Chaos Game | Fractals emerging from chaos

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Chaos Game | Fractals emerging from chaos | Computer simulation |

Рет қаралды 331,993

Күн бұрын

I wanted to make a video about this topic for a long time, however without knowing how to code it would be really inefficient to animate every single dot by hand. So for the past few weeks I've been learning how to program animations and so I managed to simulate a chaos game.
Thanks for watching :)
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Thank you everyone for 1000 subs. It's really cool that over a thousand people are actually interested in my content. I'm thankful for all the comments and support that I've been getting from you. I decided to create a Patreon page. If you feel like you would like to see more of my content as well as get to know more about me consider becoming a patron at :
/ think_twice
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Any further questions or ideas:
Email - thinktwiceask@gmail.com
Twitter - / thinktwice2580
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Overall render time: ~ 4 hours
Programs used:
- Processing (Game simulation)
- Cinema 4D (shape outline)
- Adobe Premiere Pro (Editing)
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Music by: Tom Day
Hiatus - • Video
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Пікірлер: 475

@cinfdef 3 жыл бұрын

Me putting a point directly in the center: *Reality can be whatever I want*

@struanpeat5116 3 жыл бұрын

Okay, but seriously What happens if you put a point in the middle?

@struanpeat5116 3 жыл бұрын

@Sai Sasank thanks, just watched it, I actually understand the phenomenon alot better now

@BakedAlaskaTM 3 жыл бұрын

since it's technically forming a one dimensional shape, it's pretty much impossible to not put the dot in one of the blank areas, since the points have zero length and width.

@NStripleseven 3 жыл бұрын

@Sai Sasank link?

@user-ov1fe3xu7f 3 жыл бұрын

@@BakedAlaskaTM They're not one dimensional, just more like 1.3 dimensional or like that. You can understand that with areas, if you will try a square and a line, i can give you link to one video about that.

@kaninchengaming-inactive-6529 6 жыл бұрын

The Legend of Fractal - Breath of the Chaos

@getbottedkid11whocares97 3 жыл бұрын

@123cityperson 3 жыл бұрын

underrated comment since the last 2 years

@lukasmaclean9885 3 жыл бұрын

Lol

@pawn1234 3 жыл бұрын

triforcw

@asheep7797 Жыл бұрын

@@123cityperson now 4

@dubarnik 6 жыл бұрын

Long time ago, I was an 8th grade math teacher. At the end of the year, after all the state testing was done and we were counting down days until summer break, I did a unit on fractals. I spent a couple of lessons teaching my students how to program their TI-83s to graph the Serpinski Triangle. The program is about 12 lines long and uses the algorithm at the start of this video. The program is still available and works on all of the TI-84s.

@ThinkTwiceLtu 6 жыл бұрын

Oh, that's nice. If you'd like to check out my code it is at the bottom of the description however keep in mind that im new to coding so feel free to share any tips.

@Tondadrd 6 жыл бұрын

Hey! I didn't find your code mentioned in the description... Maybe you standardized video descriptions and overwrote it?

@silentobserver4839 5 жыл бұрын

*Sierpinski

@silentobserver4839 4 жыл бұрын

@kurz1000 I bet that in English it is „Sierpinski Triangle", while in Polish "trójkąt Sierpińskiego".

@FeedMeChaos1 3 жыл бұрын

Yes but why?

@ethangilchrist3534 5 жыл бұрын

"Is this a fractal?" "No, this is Patrick"

@chaosxdani 3 жыл бұрын

Lol

@spacekid9680 3 жыл бұрын

Patrick

@simonmultiverse6349 2 жыл бұрын

Is "Patrick" a trick performed by your father? That would mean that "Matrick" is a trick performed by your mother. Then we have Brotrick and Sistrick (obviously) and, if you have a four-legged pet, we have dotrick.

@hyperclipse Жыл бұрын

@@simonmultiverse6349 dotrick = two tricks in 1 tritrick =

@simonmultiverse6349 Жыл бұрын

Woof. Furthermore, *WOOOOOF* also bark, bark!

@Blananas2 5 жыл бұрын

"Oh, just random points, it can't be that intresti-- Wait what?"

@yamansanghavi 5 жыл бұрын

Can we now say that math is art?

@loreleihillard5078 3 жыл бұрын

Who ever said the two weren't linked?

