Chaos Game - Numberphile

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[@numberphile]

Catch a more in-depth interview with Ben on our Numberphile Podcast: kzbin.info/www/bejne/Y6Wqn5xvhMd9jc0

[@CC-hx8gj]

based

[@WhatWhy42]

Are all the dice rolls done by hand or is it simulated random?

[@Cragoon]

This is the shape of procrastination, always changing your mind halfway through different task.

[@pinkraven4402]

So prpcrastination is a fractal

[@tompeck5495]

As someone who is watching this while procrastinating from homework this hit closer to home than I'd like.

[@neerajkrishnang3916]

So that's how you end up Jack of all trades and master of none, because of all the holes in your skills..🤔

[@blitsriderfield4099]

so the shape of procrastination is the triforce...makes sense

[@paulrussell1207]

See the big triangle that gets left out? Your 1st class honours degree is hiding in there!

[@acetate909]

"I'm not going to go into the details. It's worth looking up". Dammit, you are the "looking up" process.

[@STOG01]

This is suspiciously interesting.

[@Triumvirate888]

One might even say Auspiciously interesting...

[@STOG01]

Anything mathematical is eventually auspicious. Just a matter of when.

[@Lucerne9]

It makes me fascinated to how mathematical reality is. Many assume biology and math are far apart, but then there's the affiliations like that in the video

[@christosvoskresye]

That idea can be taken too far, Tate H, and often has in the past. It is possible today to extend the ancient fascination with numerology far beyond the positive integers.

[@seamusandpat]

Spaciously interesting!

[@timchallenge]

Chaos theory is literally my favorite area of mathematics, I would honestly love an entire channel dedicated simply to that one field.

[@DLCS-2]

Same

[@highlewelt9471]

Please more vids with this guy about chaos theory!

[@hierro09]

Ben: "this is a familiar shape" Me: Yes of course it is, after many years of gaming I know it's a trif- Ben: "it's the Sierpinski gasket" Me: -erpinski gasket, yes... Worked with it a lot, sure.

[@arthurthekyogre9155]

It's a triforce fractal

[@anawesomepet]

What is a Sirepinski triangle? I only know about the triforce.

[@ragnkja]

SPSheep It’s what you get when you replace each filled triangle in the Triforce with a smaller Triforce, and repeat infinitely many times.

[@brandoncarson905]

@@ragnkja It's triforces all the way down...

[@simonmultiverse6349]

(1) those three points... move them a bit, so that it's not an equilateral triangle. Make it a long, skinny triangle if you want. Then the resulting Sierpinski gasket will be long and skinny. (2) OK, you want a SQUARE?? Easy: have four points, not three. Go halfway towards a (randomly-chosen) corner point, and plot your position as a point. Repeat it all day. (3) How about a distorted square, one that's seriously knocked out of shape. The same rules will result in a bent/twisted quadrilateral version of the Sierpinski gasket.

[@alejandronq645]

This is probably my favourite numberphile video ever made

[@TheSunriseAnimation]

After the "Amazing Circles" one, its my favorite!

[@alejandronq645]

Lila_Kuh98 that one was great too

[@benjamindavid4360]

The Feigenbaum Constant video is as mindblowing as this one.

[@jacoblpeterson]

SAME!

[@abhinavNJ]

Interested

[@sirstar45]

When that simulation ran, my jaw genuinly dropped. That is so amazing I have no words.

[@jamiedonaldson794]

same I thought how? just how?

[@AribZeeshan]

@@jamiedonaldson794 Yes Bro.....This Is Crazy

[@qwertyfinger]

THIS IS THE WEIRDEST MATHS THING IVE EVER SEEN

[@Triumvirate888]

I read a book a long time ago that had a quote in it that sums this up perfectly. "You look around and see the whole world falling apart. But you are wrong. The world is NOT falling apart. It is falling into place."

[@DaveGeelen88]

look for = hanoi binairy :D You will be amazed again

[@JayTheYggdrasil]

lifeinsepia just look up iterated function systems, it's almost the same thing but better

[@michagrill9432]

DeutschMaga Was isn dat? XD

[@DaveGeelen88]

towers of hanoi and the link to binary counting very amazing :D

[@santinxt]

If you play the same game in 3D with the vertices of a cube and the midpoints of the edges and instead of dividing the distances by 2 dividing them by 3, after enough iterations you will get a Menger sponge.

[@Benny_Blue]

Would that be going one third the distance, two thirds the distance, or either/or?

