The most unexpected answer to a counting puzzle
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- Опубликовано: 12 янв 2019
- Solution: • Why do colliding block...
Even prettier solution: • How colliding blocks a...
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An equally valuable form of support is to simply share some of the videos.
Special thanks to these supporters: 3b1b.co/clacks-thanks
New to this channel? It's all about teaching math visually. Take a look and see if there's anything you'd like to learn.
NY Times blog post about this problem:
wordplay.blogs.nytimes.com/20...
The original paper by Gregory Galperin:
www.maths.tcd.ie/~lebed/Galpe...
Evidently, Numberphile also described this problem (I had not known):
• Pi and Bouncing Balls ...
You'll notice that video has an added factor of 16 throughout, which is not here. That's because they're only counting the collisions between blocks (well, balls in their case), and they're only counting to the point where the big block starts moving the other way.
Thanks to these viewers for their contributions to translations
Bengali: Prayas Sanyal
Hebrew: Omer Tuchfeld
Italian: @Deye27, @hi-anji
------------------
These animations are largely made using manim, a scrappy open source python library: github.com/3b1b/manim
If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind.
Music by Vincent Rubinetti.
Download the music on Bandcamp:
vincerubinetti.bandcamp.com/a...
Stream the music on Spotify:
open.spotify.com/album/1dVyjw...
If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people.
------------------
3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with RUclips, if you want to stay posted on new videos, subscribe: 3b1b.co/subscribe
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![NOSFERATU - Official Teaser Trailer [HD] - Only In Theaters December 25](http://i.ytimg.com/vi/b59rxDB_JRg/mqdefault.jpg)
Pi has no business showing up literally everywhere in math.
Wait, it's all pi?
@@onebeets always has been...
What goes around comes around and voila: pi.
Nobody expects the Pi inquisition!
@antiscribe it’s like that one guy who always seems to be at every party but no one knows who invite him. He just shows up no matter where you are.
Originally discovered in 1995, published in 2003. maybe he DID count the clacks?
lol
Alem adamsın
Lmfao😂😂😂
lol
Probably a computer can do this simulation and count the clacks
2:37 I was watching in the middle of the night and got absolutely flashbanged by the sudden swap from dark coloured example to bright white paper.
same
the spherical cow
Another interesting observation : When the masses colliding are powers of some other base (say 3), the number of collisions still equal the digits of Pi, but in the same base.
Eg : Pi in base 3 is 10.010211012222010211002111110221222220111201212121...
If you run the simulation with masses of 1, 3^(2 * 1), 3^(2 * 2), 3^(2 * 3),..., then the number of collisions will be 1 (base 3), 10 (base 3), 100 (base 3), and 1001 (base 3) respectively.
Number of collisions for 1,3^(2 * 50) will be first 50 digits of Pi in base 3 : 10010211012222010211002111110221222220111201212121 , or 2255343044159619899886237 in decimals.
Now that is very very cool!
this is what I thought... awesome!!!
yes
Good thinking
🤓
Teacher: "gimme some digits of pi"
Me: "clack clack clackclackclackcla... clackclack clack clack... Wait for it"
Teacher: "what on earth is that supp...?"
Me: "... clack"
3.1415926535897932
There take that
No, it's:
clack clackclack claclaclclcl reeEEEEEeee clacla clack clack... clack
I demand immediate satisfaction!
I baked you a pie
Damn, this made my day! 😂😂😂
Physicists: "Noo! You can't have ideal collisions make a sound!"
3B1B: "Haha, blocks go brr"
Don't you mean clack
@@midlanismail416 in the 100000kg one it went brrrrrr
The sound also goes hypersonic because the frequency of clacks is so high
@@aa01blue38 lmao what, that’s not how that works dude
@@aa01blue38 "Hypersonic" means "faster than sound"...
You literally just said "the sound goes faster than sound".
This is why I love math. You always look at a problem, read it out loud, then discover something about that problem. It's like there is always a hidden puzzle in math equations. For example, in 7th grade, we were learning about circumference. My teacher showed the class a video which said that if you take the diameter and try to wrap it around a circle, there's a tiny bit left, to which I realized that that tiny bit looked EXACTLY like pi, or 3.14. It's so cool finding small details that make so much since!
