- Intro
oh the coolest math video I've ever seen?
anyone else felt nostalgic from the reference
"to link to link to"?
3 blue Pis and 1 brown Pi
advance Happy pi day!! Every one 🎉🎉
six seconds in utter fear of cutoff released by instant gratification at
- Recap, the surprise pi
0 = 215
00 = 74
0000 = 455
= 21
STANDING HERE, I REALIZEYOU WERE JUST LIKE ME,TRYING TO MAKE HISTORY
Imagine the cow as a sphere ahh problem😆
Spherical Cow!🐮 ^.^
what did the cow do
I’m a physicist. I feel seen. Thank you for this point cow.
wait.. im dumb. is sound is a sign of energy transfer? because why no sound?
wow you reed all the comments
love that you knew someone would complain about the sound even though it's an obvious simulation where you add "light" flashes and sound to demonstrate collisions
Do you know what else is massive?
U know what else is MASSIVE?
This is the first time I notice that the word "massive" is actually derived from the word "mass".
do you know what else is MASSIVE?
massive
you know what else is MASSIVE?
you know what else is massive?
you know what else is massive? LOOOOW TAPER FA-
Do You Know What Else Is Massive?
ultrakill door sound
really caught my attention i would like to see what relativistic effect exactly would happen
It's so funny how 1 kg block is starting to break a time-space continuum because of how fast it's being squished
Finished watching at exactly here. Toucheé, world
If you puttes the time at but thanks! Happy pi day
I feel like the small cube is screaming in pain when I hear that sound
bruh you think that i have 10 billion kilograms of mass hanging around with me?
@ just noting, (as explained in Matt's video) that the result shown here wasn't exactly a 100:1 mass ratio, and that they started at 100:1 but it didn't give 31 collisions, so they kept changing the mass of the larger object to "tune" it until they actually got a result of 31 collisions - this had to do with the fact that the problem wasn't perfectly translatable into the real world (still non-zero friction, not 100% elastic collisions, and I believe the actual biggest issue was the imperfect coefficient of restitution). In the end they used a mass ratio of 93:1 in order to get that 31 bounce footage.
Was that a parker square
it never gets old :D
that parker square damn😂
the brief parker square was perfectly timedI really like these collaborative videos you guys make, keep it up :3!
- The game plan
@ just make an attempt. I love it!
its the parker square!!!!
damn you, I was about to go
3Blue2Brown
- How to analyze the blocks
"use the defining features of the problem" do you also put in this category the fact of generaizating or restrenght the problem (opposite to "simpler version" but strangelly efficient sometime)
edit: Machine Mentioned
the ultrakill brainrot is getting to me
mildly accidental Ultrakill reference
my ultrakill brainrot is so bad i had this playing in the background and i actually had to pause and rewind when i heard him say this
V1 and V2…. Is that… IS THAT AN ULTRAKILL REFERENCE?!?
ultrakill reference
at instead of 5.00. The 1 kg block velocity is also wrong. Correct values start at
I'd also list the comparison of m1 >= m2 for the sake of simplicity - if m2 > m1 it may be too difficult to compute at this time as it's a small block pushing a big one into the wall.
ULTRAKILL reference???????
.
"OH MY GOD IT'S A CIRCLE!!"
Not letting the square hit the wall at made me so upset. Great video though, really interesting.
Why didn’t you let the square touch the wall 😭😭😭 that felt so unsatisfying
Wow, wow, wow...@ min plotting the velocities! 20 year physics teacher and never seen that display. Brilliant!
Amazing! Gosh, this is so good ❤
*[****]:*That would explain why the omni-directional symmetry of a circle is so powerful, as well as the fact that any triangle is one linear transformation away from being equilateral.
Nobody:Math: RESPECT MAH SYMMETRY!
"and pi has everything to do with circles, relating distances around the circumference of that circle to the radius"Oops, seems you meant to say TAU here ;)Pi would be the diameter.
I think my brain exploded at around when I realized the equations for x and y were used in the project currently open on the computer in front of me as I'm writing this.