[@santinxt]

@@Benny_Blue one third the distance. I'm actually not entirely sure what would happen if you took 2/3 the distance instead.

[@emdadahmed5592]

Thank you, numberphile, for another great video. Every video my fascination for fractals grows stronger

[@fossilfighters101]

+

[@heeheehawhawheehee]

Moustache

[@knotwrite]

Watching this video is like peeling back the curtain on reality. I need to go sit down for a bit.

[@imaclock8144]

what did you ever think math was in the first place?

[@uniqueusername_]

im a clock I couldn’t have put it better. People who don’t like math confuse me.

[@theliamofella]

I'm a clock and unique user name, so this is just obvious and not profound in any way? So what has this video got to say about mathematics, if you are fantastic at maths be humble about it

[@theliamofella]

?

[@theliamofella]

Actually, I take that back, it doesn't mean anything profound about reality, it's just a shape that has to happen given the rules etc

[@boalollal242]

Disappointed there wasn't a giant eye in the middle.

[@theliamofella]

Or a ganja leaf lol

[@raphaelkelly861]

@@theliamofella honestly though imagine something like that emerging from iterated randomness. Probably is some way to do it...

[@cobaltbluesky2276]

“Slightly disturbed about reality” prefectly describes how I felt when that shape appeared. I am a changed man

[@razielhamalakh9813]

This man can't roll dice to save his life.

[@IllIlIIllIlIl]

It's because of the paper. It cushions the dice faling and makes it slide instead.

[@z-beeblebrox]

I don't know, if his survival depended on rolling a 1 or 2, I think he'd be pretty safe!

[@ZXLegend1]

Is this a Kaiji reference?

[@MenacingBanjo]

I guess that means he's going to... die.

[@2dividedby3equals666]

That is not a dice it is a camouflaged Parker cube.

[@Glumurinn]

The bit starting at 3:15 is almost magical, and the music really makes it even more so!

[@craken1566]

4:56 the fact that also the starting point is replicated in the smaller triangle at the bottom right just blow my mind that's Crazy

[@mighty8357]

I'm genuinely baffled by this result :O I would have never guessed that rolling a dice could be linked to fractal theory!

[@DarkTF2Director]

it has to do more with he fact hat he always moves half the distance and he only has specific point where to aim

[@gavinhowe9897]

the rules are the fractal, not the randomness, the rules chosen in these scenarios are more like stencils, with the dice being more of a spray paint.

[@hhouse1234]

cool analogy! :)

[@benbooth2783]

There is no information that can create structure in the random set of numbers generated by the dice. The structure comes from the number of dots and the rules.

[@simonmultiverse6349]

The die just takes you on a random walk through the possibilities. The rule (i.e. move half-way to one of the three fixed points) defines the shape. The die just causes you to sample all possible combinations of moves. You could EQUALLY WELL systematically try moves 1,2,3 then 11 12 13 21 22 23 31 32 33 then 111 112 113 121 122 123 131 132 etc. then all combinations of 4 points then all combinations of 5 points, etc. That would be a SYSTEMATIC sample, but you'd visit every possible combination eventually. The die makes you visit every possible combination/point but in RANDOM order. One way makes you visit every point systematically; the other way makes you visit every point in random order.

[@pietro9801]

5:20 if you are wondering why it doesn't work, the rule is move 2/3 of the way

[@simonmultiverse6349]

In the Sierpinski gasket, there is a very simple way of proving that the central large triangle is always empty. You imagine that a point lies in that central triangle, i.e. it has started somewhere and it has gone half-way towards one of the points. Pick point A as an example. If we go half-way towards point A, and land in the central triangle, all you do is ask the question: WHERE DID WE START FROM? You will find that, if you land in the central triangle, you must have been outside the overall shape, in order to land inside that central triangle. Since the rules are that you always start inside the triangle, you can never get out of the triangle, therefore you will never land in that central triangle. Ergo, the central triangle is always empty.

[@ivanberdichevsky5679]

3:49 "This is a familiar shape" , of course it is! That's the Triforce from The Legend of Zelda

[@WayneStakem]

_"You unlock this door with the key of imagination. Beyond it is another dimension - a dimension of sound, a dimension of sight, a dimension of mind. You're moving into a land of both shadow and substance, of things and ideas. You've just crossed over into the Twilight Zone."_

[@nghiepluu6939]

Wayne S t

[@I_AM-MICHAEL]

Sounds like something the pot gods would say.