🤓🤓🤓
@@user-rc1mv2zy3r thanks :)
@@user-rc1mv2zy3ryou didn't come to a mathematical video to just spot "nerds" now didnt you
🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓
the clacking sound is so satisfying i want it on repeat forever in my brain
1:40 Me opening the door at 1:43 am
3:14 am
PRODUCES SLAP BASS MELODIES，SO TRUE
hahahahah
@@orvillevroemen3956 3:14
Sounds like a sound effect on a zx spectrum game
That animation of the spherical cow actually made me wheeze. That was unexpected
Selicre [Hyper] it’s my favorite picture on wikipedia
that's a great image
@@NickiRusin I really, really, really like this image.
@@Selicre a long time ago my dad told me a great physics joke. I don't have the patience to translate it from Russian, but the punchline boils down to "a spherical horse in a vacuum". For some reason I never tried to visualize that, but now it's crisp in my mind thanks to this video.
ASSUME THE SPHERICAL COW!
I thought your video on relating the Basel Problem to the circle was simply gorgeous, astonishing and unforgettable. These three surpass even that! Thank you so very much!
i love coming back to this video every once in a while because it's just so mind boggling that it reblows my mind every time
Highest quality RUclipsr out there. And I mean that in every dimension.
Make that in concurrent parallel dimensions 😉
Even the fourth?
@@jacobkleeman8546 All spatial and abstract dimensions up to infinity, even the temporal dimensions (if 2D time and up makes any sense). Even the fractal dimensions where you can have non-integer values (like 1.3425 dimensions). Even any system, existing or to-be-discovered or purely nonsensical for the sake of argument, that calls for negative value dimensions, or imaginary number values, or complex values, or quaternions, or octonions, or sedonions, and beyond.
@@ariqahmer we need to talk about parallel universes
@@papi1050 Agreed 🤔😎😍
*gets this on recommendations for the 10th time*
Brain: click on it.
Me: but I've already wa-
Brain: *do it.*
this is probably like my 6th time
Its my 5th time
its my 4th time
This is my 12th time…
Ah yes..
𝗥𝗲𝗹𝗮𝘁𝗮𝗯𝗹𝗲.
Im not here for the math stuff. Im here for the colliding noise..
And im here for cow sphere
I like how the speed of the last collision is an expression of the remaining digits. So when it's 314(15...) collisions it juuuust reaches the line, but when it's 31415(92...), it gives the moving block a proper final spank to send it on its way.
wait, if what you're saying is true, we can get more digits by analysing the speed more intuitively?
Pi is a creep. I'm gonna file a restraining order on him. He has started to show up on my integration problems now. He's gone too far.
pi is a cursed number, way more cursed than 13 or 666.
@@oblivion2755 whats wrong with 13 lol it's my lucky number
@@iqbaltrojan oh the irony
@@oblivion2755 *4* is the worst
@@oblivion2755 indeed four, or, in Japanese, shi, which us also the Japanese word for death, is terribly cursed
Clack.
[ I donot know. Let us ask someone. ]
At some point the clacks would be so rapid that the frequency created would be too high pitched for human ears lmao
@@finalftl732 so ideally you'd find the ideal magnitude that over a 10 hour total period would at some point get closer to the highest averagely humanly perceivable pitch than the next magnitude, which breaks that barrier. And there's your 10 hour video.
MrBeast can, ofc.
Just loop the system - add an opposing wall the other side of the masses/objects; masses between 2 walls. Then that system cycles, repeating (to infinity). Under the special ideal conditions. A truly closed system, with only total motion & energy transfers, where all is constant & only velocity can change.
3Blue1Brown never fails to make me question reality!
thank you for giving me a math project! this was very fun to work on and you explain this very well.
Small brain: Memorizing digits of pi
Galaxy brain: Having blocks of precise mass on hand and counting the collisions
Also me: destroying earth to find Pi to the 20th digit
Don’t forget about removing friction!