Hang on... if we look at the 1:1 mass case, that gives a slope of -1, meaning we go from (-1,0) to (0,-1). The mass on the right stops while the one on the left moves... so far so good.Then we jump up from (0,-1) to (0,+1) when the left mass hits the wall. But there is a 90 degree angle around the circle between each of these. So there should be a 4th point at (+1,0), following the -1 slope from (0,+1). This implies that the first digit in the 1:1 approximation should be equal to 4.Indeed, the only way the first digit can't be 4 is if the angles are a miniscule fraction above 90 degrees, such that only 3 segments can fit in the circle. But then we're saying the intercepts are (-1,0) -> (>0, -1) -> (>0, +1). But think what this implies - the right mass never goes stationary at any point, and in fact starts to move a miniscule amount to the RIGHT upon first contact with the left mass.EDIT: Best guess why I'm wrong is that, if we sum to precisely 360 degrees, that implies we can exactly equal pi with this approximation. But that implies pi is a rational number (finite length), when it is not.
That was clean
Me, having watched þe oþer video, at_”þe sine what now?”_
half way point
I'm confused. You said some momentum was lost to the wall, but the small block approached it with -1.82 and rebounded with +1.82. Where's the loss?
This is amazing. But I have a question for . I thought we made an assumption that no momentum went into the wall (was negligible or that our system was completely ideal) and was conserved between the two blocks. This seems to be confirmed by the graph, since the magnitude of velocity is the same for the small block after a collision with the wall. Why then does total momentum change?
at this moment I realized that Einstein (yeah, that guy again) shows us we put the momentum back into the wall, which is the whole universe. If we ‘zoom out a bit’, we would see our whole reference frame starts to move to the left. This makes me wonder what we would uncover if we take note the wall as our steady state of reference, but the large block?
I would like to mention here that also geometrically this end zone is forced, because whenever a point lands there you cannot hit the circle again with the slope going downwards or the vertical line going upwards. So the end conditions were already baked into our geometry the way we created it. I suppose you also show this at
There's a programming technique for making collisions(I forgot the name of),similar to the figure generated in , Is there any connections to this study and that programming technique
- The geometry puzzle
@ Strange that the longer the slope line, the faster the small block is hitting.
That just sounds to me like you're missing a dimension. Like you went off the Eastern end of the map and you popped up on the Western side, only because you were moving on a 3D sphere and not on a 2D map. So this circle is probably the projection of a sphere, maybe.
Wow, thath's trippy!
Having seen this I now realise there's easier ways to draw circles in graphics 😅
Around , it became apparent to me where the Pi comes from. It's basically like the infinitesimal slices used in building an integral, except instead of working with areas, it's lengths.
in the video. Cool to consider and makes me imagine what similar math is computed in modeling / rendering software programs when adjusting your cameras viewpoint to your
It can be proved easier. The vertical lines are parallel, therefore the vertically-opposite arcs have the same length. Also, the sloped lines are parallel, therefore the sloppy-opposite arcs have the same length. That is, each arc is equal to 2 opposite arcs: the vertically opposite and the sloppy-opposite. Because it applies to all the arcs and they are contiguous, all of them are equal to each other.
cue Circle of Thales as a special case of this.
I do not know if it’s true but maybe drawing parallel lines crossing the circle can also explain why the length of the arcs is always the same ?
, but to me it feels a bit more obfuscated there.
OMG this update help me to understand truelly/deeply why it's related to pi ! thanks you !
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this is the exact moment when he knws we dont understand shi1 about what he's saying
I love that the navy π creature goes from delighted to angry when it is revealed that he was wrong at
i like how the pi creature got mad when he was told he was wrong
lil blue pi bro be mad 😂
- Small angle approximations
What is slope? Baby don't curve me, don't curve me, no more
Oh my god I’m so glad to see this.. Thank you professor!
lol, this has been mentioned under your previous video, no, just because arctan(10^-k) ≈ 10^-k does not at all imply floor(π / (10^-k)) = floor(π / artcan(10^-k)), effectively it's saying "the interval [π / 10^(-k), π / arctan(10^-k)] doesn't contain any integers just because its length is tiny" it might as well be that for any k it doesn't and I'd assume it's probably the case, but it is a non-trivial fact about the number π, but it'd be honest to a) mention this caveat once you know about it b) give some context on whethere this has actually been proven or that it's an open problem or whatever is this the status of this statement
"These numbers are so close that they might as well be the same" is what I always say
sir you could simply say as n increases (0.1)^n gets smaller and approximation of arctanx=x becomes better and thus error reduces and for equal masses ie n=0 , we verified ourselves thus for next values of n we obviously wouldn't get any error
@@babyeatingpsychopath *
okay this law is actually so good though 😭😭😭
I laughed out loud; the delivery is PERFECT.