[@boobyjustin]

That gave my goose bumps' goose bumps goose bumps

[@danielodors]

Justin Booby your goose bumps became fractals.

[@simonmultiverse6349]

@Stone H Here's a fractal: bufbufbufbufbufbufbufbufbufbufbufbufbufbufbufbuffalofalofalofalofalofalofalofalofalofalofalofalofalofalofalofalo

[@whoeveriam0iam14222]

can you share that drawing dots program with us? it looks fun to play with edit: it's in the description now =D

[@tsugua001]

It's called geogebra, I have no idea how he did it though

[@turun_ambartanen]

then he should share the .ggb (or whatever) file. the fern doesn't look like it was in geogebra though.

[@Brotcrunsher]

whoeveriam0iam14222 I could code this quickly and i probably will. If enough people are interessted then I will share it and make it open source on github.

we are very interested

[@turun_ambartanen]

I am very interested.

[@Alexander-is9jo]

This is absolutely mind-blowing

[@theorist-qx4vz]

Wow wow. Predictable in the relative macro-scale, chaos system is insane, and beautiful.

[@khan_msj5680]

how dislike the video, this is the most interesting video ever...

[@bingobangini]

because it's kinda misleading...

[@iammaxhailme]

▲ ▲ ▲

[@oreole9608]

▲ ▲ ▲

[@oreole9608]

hmm

[@oreole9608]

. ▲ ▲ ▲

[@Ronitrocket]

▲ ▲ ▲

[@Ronitrocket]

..▲ ▲ ▲

[@HunterJE]

The origin of the pattern gets easier to intuitively grasp if you think about the problem backwards, e.g. note that if a midpoint is in the central void, all the points twice that distance from each of the three corners are outside triangle ABC, so no point inside the triangle will result in a midpoint inside the central void*. Once you have that you can see that the next order of voids can only be reached when stepping from points inside the central void and so on down from there.... *there's an extra zeroth step if you want to consider the "outside start" version, which is to convince yourself that an outside start will result in points moving steadily towards the triangle and once they are inside the triangle will stay there forever

[@matthewd759]

I couldn't resist but stay late and create this on Excel after work... on a Friday....

[@corpsiecorpsie_the_original]

I was just thinking that this could be created in excel

[@marklafrentz1070]

Me: *furiously opens matlab to create this so I can play with it*

[@JohnSmith-kc6ov]

This is one of the few math things that made me go "holy sh*t"

[@swingardium706]

*Vi Hart screaming in the distance*

[@Isaac-lb1gl]

yes!

[@minnarewers3573]

SbAsAlSe HONRe yey

[@interrexclamacion]

Yes!

[@necromancer2367]

I WAS LOOKING FOR THIS COMMENT

[@quinn7894]

*whispering in the distance*

[@SorcerorNobody]

I'm very familiar with the chaos game generating the gasket, and that changing the vertices and distances generates other similar geometries. But the two triangles ruleset is new to me and my mouth genuinely dropped open as soon as I saw what it generated.

[@peepock7796]

3:14 I immediately recognized that as serpinsky’s triangle. I drew it in my grade 7 science binder!

[@SocksWithSandals]

Loved the mystical sitar chord during the computer fractal render.

[@frankfranksen204]

Cool things happen if you go twice the distance instead of half. Doesn't diverge and looks different every time, kind of like DNA in its densely packed state in the nucleus.

[@fakename3344]

Could you link to an example of that? And maybe comment after you do? KZbin likes to tell you that it's posted comments with links when it really didn't.

[@nonazjr5120]

@@fakename3344 You can easily do it in the link I posted as another comment and on my Twitter (@Nonaz_jr). You can DM me if you need help.

[@567legodude]

Before you even mentioned the part about the computer, I wrote a script to do it and I was extremely amazed at what was showing up on my screen.

[@props3311]

Triforce confirmed

[@DarkTF2Director]

GermanLoLCaster my first thought was "oh it looks like a triforce"

[@GuyWithAnAmazingHat]

Seems like the Triforce is indeed the Power of the Gods.

[@ale_schneider]

It specially reminds me of Deus Ex. Like, this is the most Deus Ex thing I've seen in a while.

[@fricktion01]

%triforce

[@megatrix500]

you mean infiniforce

[@spencerallbritton9459]

I'm majoring in Electrical Engineering but I love Mathematics. It's logical, precise, absurdly useful, and highly mysterious at the same time.