@BungerBurg Жыл бұрын

It's always been art

@xoiyoub 3 ай бұрын

Depends on your definition of art. Math is related to art? Yes. It IS art? I don't know. As I said, it depends on your definition of art.

@FelineFlame-dj1xl 3 жыл бұрын

I love how it pulls a 64 bits 32 bits 16 bits 8 bits 4 bits 2 BITS 1 BIT HALF BIT *QUARTER BIT* at 1:07

@chisinau1302 3 жыл бұрын

Yeah, cool animation

@Tondadrd 6 жыл бұрын

Cool, A bit of research, putting the first point into proper position and it will become a rotation similar fractal. Right? Btw. I came from *3b1b featured creators #1* and I'm happy I have. Your videos are showing pure beauty of math!

@brentonwang6771 6 жыл бұрын

MIMIK 100th like

@Mynameisevanbro 4 жыл бұрын

Same

@masonhunter2748 3 жыл бұрын

It’s an error in the code

@ragnarock1008 3 жыл бұрын

I have a Guide Voodo, and imma throw it to the lava

@haxel96 3 жыл бұрын

@@ragnarock1008 Andrew the Guide was slain. Wall of Flesh has awoken!

@paithanq 6 жыл бұрын

Why doesn't the last one have rotational symmetry?

@ThinkTwiceLtu 6 жыл бұрын

I was surprised by that as well. I tried to check if I gave made any mistakes in my code but everything seemed fine. Let me know if you find the answer:)

@aaaa-hj9vv 6 жыл бұрын

Either there's a problem with your code, or your RNG isn't working properly. I did this same thing in Java and the generated shape had rotational symmetry.

@ThinkTwiceLtu 6 жыл бұрын

yes I guess there was a problem with the code. My bad:(

@aaaa-hj9vv 6 жыл бұрын

It was a great video nonetheless, and your new videos are also great! Please continue making more!

@ThinkTwiceLtu 6 жыл бұрын

thanks for nice words:) I will try to upload whenever I can. Hope this community will keep growing.

@marco_gallone 3 жыл бұрын

I think “chaos” is interpreted incorrectly here, chaos essentially means *extremely sensitive to initial conditions*: change the initial randomly selected point marginally different from the first attempt and see a completely different pattern emerge. In this case it is more like “randomness”, and is opposite to chaos in that despite any random input conditions will always form a result with order.

@tszheileung4410 3 жыл бұрын

When you are just having fun with shapes but end up summoning a demon

@paramrathour441 6 жыл бұрын

You missed out on the fern one Its really beautiful, a perfect fern just from chaos

@ThinkTwiceLtu 6 жыл бұрын

Param Rathour ya i was gonna make a separate video about it

@paramrathour441 6 жыл бұрын

Think Twice I would be waiting for that. I like your way of explaining visually.

@LittleFeet21-e5h Жыл бұрын

We love you Think twice! Thank you for creating this iteration animation!

@jrtao3709 6 жыл бұрын

why pentagon does not generate a symmetrical figure ? initial condition ?

@ThinkTwiceLtu 6 жыл бұрын

jr tao i messed up the code, sorry

@jrtao3709 6 жыл бұрын

its strange ! github.com/appleminis/PolygonDreamcatcher

@ThinkTwiceLtu 6 жыл бұрын

the rule was that the same vertex can't be chosen two times. Or the neighboring vertices can't be chosen. I can't remember from the top of my head right now.

@jrtao3709 6 жыл бұрын

random walk what iam looking hehe

@jrtao3709 6 жыл бұрын

mine don't break symmetry, i will try to understand something, thank you twice ;)

@weakamna 6 жыл бұрын

What you are describing here is an Iterative Function System, more specifically the "fractal flame algorithm", if you google that there is a paper as the second result. The first triangle algorithm you describe is the first one described in the paper. The paper goes on to describe techniques on how to improve the fidelity of the images generated (which can become very vivid and intricate!). These are used by programs such as Apophysis(7x) or Fractorium (which is a gpu based implementation, which runs many many times faster on basically any computer with a graphics card). For people interested in this, I highly recommend reading the paper and then trying the programs out, maybe looking at more informal tutorials as well.

@justapassie Жыл бұрын

ty!

@sparshsingh3991 6 жыл бұрын

After all that I have learnt about fractals and their formation; (fixed initial condition and a fixed algorithm put on loop) the video title intrigued me beyond words. This concept further pushes us from our understanding of fractals.