the small angle law - so true 😂
Knowing all of this, let's look back at . Using the meaning of the tangent function that I've given here, we can come up with an alternative geometric view to the one shown in the video. When we look at that small angle measure θ on the unit circle, we can draw the line segment representing the value of tan(θ). To sum up, this line segment:
Ooh, I never ever miss an opportunity to bring up what I'm about to say.
im not a math person, and maybe this is a dumb thing to say, but if you have the arctan and the tan, wouldn't you not need to approximate? since it looks like the original line is directly in between the two all the way through, so couldn't you just average the two of them to get the original line?
I feel like there is even a more visual geometric reason: since the tangent is also the lenght of a line tangent to the circle between the horizontal line and the prolongation of the radius you can see how for small angles it's very close to the arc length
I think it is related to one of Taylor sequence formulas for Pi. Edit: Wrote this before watching to
Isn't that exactly the caveat he talks about starting at ? If π / 10^(-k) is smaller than and sufficiently close to an integer (the decimal expansion contains enough 9s in a row behind the decimal point), then the interval you're talking about does contain an integer and the number of collisions does not match the digits of π. That's what he's talking about, right? He explicitly states that this is an open problem. Did you not watch the full video before commenting this?
Interestingly, when I was young and learned about the existence of statements that are true but not provable, I thought up an example of something that would very likely fall into this category (though of course not provably so): the statement that the decimal expansion of pi, some finite initial (and non empty) sequence of digits would repeat itself identically immediately after its first occurrence (after which it must at some point start non-repeating, since pi is not rational). I never imagined such a statement would be of any use, yet it comes eerily close to the condition mentioned at in this video! (But the way, it would seem you need about twice the number of digits 9 in order to risk breaking the pattern of digits, and even then it would not seem a necessary and sufficient condition.)
At whenever I see pi written out I pause and check.In high school (over 40 years ago) I had a contest with a friend to see who could memorize the most digits of pi over a weekend.I memorized 100 digits and went back to school confident in a victory. He had memorized 250. (Hello Ed!). It was a good experience, I like to stay humble.But I still have my 100 digits memorized and checked Grant's approximation. Everything is square: Grant's digits were correct!
but pi is infinite right? So there has to be a point where there is a sequence of 9ns is as large as the numbers that come before?
Dear Grant, thanks for the great work! When trying to derive for myself the condition for the accumulated error to really make a difference in our counting process, I think it shall be more strict here: Consider the first (2n+1) consecutive digits of pi, if the last n of them are all 9s, we’ll get the off-by-one error. The condition shown in the video is more loose thus would not necessarily guarantee an error.
the worst part of this is that because of the nature of Infinity, this scenario does exist. 😭
Just for context, a string of nine 9s (999999999) occurs at position 590,331,982 and the next one at 640,787,382.
when the newly introduced vertical line of the cubes shows in this animation, it is almost as if you are observing the cubes from a camera panning around them - rotating in a circle around the cubes at a distance that appears to be equal to the segment of the circumference of the circle that was calculated earlier at 3D model or that of a video game character in a game.
🔍 “The fact that a simple physical system like colliding blocks can reveal something as profound as π is just mind-blowing. It's a beautiful reminder that math isn't just numbers — it's hidden in the motion, rhythm, and laws of our universe.” 🧠💥
. We can fiddle around with it in order to determine the number of collisions in the experiment; indeed, the video contains a formula to do so at . So what if we make the number of collisions 10, and then 100, and then 1,000? Nothing's stopping us from doing that. As we continue, we get the decimal expansion of the number 1. This still follows the reasoning present in your comment: at each iteration, the number of collisions increases by a finite amount, the angle of the end zone decreases by a finite amount, and we get more and more digits as we go further and further. But the special thing about an irrational number's decimal expansion isn't just that it goes on forever, which we've already established doesn't mean anything; it's that it goes on forever without ever entering an infinitely repeating loop.
technically because one hundred is just your base number squared this is something special about 100
EDIT:
NO WAY i actually got my question from SIX YEARS AGO answered
i was wondering if it worked in other bases.
- The value of pure puzzles
"I just wanna leave you with one point".
this outro was so beautiful.
Yes, I got a point!
, the point in question:
