - though in math's case it's probably not nefarious, though maybe equally elitist.

[@danwillits7954]

I never knew how one got the dimension of a fractal until this video. Thank you.

[@dudeman3981]

I would love a video on the Fourier series and transformation. Your animations would make it look so beautiful and intuitive!

[@federicovolpe3389]

Thank you! You've explained clearly a difficult concept! I'm happy I discovered the world of fractal. Subscribed!

[@jacobkantor3886]

best channel on youtube

[@knightlypoleaxe2501]

11:20 It's like circumscribing and inscribing shapes based on a set circle size and then measuring the area of each, the shape's area slowly approaches the area of the circle it is inscribed/circumscribed around.

[@ultim8yeetr708]

Wow I didnt know that dimensions can be decimals

[@gigglysamentz2021]

6:34 I stopped the video and spent a thousand years trying to calculate it. It was great fun :') Also I got it right. And that was AMAZING OuQ

[@mistermessy735]

I was like 1.585,1.585 even though im in 6th grade and have no clue whats going on because i heard him say it in the begenning lol

[@trickytreyperfected1482]

GiggitySam Entz Shouldn't have taken too long if you have a simple understanding of how logarithms work. But hey you might not have that understanding; I'm still trying to understand which number goes where in the log function (I'm doing it indepently though, I'm not in calculus yet).

[@私-c9j]

Thomas R lmao, dude wtf.

[@austinfeng735]

world record for oldest youtube account right here!

[@dp-zn8bd]

@@trickytreyperfected1482 It depends on what you mean by simple because working out what exponent makes two equal three isn't exactly easy. Unless, of course, you mean just plug it into a calculator.

[@fossilfighters101]

So then... How many dimensions is a human?

[@luongmaihunggia]

3

[@goldengeek3320]

Actually a human has a fractal dimension not between 3 & 4 but between 2 & 3... For an object to have a fractal dimension above 3 would require their to be more "mass" or detailed/structured points in the shape of a human then what is physically possible because we would need a fourth perpendicular spacial dimension for those extra points to reside in. If you have not noticed, humans are full of empty space in our teeth (maybe :o ), our natural orifices, and other places such as in our viens. If we had a fractal dimension of exactly three we would have no mouth, blood, or any other movable parts. So our fractal dimension is between 2 and 3. For humans I would assume our fractal dimension is between 2.5 and 2.8 but that is just a wild guess. Look up menger sponge to get a better idea.

[@trangium]

I dunno, about 2.9999999999999997

[@Gicopiro]

@@goldengeek3320 Read the description it must be greater than 3

[@neoxus30]

e)

[@back2d_lobby]

Incredible presentation. I wish I had have these resources when I was at school