@ineedtostudy6853 2 жыл бұрын

I just woke up and I am not ready for this. I will come back when I am actually awake

@linsproul3548 6 жыл бұрын

on the triangle, what if you put the first point in the center of the triangle? would that ruin the fractal?

@xtehmoonlight7400 4 жыл бұрын

@leophysics 2 жыл бұрын

No it's somehow manage to get its shape

@aidenwallin3523 Жыл бұрын

It's interesting that if you ran infinite cuts, there would basically be straight lines from the corners to the center, except for an infinitely small point in the very corner of the triangle. The middle points on the edges of the blue triangle create the midpoint on the edges of the cut out triangles, as the midpoint of the edge midpoint and vertices create triangles.

@lscskoolnbownz5555 Жыл бұрын

@@aidenwallin3523 fddd ddo dddo did did didnt 😮😂do Z

@giuseppepapari8870 5 жыл бұрын

Nice! Have you tried to use an initial point that lies inside one of the empty areas in your fractal? What will happen? Will that be the only point outside the fractal? Or maybe the fractal only appears for certain initial points?

@sparshsingh3991 6 жыл бұрын

The title deserved a long ponder and a like STRAIGHTAWAY

@accursedcursive4935 3 жыл бұрын

For those wondering why a fractal is produced, think of it like this: For the large middle black triangle, there is no random vertex and random point in the triangle you could pick such that the midpoint lies in that triangle. So, the random movement of the point will never land there. For the three smaller triangles, the only way for the midpoint to land there is for the starting point to be in the middle black triangle; however, as said before, the point never lands there, so the point never lands in these three triangles either. And then for the nine yet smaller triangles, their starting point would have to be in one of those three smaller triangles, and so on.

@justapassie Жыл бұрын

ohh omg huge thanks for the explanation ! so for the midpoint to land in the middle black triangle initially, a random point must be outside the triangle (and that's not the case here) and this would lead to the approach that our 3-sided triangle shown is adjacent to 3 larger black triangles outside on the screen

@Aethos79 3 жыл бұрын

this is the best thing to watch when sleep deprived

@Forsajak Жыл бұрын

0:58 Zelda logo complicated 1:45 snowflake 2:20 summoning a demon

@andistansbury4366 3 жыл бұрын

2:20 I don't know what that fractal is called so I'm going to call it the serpenski rocket

@coreymallard6650 2 жыл бұрын

mathematicians: *put together an elegant closed system that works beautifully in creating an image that infinitely repeats itself over and over, evolving the world of mathematics and our understanding of it* me at 2:20 : hehe patrick

@lucyluo497 6 жыл бұрын

Looooove your choice of music. Will be looking forward to hear your own music as background, would be cool ❤️

@ThinkTwiceLtu 6 жыл бұрын

thanks Lucy :) I'll try to write my own song

@lucyluo497 6 жыл бұрын

hope you feel better soon x

@sjoerdstougie 5 жыл бұрын

If i were an artist, a musician I would buy the rights to the animation of the triangle and use it as a cover, its soooooo satisfying!!

@JJschannel255 Жыл бұрын

2:26 is this a fractal? No,this is Patrick

@sayr2312 Жыл бұрын

This is one of the best vídeos that i ever seen

@IWantToStayAtYourHouse 5 жыл бұрын

But for the triangle one, what if the intial point is in the middle?

@tapioms 4 жыл бұрын

It doesn't matter, the point could be even outside of the triangle and the result will be the same

@loreleihillard5078 3 жыл бұрын

Not EVERY point is on the Sierpinski Triangle, but the further you go, the more it approximates it.

@supreetsahu1964 6 жыл бұрын

I love this chaos game because I can't code, but these are so simple operations you could run it in MS Excel or any other spreadsheet app.

@ThePharphis 6 жыл бұрын

nice alternative

@Brahvim 3 жыл бұрын

Desmos too?

@arealfluffycookie 2 жыл бұрын

I am thoroughly terrified yet incredibly shocked at the same time

@jlpsinde 4 жыл бұрын

Amazing, this man deserves more people supporting on Patreon!

@ThinkTwiceLtu 4 жыл бұрын

Thank you, really appreciate your support.

@jlpsinde 4 жыл бұрын

@@ThinkTwiceLtu I raised from one dollar to two is not much but I can't afford a lot, is just you, Veritasium, 3Blue1Brown, Science Asylum and Eugene Khutoryansky. So you are one of my top five! I'm a physics teacher in Portugal.

@ThinkTwiceLtu 4 жыл бұрын

@@jlpsinde No worries, I appreciate any kind of support whether it's monetary or not. I am happy that you enjoy my videos it keeps me even more motivated to create more high-quality content. Thanks a lot:)

@sebastianzaczek 5 жыл бұрын

The Pentagon one looks like an Evil Version of Patrick star

@k_0_n_t_i_n_w_m 8 ай бұрын

Nice! Good clean idea, well presented.

@xyz.ijk. 6 жыл бұрын

Your work is brilliant. It's so good that I even let the ads finish just to support you. And, yes, I'll hit up Patreon Good work.

@ThinkTwiceLtu 6 жыл бұрын

Really appreciate your support~

@yueshijoorya601 Жыл бұрын

"Total chaos" lmaoooo. The amount of misconception in the layman is just absurd.

@sparshsingh3991 6 жыл бұрын

THIS CONCEPT ITSELF IS TOO AWESOME SO AS TO DESERVE A MILLION VIEWS.

@benjaminr1660 6 жыл бұрын

Absolutely magnificent I've been working on this Serpinski triangle and I never imagined such a thing

@shilohschwartz8671 6 жыл бұрын

This place NEEDS more views, you are awesome!

@musicismoreimportantthanyo9246 2 жыл бұрын

I was obsessed with fractals a few years ago

@Mynameisevanbro 4 жыл бұрын

That is so fascinating

@technowey Жыл бұрын

I have done this simulation and created Sierpinski triangle images. I've also created a Sierpinski triangle using another algorithm, and I know a third algorithm. These are cool.

@snaukball8764 3 жыл бұрын

Everybody gangsta until he makes Mandelbrot's set with this

@crazedtech7488 6 жыл бұрын

Link's been hallucinating.

@jaybretherton6246 6 жыл бұрын

Why is the pentagon one not rotational symmetrical? Why is the bottom side the only one that fills?

@fazaazafg 3 жыл бұрын

doing this on 3d, in a tetrahedron, will actually result in the 3d version of the sierpinski triangle which is the sierpinski tetrahedron

@hpp6116 6 жыл бұрын

No way! It's absolutely fascinating! Thanks for sharing.

@TheJulioToboso 6 жыл бұрын

How do you make this animations? It’s awesome :)

@mathieugouttenoire9665 6 жыл бұрын

Julio Toboso García he said in the description that he used a programming environnement called Processing, so I gess he programmed it himself

@marcosrios3626 6 жыл бұрын

Processing is a really powerful visualization language. Look for The Coding Train on KZfaq, you will find a lot of tutorials.

@nahfid2003 4 жыл бұрын

@@mathieugouttenoire9665 please remind me in a yearrr

@NeilRaouf 3 жыл бұрын

processing.org

@morgandavies9685 3 жыл бұрын

@@nahfid2003 it’s been 7 months, why did you want reminding in a year?

@smol9930 3 жыл бұрын

Watching a one hour ad on a 2 minute video lets gooooooooooooooo

@dirt_dert_durt 3 жыл бұрын

Nightmare patric wants a hug

@carsonm6140 6 жыл бұрын

Cool. Mathematically makes sense, but awesome to see it in action

@drbelli 3 жыл бұрын

i believe this concept sparked a notion in my mind, that... this is one of the ways the Cosmos has to create reality, like a rule* that particles and atoms arrangement , a building block of the physical realm can never be in the same spot twice... a quick example, the snow crystals, but reality's matrix is 6 triangles facing each other or hexagonally based... 🤔

@Haapavuo 5 жыл бұрын

This is the scariest video of 2018! :o

@mixbaal0 3 жыл бұрын

Randomness is the name we give to the ocean of unknown.

@angelicplace5615 Жыл бұрын

Woah nice Ultra Triforce!

@kaustubha7371 6 жыл бұрын

Yet again the one didn't fail to impress me This time my mouth was open for 2 mins. Legendary!!! Just wait for me to get a job and I will pay half my salary💚

@ThinkTwiceLtu 6 жыл бұрын

really appreciate your comments. thank you for the support:)

@kaustubha7371 6 жыл бұрын

Your videos are amazingly exciting for me. I could never imagine Maths could be so fun. Thank you.

@Overthinkingerrors Жыл бұрын

The first one is called sierpínski’s triangle and it’s very cool

@Ivan_1791 5 жыл бұрын

Why does that happen?