Never have I ever thought that the small explanation bit with words sliding away at 3:59 from a math video would get me giggling like a kid at six in the morning
Doctor: it’s not gonna hurt!
The kid in the next room: 2:22
Lmao
Saw this at least 5 times. Still amazed at the quality of the explanations and the correlation itself. You are truly one of the best out there.
Watch it again
@@marcgonzalez5628 aight
again
Again
@@destroyer100onblitz ayo it's only been 4 days
I don’t speak English.
So I don’t get it well.
But when I got that the collisions number turned near π, I was like “！？.”
It was so beautiful phenomenon.
cool
Hi (sorry for my bad english)
!!コメントだけで登録者10万人を目指す すうがくはすばらしいですね
Blakee
Yes, I believe math is beautiful and also amazing
!!コメントだけで登録者10万人を目指す You write English wonderfully! ❤️
0:19 Love the spherical cow animation btw.
i truly understand why people loves so much mathematics, all makes sense and everything is explained , thats just magical 😊
Any maths/science/engineering problem: *exists*
Pi: aight imma head *in*
2:32 ' Credit to the viewer Henry Cavill.' Of course Superman would know the answer. He's brilliant at math. And physical education.
*kavle
@@andrewzhang8512 guess who didn't get the joke
@@mono6359 ohhhhhhhhhhhhhhhhhhhhhhhhhh mb
Lol
Silver for physics, steel for maths
Thanks for showing me math can be fun and interesting. Great video
I think the large cube motion represent as a semi-circle or half sine wave. Maybe something has to do with that which would be half of pie no collision to the big cube and half value are from just the end wall.
I always COUNT my blessings whenever 3b1b uploads an UNEXPECTED video. As usual, great work!
We feel the same way about you. :)
I love the videos you put up! Great content that is nearly impossible to find elsewhere :D
I am also making video on physics
@Just A Random Dood Shhhh let me make my lame puns on the title
I Subbed to Ur channel and really glad to have found you
2:31 when the actor for Superman helps out 3Blue1Brown
LUL
But Gregory Galperin is the real Superman here ❣
When you need to move blocks weighing 10^(20-1) kg, you'd better call Superman to help out!
Is it really the same person ie superman?
True LMAO
I’m just here to say that I’m glad that there was sound on those simulations, they were satisfying
0:38 the best decision ever took place on the planet
WOW!!
Well hello there :)
Hi BPRP! Love your videos
Isn’t it?
Congrats for the 200k man
1/1-x
me thank you!!!!
Sliding off to infinity, never to be touched again- so sad
*adding sad comment about you and your ex
my dad's hand and my crtoch
r/cursed comments
@AssmasterFlex69 until the big crunch, and everything will be once again, at once place, at one time
Bobby P the blocks will forever remember how their first and last touches were
"I highly encourage you to take a stab at it yourself" is the most someone has ever overestimated my abilities
I love how the pi creature goes "this doesn't seem like me..."
What the what!!!!! That's so cool
Wow I didn’t know you were interested in this kind of stuff!
Yo big fan dude keep it up
Fancy meeting you here, woodsman.
What the what
wood
“Like a satisfying game of breakout.” Is my favorite analogy on this channel so far.
The ideal cow takes the cake for me
"Sailing off into infinity never to be touched again"
I felt that.
I loved this video but lost it for 3 years I am so glad I found it again,
comment "salute" for those who still haven't refound this
salute
3:56
Why is this cow your idea of "way over-idealized" and why do I agree?
i don't know about a spherical cow, but there is popular joke about spherical horse:
Some man hired physicists to calculate which horse will finish first in the upcoming races. They gave him their results. And that horse didn't win. Angry man asked physicists why is that so, and they replied, that they calculated race results for spherical absolutely black horses in vacuum.
So this is a joke about over-idealized conditions that physicists use in their calculations.
DifferentStuff Yeah, that’s pretty much it. Both my engineering teachers in the past enjoyed the phrase “spherical cow in a vacuum,” which just says how engineers and physicists would assume the cow is a sphere so that calculations are much easier
@@fordsquared537 In my language it's a horse.
in topology, you count holes. A cow (assuming it had no holes) would be the same as a sphere from a topological perspective since they both have no holes.
It's from a Neil deGrasse Tyson's joke. It's about how physicists love to see the things in universe to be a perfect sphere. If u ask a farmer what is a perfect cow, he will answer a perfect cow is the one which will give lots of milk, a butcher will answer a perfect cow is the one which is fat. But a physicist will answer that a perfect cow is a cow which is spherical.
Pure poetry! ❤️
How has noone noticed you lol?
Olha, brasileiro na área
O RAFAEL
Krlh a lenda aqui.
yo
3blue: Quick! I need some visual way to show the audience how over-idealized this simulation is!
1brown: Cow sphere
3blue: w h a t ?
1brown: _c o w s p h e r e_
cowsphere is one word actually
@@shadesmarerik4112 hiw do you know that???!?!?
did he stutter 3blue?
@@achtsekundenfurz7876 Oh so it’s just the humor for simplified models such as a spherical cow because the shape of a cow is too complex lmao
“Assume a spherical cow in a vacuum”
This is amazing, physics teachers never taught this, and everything we learned was designed to solve problems, from conservation of momentum to conservation of mechanical energy, and we even came to a conclusion E（lose）=1/2 (M1•M2)/(M1+M2) • V（relative） how fascinating physics is now
something about the collision sound is so satisfying
Everybody gangsta till the blocks start pullin’ out autotune
supertone lol
Me: Hey that looks like Pi lol what a coincidence
Me: Ah
Exactly what I was thinking 😂
It probably appears because the velocity gained and lost between both cubes which then represent aceleration diferences which then represents the variation between sen and coss
The transient of any natural movement will be e^n where n is x*jw, which is a frequency.
Then, any natural movement has “e” implied and a natural oscillation associated. We know from Euler that there is a relationship between e and Pi.
Great video 👏
Originally discovered in 1995, published in 2003.
Galperin (for 8 years): "Unbelievable. Unbelievable. Unbe... Well, time to publish then, I guess."
that's what i was wondering. why wait almost 10 years to publish it?
The time he took to count the no. of collisions
@@hiransarkar1236 Gralperin: "956...957...958..."
His wife: "Honey. Dinner is ready!"
Gralperin: "Sure, I am comming soon...fuck...1...2...3..."
Xd
no
1:40 what a cool sound effect
Kinda sounds like the beginning of that one Crystal Castles song🤔
Sounds like a radio
the beginning sounds like a geiger counter
sounds like a creaking door
@@hishykotCrimewave? Yeah I thought the same thing.
i dont know how but this guy makes math actually interesting
Thank you for letting me learn what “ 20 to the power of 5” is
I really, REALLY, appreciate you leaving in that last bump at 3:53
Its just so amazing when the 1 kg block is moving slightly faster than the other block and you are just like ooooooooooooooooooooooooooooooooooOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOHHHHHHHHHH AAAAAAY
@DON'T I WONT 😶
Woah. Freaking woah.
@Laquelectro woah
s
mr beast give me moners
@@ignacio6851 this is not Mr Beast, but Mr Beat. Instead of giving you money he gives you a beat down
Mr Beat on a 3b1b vid? Worlds are colliding
Thanks for the video. Enjoyed it.
perfect explication of this fenomen. perfect video continue like this bro.
1:34 The sound is perfect
134 rearranged is 314
bambi fantrack
Duck
Content - 💯
Editing - 💯
Voiceover - 💯
That's the definition of 3 blue 1 brown. Keep up the good work. U will definitely hit 10M subscribers soon
I would give another 💯 for the colision sound kkkkk
I wouldn't say soon, because for some reason most people hate math for no apparent reason. If they were to see the true beauty of math I believe there would be a lot more people loving math!
@@guilhermegondin151 true, how could I forget that?