[@DashedSimpusMaximus]

"The rules of the universe can be written down on a single piece of paper" Math is the language of god it seems.

[@Bleagle]

It's the language of structure/patterns, and without those (think pure chaos) life can't form or exist long enough to become intelligent. no intelligent life => noone to discover math

[@DomCurtis2023]

Look up nikola Tesla and his 369 theory

[@lonestarr1490]

Makes you think. The universe could have been invented one evening in a bar, as a brief and sketchy calculation on a napkin.

[@sillysausage4549]

S

[@spencerwadsworth7024]

This is one of the most intense surprises I have ever experienced

[@DonGeritch]

At first I thought - oh, this seems a lot like my life (changing goals all the time and not reaching anything, no matter how close I get). But seeing the result got me thinking..

[@loughkb]

The easy way to get your mind around the first 'half way' patterns is this. The voided areas do not lie on a line that's half way between any two of the points.

[@loughkb]

Draw three dots, grab a ruler. No way to draw a line between two of them that crosses the voided area.

[@IMadeOfClay]

Years ago I was playing around on my calculator in maths class and I noticed something strange: tan 89= 57.28996163 tan 89.9= 572.9572134 tan 89.99= 5729.577893 tan 89.999= 57295.77951 tan 89.9999= 572957.7951 tan 89.99999= 5729577.951 If you compare this to how many degrees in a radian (57.29577951) you notice two things: (i) the order of the digits get closer and closer to the order of 1 radian (in degrees), (ii) each time the numbers increase by approximately a factor of ten. I'm not a professional mathematician. I wonder if anyone can give me an answer after all these years. Then I can die peacefully lol.

[@glenneric1]

As you keep going to 1/10th the remaining distance from your angle to 90 degrees you keep multiplying the tangent by 10/1... 10/1 and 1/10 are reciprocals, as you might expect for a line approaching infinite steepness.

[@dieselguitar1440]

I hope you still get this notification after 2 years. As x (in degrees) approaches 0, tan(90-x)/tan(90-(x/n)) approaches n, n=10 in this case. This is because tanx=sinx/cosx. As x approaches 0 and 90-x approaches 90, sinx is basically 1, while cos x becomes nearly proportional to x (zoom in real close on cosx at x = 90 degrees and you'll see it's basically a line going straight through the origin, that is to say, y is x times a constant.) So, when you divide the difference from 90 by 10, sin stays pretty much the same, while cos goes down by a factor of pretty much 10, and so tanx=sinx/cosx gets multiplies by 1/(1/10)=10. If my statements about the behaviour of sin and cos don't make sense to you, you should know that their values correspond to the y and x axes respectively for sin and cos, which should serve as an intuitive/visual basis for understanding all this. I didn't look this up, I figured it out from intuition of the trig functions, it's a valuable skill.

[@redtoxic8701]

It's just an exercise of limits. If you want to see the proof to your question just tell me, I'll post it on imgur

[@FlyingSagittarius]

@@dieselguitar1440 hey, you never know when someone stumbles upon this two years later and wants to know the answer too. 😋

[@MrHeroicDemon]

He is one of my favorites because the things he chooses to talk about are like the secret low key super interesting ones.

[@raiyanbasher9529]

This is the best video of numberphile Together with the pizza one and minus one by 12 one.

[@rukia3947]

This is soooo cool!!! I wrote and tried out the program at home and it came beautifully! It never ceases to fascinate me!

[@aleratz]

I am pretty sure that if you do it too much you end up summoning some kind of demon.

[@mementomori7160]

Now(I'm watching this 2nd or 3rd time) I understand this and see the beauty. Beauty in randomness. π, triangles, everything is hidden in randomness. Math is the one and only true art.

[@neingeben9510]

I've seen this shape before when working with Cellular Automatons

[@simonmultiverse6349]

Can we have that argument about "automatons" v. "automata" ? Please? Please please PLEEEEEEEEEEEEASE?

[@ferax_aqua]

The last bit about ferns is certainly fantastic and inspiring for the computer folks out there, myself included. A million upvotes I would like to give to this one, if possible.

[@dgtlrn]

But what if Wil Wheaton rolled the dice?

[@bobrobert1123]

This comment

[@nblack3879]

A straight line that infinitely points towards one.

[@qwertyquazo673]

Excellent question.

[@bravojr]

I can think of no other man who fails so hard he breaks chaos... Brilliant!