@cubicardi8011 6 жыл бұрын

You need more Subs!!!!

@ThinkTwiceLtu 6 жыл бұрын

appreciate your support:)

@saurabhsharma7123 6 жыл бұрын

I completely agree with this!

@chaddusmaximus643 3 жыл бұрын

2:22 fancy patrick

@ErnienatorLego 3 жыл бұрын

The first triangle is very interesting. If you look at the pascall triangle. And always draw a dot for an odd number than you will get the exact same triangle pattern. Math ist beautifull

@vpambs1pt 6 жыл бұрын

It's rare to find youtube channels that produce the "style" of videos(which I enjoy) that you produce, not too long neither too short, a good explanation of what are you going to do/happening and I particularly liked the fact that you showed that the rule that we use at the triangle doesn't work with the square and showed how to correct it! And as always you choose a good song to follow the video with (:, for example I think that a piano song(Erik Satie for istance) wouldn't go as good as this song that you choosed went. Edit: I forgot to mention that that last few videos that you made, we can actually try it by ourselfs (and not only see a different/cool proof, say that's nice and close the video/investigate about it), one example is "Cutting a Möbius strip in half" ahahha. Sorry for my mediocre and redundant English.

@ThinkTwiceLtu 6 жыл бұрын

Thanks:) It's good to hear from people like you that they enjoy watching my videos. I don't think I would be making a lot more videos if not for feedback like yours. Glad you like the music choice as well. Tbh I've been really struggling in finding songs that sound good for the background of my videos. Most of the royalty free songs don't sound that well, so most of my videos get copyrighted for the songs I use. If you have any songs in mind that you think would be suitable for my future videos feel free to share haha. Thanks again for watching:)

@vpambs1pt 6 жыл бұрын

I should be the one who thanks for posting the videos! I will try to finds the songs, but how can I "share" them?

@ThinkTwiceLtu 6 жыл бұрын

Nuno Mateus You can always email me at thinktwiceask@gmail.com , but you don't really have to look for the songs, dont want you to feel like you're being forced to do that. Thanks again:)

@pumpkinite4889 Жыл бұрын

OH MY GOD WHAT?! I *never* knew that

@hobowithabow9865 6 жыл бұрын

The first was sum good Zelda's lore shit

@kale936 5 жыл бұрын

The math with the triangles came out perfectly matching what our universe probably would have done. The list of dots per size range from 1XXL to 3XL to 9L to 27M to 81S. To see how 9 plays a major part in our universe, check out Marko Rodin; or take the 1-9 multiplication table and reduce all the numbers through addition, for example 10=1+0=1, 12=1+2=3, 14=1+4=5 ect. This produces what I'm dubbing the Layman's Fingerprint of God because you can see how 9 exerts its dominance over it all. If you add it all together you get 459 or 4+5+9=18=1+8=9 and how it is surrounded by 9 and has 9 running though it. Making 9 very similar to our description of God being one who is equal to it all, yet outside of it all, yet running through it all.

@maurolionelmipianoyyo11 3 жыл бұрын

0:53 wow look it’s the sirpenskiri triangle

@ThatLonelyMeatball 3 жыл бұрын

the fact that there are no Zelda fans here saddens me

@michaellight6981 3 жыл бұрын

it's not exactly emerging from total chaos, since there are rules guiding where dots get placed, but nonetheless this is fascinating.

@LordDoome 6 жыл бұрын

Order is chaos at its most chaotic point.

@GabrielsEpicLifeofGoals 6 жыл бұрын

Wait, but what if I choose a part that is NOT part of the fractal in the first place??

@ThinkTwiceLtu 6 жыл бұрын

Gabriel's Epic Life of Goals you will still get the same result

@trangium 6 жыл бұрын

The first few points will not be on the fractal, but all future points will get exponentially closer to the fractal. So after 100 iterations, the point will only be less than 1/(2^100) of a unit away from the fractal.

@weakamna 6 жыл бұрын

You hit on a really interesting point there btw: these images are technically just very very close approximations of the fractals they describe. They are what is called strange attractors. Rulesets that somehow make points chosen at random "attract" into a shape. If the shape is a fractal, the attractor is called strange.

@mathiasbarreto5648 6 жыл бұрын

Interesting!

@lordofsandvich3630 6 жыл бұрын

Would you always get the same fractal given any point, or does the initial point matter? It seems like getting a perfect fractal would be impossible if the middle was selected.

@kitanotatsu Жыл бұрын

I'll just note that the systems shown in the video are not chaotic: even though we might casually understand chaos as "random" (for example, in the context of a mob), mathematically speaking, that view is completely incorrect. Chaotic systems are fundmentally deterministic: if you know exactly how a system evolves and know the exact state of the system, you will be able to compute all past and future states of the system. Chaotic behaviour refers to how small changes in initial conditions of the system can produce vastly different behavious as time goes on. By contrast, with the systems shown in this video the next state is chosen at random: you can't predict with 100% certainty what the future states will be. That makes this video a demonstration of how stochastic processes - systems which evolve randomly - can sometimes form structures which are also interesting for other reasons.

@tophu7903 6 жыл бұрын

Out of chaos... Comes order.

@bharasiva96 5 жыл бұрын

Numberphile made a video on the same topic too. All those who liked this video should check that too. Just type in "Chaos Game Numberphile" on your KZfaq search and you should find it.

@J7Handle 2 жыл бұрын

Ok, but this isn't chaos at all. It's randomness. Each starting point is mapped to a location 1/2 closer to a random vertex. If you map each point simultaneously to be 1/2 closer to each vertex, and then apply that mapping to every point in the starting triangle, it becomes obvious why the sierpinski triangle emerges. With the randomness added back, it becomes clear you're just taking a random sample of the sierpinski triangle. Take a large enough random sample of a sierpinski triangle, and ofc you'll get a sierpinski triangle. That's not chaos.

@duck3892 6 жыл бұрын

*THE TRIFORCEEEEEEEES*

@theshuman100 6 жыл бұрын

Im calling you out on that triangle. If you drew a midpoint from a point to any random vertex it would pretty unlikely to be so close to the middle of an edge unless that point is in the corner

@snowman7514 6 жыл бұрын

beautiful

@nicholasauwaerts2280 6 жыл бұрын

Really Really good vids! As a sience teacher this is gold! Thank you!!

@nicholasauwaerts2280 6 жыл бұрын

Btw Also came through 3B1B...

@ImaginaryMdA 3 жыл бұрын

What's the rule for the pentagon? It's pretty suspicious that it wasn't rotationally symmetric.

@thingsfromspace 6 жыл бұрын

Awesome! I think I saw this on numberphile, but this video took it further. Thanks! Btw, it's cool to see stuff done in processing. I've used it a few times, but mainly I just stick to python.

@ThinkTwiceLtu 6 жыл бұрын

Thank you:) I haven't really tried coding in python so I don't really know the difference.

@MagicGonads 6 жыл бұрын

I did it in p5.js so it was easier to share with others and edit on any PC

@ryanbell3704 3 жыл бұрын

this channel still doesn’t have 10billion subscribers? tf

@mescale 3 жыл бұрын

Wonderful!

@angelscupboard 6 жыл бұрын

*the ultimate triforce*

@islandmonusvi 6 жыл бұрын

Inspired to program these examples in Mathematica CDF, thx

@brenomoraes1 6 жыл бұрын

Excelente video! Good job, keep up with the good work, please!

@ThinkTwiceLtu 6 жыл бұрын

Thanks a lot:)

@brunnomenxa 3 жыл бұрын

Chaos, but it quickly creates a rule that iterates. Iteration is the beginning of a fractal.

@smogy001 5 жыл бұрын

The star-like shape is supposed to be symmetrical

@NoNTr1v1aL 6 жыл бұрын

I absolutely love your videos!

@ThinkTwiceLtu 6 жыл бұрын

glad to hear that! :)

@RafaelBarrientos 6 жыл бұрын

I love your animations!!!

@ThinkTwiceLtu 6 жыл бұрын

thank you:)

@Snowy167 6 жыл бұрын

Pretty cool

@annazarovskaya2366 6 жыл бұрын

Incredible! thank you! please keep it up!

@ThinkTwiceLtu 6 жыл бұрын

thanks a lot~

@richardnormantaylor6556 Жыл бұрын

Reality cannot be whatever you want because other people have different ideas of what is the center . The relation between your Center and other centers is what makes "reality"

@GeorgeCowsert 6 жыл бұрын

Oh. At first I thought this would be a video about making continuous triangles inside triangles, so you could zoom in forever and always see a tri-force somewhere.