@@enverko Yaa you are absolutely right
Date format - 0
i have rewatched this at least 10 times, still entertained
every now and then, this video comes to my feed, and i watch it just for the sound
Wrong
The 100²⁰ would have destroyed our slippery floor
And our tiny cube, either that or the bigger cube itself collapses into a black hole lol
@@carltonblend And eats the Tinny cube
@@carltonblend and even if it has no enough mass for a black hole its gravity influences purity of the experiment )
what about a 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 kg
mass
@@carltonblend What if the cube is made out of bedrock?
Dr. Galperin was my geometry professor at University. I have never enjoyed geometry so much in my life. The man knows and can prove an incredible number of astounding, non-obvious facts. Thanks for sharing his work!
Common core lessons in a billion billion billion billion nutshells,
Clacks is in session
I was with you until you mentioned the Black Hole lol. Yet; as someone who was not taught math in a fun way, I'm very grateful to you for creating this channel. Thank you! ✖➕➖➗〰🟥🔶❇🔘💜
That "clack" sound you added is apparently ASMR to my ears, so it's very appreciated.
Ah so 3blue1brown is blueballing me. Figures 😂
you mean 3blue1brownballing you..?
3blueballing
Would you rather they brown balled you?
tfw you came into the comments to make this exact joke xD
Well at least you can see where the circle comes from coz the equation of a circle is (x-a) ^2 +(y-b) ^2 = r^2 so 1/2m1v^2 + 1/2m2v2^2 = const. is clearly a circle equation
do you have any update on this? I found an interesting correlation between pi number, circle and perhaps collisions of all particles in the universe? Is there any concept of collision force? Excuse my questions I was not good at physics during school I am just being curious and imaginative
Question: Would this same algorithm compute TAU if we switched the walls once after the initial series of collisions before the larger block is ejected back out from where it came? Let PI be defined as C/r (circumference of circle divided by the radius), AND TAU which is C/d (circumference / diameter).
Can we reverse the roles of the walls to block the heavier block from leaving, so that the "high impedance" wall and "open circuit" walls are switched?
3:56 me after watching this video....
YEAH PHOENIX
3B1B's homework best homework
The next day...
Math teacher: Have you done your homework for today?
Me: No, but I know why if you shoot a moving object to a still one with a mass ratio of 10^k under no friction conditions you get the digits of pi!
I know, I'm gonna do this instead of my actual homework
I find it very interesting that in the first case 100% of the energy is transferred back onto the initial 1kg while for high masses a certain amount of energy will stay on the left mass which will never catch up with the larger mass again
3:56 why did this make me love the whole video
Reading comments section:
.
.
Expectations: people discuss math
reality: clack clack clack
Okay how many collisions if it was 10^1,000,000 times the weight of a 1kg object?
Me: C L A C K
1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000^10 years later " ... Clack.. I freaking finished to count that, oh boi!!"
It will become part of it.
Imagine gravity.
Or it would get so hot. May be it could melt.
@@full5339
I prefer to read this comment without the context of your other two.
Hello.
_Imagine gravity._
1:38 open the 0.25x speed
Nice
когда не мог определиться, что сдавать и купила на EL курсы Кико вместе с курсами физика и теперь ты здесь
me: **moving frame by frame at **2:29** and seeing 59 instantly become 313,979** **doing the same for **3:12** and seeing 941 become 314,159,265,136** this looks so fast... gotta know how fast it was...
So true. I wanna know too. I could not even pause it between 100 and 3 hundred trillion
@@AspectOfTheBlade He literally said the rate in the video.
literally did the same thing bruh
@Fernando García salazar i already knew that
Unfortunately it will be faster than the frame rate of the video. You would need a 314 million fps youtube viewer
3:58
I like this representation of how this is just theoretical, not practical.
Creative.
This is so cool bro I am amazed by this
This is mind blowing.
As a 6th grader, I don't understand the terms you use in your video but I so much hope to learn them throughout my coming years. I find your videos very intriguing, keep up the amazing work!
Keep being curious bro😎🔥
@@PritamDavis until being killed by the education system of the country
@@e2532e agreed bro.. it really sucks at times
@@e2532e well 😔
@@PritamDavis im 7th grade rn
Perhaps you could set up analogous situations for any other base:
0. You have two blocks with masses of 1 and b^2^(d-1), where b is the base of the number system and d is the number of digits in that base you want to compute.