[@bobrobert1123]

Is that a pit of acid?

[@Benny_Blue]

Two questions about the “attractors” you mentioned: 1) When you put the point in the middle to start, it’s obviously NOT on the Sierpinski Triangle in that case. But very quickly, it appears to reach it, and stay there. Does it really reach it that soon, and then always hit points on it? Is it constantly getting closer and closer to the triangle, but never reaching it? Or is it constantly very close to it, but never touching it until the point “almost certainly” gets lucky and lands on a genuine point on the triangle, at which point it stays on it forever? 2) The case you show of a point in the middle has the point move towards the same corner several times in a row. If you started with an arbitrarily long string of luck, and never moved towards the same point twice in a row, would it still approach the triangle? Or is that streak necessary for the attractor to work?

[@morismateljan6458]

Astounding! But what shape would appear if we tried that with 3D object?????

[@bevkcan]

the sierpinski tetrahedron

[@Jordan-mn2ty]

Or a circle with infinitely many points

[@emilianonavarro2858]

I just managed to program the thing in Octave(GUI). It's simply amazing, and the code is so simple. Breathtaking.

[@CorrectHorseBatteryStaple472]

Very interesting but I wouldn't say it's baffling. Just reverse engineer the point in the middle of the Serpinski Triangle. How do you get there? No point in that triangle is halfway between anything and one of the vertices.

[@CorrectHorseBatteryStaple472]

***** not halfway between two vertices. Halfway between a vertex and SOMETHING. Consider the middle empty triangle. What would that something have to be in order to land inside it? It would have to be outside the triangle.

[@markstoner3786]

Andrew Pearce exactly what I was thinking. Those areas are empty because they are impossible to reach with the given restraints.

[@MrDannyDetail]

@@CorrectHorseBatteryStaple472 Maybe it would have to be outside the triangle, but that is allowed and possible according to what he said.

[@suvetar]

I love the whole concept of emergent behaviour, it's the computer science equivalent of the importance of hearing someone in the laboritory repeating an experiement and saying the equivalent of "That' odd/strange/cool/unbelievable!" Thanks as always folks 🙂

[@dalitas]

I genuinely went "what the...."

[@siddhartharoy2233]

Its a strange pattern that makes me think deeply...thanks Numberphile for such an amazing video...as a math geek I will be waiting for more videos like this

[@Radar_of_the_Stars]

So close to the triforce, but yet so far

[@calvinscheuerman]

Sierpinski's triangle is actually made up of an *infinite* number of Triforces. *Infinite triforce = pretty cool.* (Also probably a nintendo-core band name.)

[@Radar_of_the_Stars]

Cool!

[@Wild4lon]

I learned this in an extra curricular maths course and about the applications of serpkinski triangles! It's honestly fascinating

[@hingheng1234]

this is crazy

[@JoshLewa]

This is my second favorite video after the sum of all natural numbers video series. So interesting!

[@peterbutterjam97]

Is this the sort of thing that's known as procedural generation?

[@ferko28]

Yep.

[@JackieChan173]

the only thing in common between all the things known as procedural generation is they go random numbers -> things

[@EmanuelMay]

Nope, you don't even need random numbers for procedural generation. Just - as the name suggest - a procedure. Does not have to be random at all (can be deterministicly based on previous events for example) , but randomness can be quite useful.

[@klaasbil8459]

6:21 that is incredible! I shouted out loud Wow!

[@jimmorrison6177]

Interesting

[@earlwirth9614]

You can get this by generating a Linear Recursive Sequence and graphically raster the binary sequence of 0's and 1's at specific widths. The function f(x) = X^15 + X^1 + 1 rastered on widths of 239 or 9538 or 32528 or 23229 will produce Sierpinski triangles.

[@LudwigvanBeethoven2]

We live in a simulation. A simulation where we live in.

[@sayanbiswas9094]

This video is arousing, literally arousing. Kudos Ben Sparks

[@rosiefay7283]

2:35 Parker randomness?

[@yujiokitani4492]

If you have 8 points around a square, four on the verticies and four on the edge midpoints, each with an equal probability, and find the point 1/3 across from the chosen point to your last point, the attractor is a sierpinski carpet

[@jyotishkaraychoudhury4762]

For the first time.... I'm seeing that a video has not got any dislikes till now. (Right now, it has got 181 likes)

[@Tiagocf2]

now it has 3 dislikes ;-;

[@jyotishkaraychoudhury4762]

Well....even the best videos have thousands of dislikes. I don't know why !!!!

[@jyotishkaraychoudhury4762]

Correct !!!! The ratio is 400:1 right now.

[@avananana]

Before I watched the video entirely, I created a Processing program to see what you were up to, and when I saw that the Sierpinskí triangle appeared out of nowhere, according to the rules, it completely blew my mind. This is some crazy stuff going on.

[@Varun2799]

illuminati confirmed

[@fergusmaclachlan1404]

I was looking for this comment.

[@georgeabreu6392]

Varun Patil Only fools can deny us now! Illuminati! ∆∆∆

[@alexakkers2859]

George Abreu r

[@Varun2799]

Nimit Dave 😂😂

[@yudhvirsingh9909]

Nimit Dave 😂😂😂😂😂😂😂😂😂😂

[@gallifrog6144]

When I saw the Sepinski triangle emerge I gasped, my mind was so utterly blown by this

[@secret12392]

What was the website used here?

[@TotodileSmile3721]

Stormageddon +

[@TheMathestar]

if you mean the software for that simulation, they're not using a website but the geometry program GeoGebra

[@retsapb6319]

Incredibly complex patterns arising from simple rules. Amazing

[@martinusny]

wow.....

[@naraferalina2308]

Math is a discovery, invention and art.

[@explodingCR33P3R]

WHAT!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

[@SK8fourL1F3]

wasn't sure I would enjoy this video at the start but it turned out to be extremely interesting!

[@SylvEdu]

Math is the language in which God spoke to create the universe.

[@SylvEdu]

Atheists literally cannot help but spill their spaghetti all over the place. Calm down, fellas. Just scroll past it.

[@calvinscheuerman]

Spill their spaghetti? That is the best thing I've ever heard.

[@SylvEdu]

"He made a universe with very specific and structured rules, of course there's going to be an underlying pattern, even if the events within that universe appear random." 6:00

[@autodidactusplaysjrpgs7614]

Unverifiable hypothesis much?

[@calvinscheuerman]

Come on, Autodidactus Communitati ; Don't spill your spaghetti.

[@michelangelodefeo2283]

Once, while I was tutoring, we built a randomizer to build the Sierpinski triangle. I have enough familiarity with fractals, but it was still exciting to see.

[@aryankumarprasad1574]

Am the only one who feels creepy ?

[@thefattestbaby420]

Yes.

[@alienmoonstalker]

I just did this with the jump factor of 2 instead of 1/2...very interesting. I encourage you to try it for yourself.

[@brandon2762]

The framing of this is totally misleading. Yes, the results are randomized, but the randomness is forced through constraints. Of course there is going to be a pattern. It definitely is interesting how nature forms patterns based on randomness but it isn't disturbing or spooky, that's just how it is.

[@olauda]

I think the fact that's how it is is what makes it disturbing or spooky to people.

[@HelpfulFlyingpig]

I can’t describe how incredible this video is

[@davidgillies620]

The first thing I did when I understood the algorithm was to pause the video, fire up Mathematica and see what happens. It's a very simple function to pick 3 random points on a circle (the triangle), a random start point, and then iterate halfway towards each of them based on a random integer in [1, 3]. It wasn't altogether surprising to see what emerged.

[@Zephei]

David Gillies I think the most interesting part of iterated function systems like this is that it gives you a way to write an algorithm to produce a fractal with certain self-similarities just by writing a few simple rules, without even knowing what the fractal will look like.

[@DarthCalculus]

The music in this episode was very well used. Incredible juxtaposition.

[@TheTechnoLocker]

This is my favourite kind of Numberphile video

[@iambranden]

This was incredible!!!! Thank you for taking the time to make this video!

[@platonitosocratico1597]

PLEASE I NEED MORE BLOWING MIND THINGS LIKE THIS!!

[@chrissalinas325]

Probably the coolest Numberphile video

[@dusty6299]

This are the moments I remember why I really love math.

[@hiimapop7755]

This is actually mind-blowing for me.

[@redsalmon9966]

It's like the sand pile thing,showing the beauty of Maths in a terrifying beautiful way.

[@lifter1000]

Thank you Ben Sparks, your subjects are amazingly beautiful 🙏

[@SteveD826]

I sprinted to my math professor with this video. He audibly gasped when the shape began to be revealed.

[@lolechi]

Almost fell out of my chair when those triangles appeared! Amazing!

[@Xetarine]

Ikr