1. Count the number of collisions in that base.
2. You now have an approximation of pi*b^(d-1) in that base.
3 years after this video is published, I discovered something(not sure if anyone found this out before)
I think the number of collisions
=Pi*sqrt(ratio of the mass of the block on the left to that on the right)
This is the most satisfying video of 3b1b😁... Btw did I fail to hear an ultrasonic frequency sound @3:44?
3:43 The sound of my brain figuring out the puzzle.
100,000,000,000_000,000,000_000,000,000_000,000,000 (10^38) (100 tripodecillion)
Solved :D
The circle in question is a circle in the momentum phase space. Specifically say m2 is the heavier block, if you rescale the momentum variable p1 and p2 by sqrt{m}, then you have a very nice circle equation. The entire process amounts to turning by angle pi from the point (p1,p2)=(0,1) to (0,-1) so that the heavier block is reflected.
Every collision between the two blocks can be written down as a mapping between points on the circle in the phase space(ie old momentum determining new momentum completely, while obeying energy conservation so staying on the circle). Every collision at the wall serves to reflect the point about the p2 axis so that the direction of p1 is flipped. Every step leaves a point on the circle, and each point corresponds to a click sound. So the total number of points on the circle(minus the starting point) is the same as the total number of clicks heard.
Since all collisions between two blocks is followed by a collision at the wall, we can consider these two events forming a single step. It turns out that as the ratio r=m1/m2 gets smaller, this step approaches an infinitesimal rotation generator, with each step’s rotation angle being 2sqrt{r}. If we let k to be the number of rotation steps to execute the full pi angle rotation from (1,0) to (0,1), then 2sqrt{r}=\pi/k, or 2k=\pi/\sqrt{r}. We identify 2k as the total number of points on the circle(involving both the rotation and the reflection, thats why there’s a factor of 2), then if sqrt{r} is 10^-2n, we have 2k=\pi * 10^n, which is exactly what we have.
The only thing left is the round off but I need my beauty sleep now XD. Can’t wait to see Grant’s solution and animations! :D
That would only be the case if the sum of the squares of the momenta (p1^2 + p2^2) is conserved, which is not the case. Conservation of energy demands that p1^2/m1 + p2^2/m2 is conserved, which is the same as what you tried only if the masses are equal
Energy is also conserved here
@@sauravchauhan4172 How so?
@@coconutflour9868 he said that in the video too , energy is also conserved , and I think circle is not the solution, it can be an ellipse also. Maybe wrong
@@coconutflour9868 I rescaled the momentum variable by 1/sqrt(m). That means given old p^2/m, I define P=p/sqrt{m}. The conservation of energy equation p1^2/m1+p2^2/m2=E then becomes P1^2+P2^2=E. If we use the rescaled momentum variables as the axis of the phase space, then states of constant energy form a circle.
5 years and I am still waiting on Matt Parker to use this method on pi-day
My 10 year old son showed me this video. It stunned both of us.
2:30 when superman passes his time doing maths.
Was about to comment something like this😂
I don't get it.
Yes I'm not the sharpest tool in the shed
@@user-ov2fc5sd1e henry cavil plays superman in the dceu
@@user-ov2fc5sd1e The Big Mass?
@@user-ov2fc5sd1e the youtuber says henry cavil lol henry cavil is the actor name for superman movie
1:50 wait for it.... WAIT FOR IT...
love it
This is the greatest video I’ve ever seen
I was watching this from the preview, and I would swear the counter of "314 clacks" hit exactly at 3:14 left in the video. Well done.
@@RIPToot it was, it says 3:14 *left*
This channel is of suprahuman intelligence
At 1:58 if youre curious
3Blue1Brown: "We have 2 sliding blocks and a wall"
Me: "I'm sorry could you repeat that, I'm already lost."
Can you link he program used to run the simulaton
This channel should be called: 3.142 Blue 1 Brown
So, Superman shared this with you? 🤯
Then this is actual kryptonian knowledge
I'm surprised more people aren't talking about this.
@@micahbradford2629 i know right?