@@numberphile this one felt like a collab more than an interview, I liked the video none the less. I hope this will give 3blue1brown even more exposure and maybe help out a few people who may not have the best of teachers, to learn calculus and linear algebra the right way.
Arvasu Kulkarni I cant remember when I saw it, but there was a demotivational poster with that as the line, and someone who had split an arrow...at the very edge of the target.
@@bentrash7885 In layman's terms: Doing things in a consistent way is only beneficial if you are consistently doing them well. There is a saying that goes something like "Consistency is key." I assume it is a play on that.
This is probably the best demonstration of a useful application of higher dimensional math that I've ever seen. Compressing a series of 2D coordinates into a single nD coordinate to get the probability of the whole set. Awesome!
"2d coordinates that can be compressed (normed) into a single coordinate" is an all but complete operational definition for the field of complex numbers (the formal name for this would be the "canonical real structure" if im not mistaken). its no exaggeration to say every STEM field finds its own wildly different favorite use of complex numbers. but more than that, the entire subject of algebra, as you may or may not know, has a pretty large hole in it that isn't actually possible to close using anything other than these same 2d numbers. so in a way, they really are the very thing ensuring any and all reasoning you do using your school textbook math will be logically rigorous; that it has the consistency to carry useful meaning by default in whatever practical context you can give it.
Two videos we now need: 1. A Healthy Relationship with E: What the factorial sum really means 2. Calculating the volume of spheres in higher dimensions
@@trickytreyperfected1482 it's kinda the same idea as with triple integrals, except it gets higher, you're still integrating independent variables nonetheless the bounds do depend on those variables, unless you go to higher dimensional spherical coordinates system, which is much easier integration is but might be harder to derive and understand intuitively
I am not a mathematician, most of the stuff on numberphile goes over my head but it is just SO satisfying to watch. For example, I have no idea what e is, I didn't understand anything past explaining what the game is, but it is so damn satisfying to watch, it's like wizardry
The way I rationalized the ratio of the volumes of a hypercube and n-ball is with the corners. You keep adding more and more corners that the ball can't reach.
Also the corners get farther apart. A corner of an 100-D cube is sqrt(100)=10 units distant from the centre (assuming that all sides are 2 units long). One visualisation of this I like uses a projection into 2 dimensions. One can cut a 3D cube with an xy plane rotated 45 degrees around the x axis. This cut gets you a rectangle of dimensions 2 * 2.82. This plane also cuts out a circle of radius 1 from a sphere or radius 1. If you draw it, you'd notice unused space -- there's a margin of 0.41 units on each side, and an even bigger margin in the corners. A similar plane cut can be performed in 100 dimensions. A cut along orthogonal vectors a=(1,0,0,0 ...) and b=(0,1,1,1 ...) will result in a rectangle of size of 2 * 19.9 units -- lots of unused space there. We get even more slack if we cut along (1,1,1,1 ...) and (1,-1,1,-1 ...) -- we get a square of size 14.1 * 14.1 units.
This kind of content really does make me realize my enjoyment for mathematics. When I’m able to break the form of rigid, applied math and enjoy some almost philosophical branches of mathematics, it makes me want to pursue the subject despite my previously held distaste for the topic.
"I'm gonna try to make this a worse shot." Grant Sanderson fails the way Matt Parker succeeds. Of course I kid. Both of those kids put out amazing content. I always look forward to a new standupmaths or 3B1B video.
I've read a number of Egan's books, and considering some of the bizarre things that goes on in them, I can totally believe he'd cook something like this up in his spare time. Furthermore, I'd just like to say that 3Blue1Brown continues to have the most pleasing math sounds around.
Love this guy and love this channel. A few years ago, when I needed something a bit more mathy than Vsause, but my teenage brain couldn’t quite make it through a Numberfile video, here came Grant, explaining the beauty of math like no teacher or textbook I had had up to that point ever could. And probably ever since as well.
So, obvi I knew Grant was smarter than me because I watch his math videos to be smarter, but I never imagined that he was also more attractive than me and this is really shaking my world view rn
@@constracted7331 It's one thing to believe you can get better which is entirely true, but it's another thing to think that with hard-work, and being bounded by the constraints of reality and time, you can become as talented as anyone, this is simply not the case as depressing as it may seem.
@@constracted7331 not sure but pretty sure he's somewhere 23-29 years old yet he's a fking mathematician from those very clever solutions to hard math problems. And what I know that I think during that age is still a time on achieving PhD on mathematics yet this guy has seemed to become a maths professor.
The small volume of hyperspheres can also be understood in a Geometric way with sections: think of the 2d and 3d case: If you slice the 3d box in half and look at the section you will see exactly the 2d case, with the same proportion of ball and empty space. But every other parallel slice will have less ball and way more empty space! So it's kinda obvious that the volume of the 3d ball is less, it would have to be a cylinder to be equal! This extends to higher dimension by taking higher-1 dimensional slices. Of course the algebraic phrasing is more precise. Also some may think that this slice thing is cheating since you can't visualize that either: well, topologists and geometers are not superhumans and have developed lots of tools such as this to think about higher dimensions, and I think that all those tools deserve to be regarded as Geometrical thinking.
Hey! I know this is really late, but I tried the simpler version of the puzzle (x1 + x2 + x3 + ... < 1) and I got sqrt(e) as the answer. Is that correct?
@@khandanish2004 Hi! I think you made a mistake, it looks to me like you've considered the volume of the cross polytope in 2n-dimension to be 2^n/n!, while it is 2^2n / (2n)!. Remember, in the original puzzle we only end up working in even dimension. (A cross polytope is the n-dimension version of the unit ocrahedron, it's the counterpart to the hyperspheres when computing the probability |x1|+|x2|+...
@@99Megaluca99 My intuition was even simpler: Look at the relation between a square and a circle thus: You get the circle by smoothing out the corners of the square. If you approach higher dimensions you get 2 to the power of dimension verteces (=corners), therefore we should have exponentially more corners to cut as wonder into higher dimensions.
Okay, maybe not simpler, but an intuition nontheless. This ultimately shows as well, when you think about how you produce one higher-dimension ball: You spin it around an axle that is perpendicular to the dimensional set that contains our ball.
"Because why not" is the definition of why math is so fun. The rules are clear, the reasoning rock solid, but the freedom to play with the assumptions is unmatched. No physics to restrain you. No "right" way to approach a problem. Another fun one is "without loss of generality". I can't count how many times these words stopped me for quite a few minutes while I tried to understand why and make sure it really is... The revelation was usually fun. Not for obvious cases of course, but some authors can make the word "clearly" work VERY hard.
@@leif1075 there are infinitely many points that can be hit, and the center is just one of them. it would be more accurate to say that the probability approaches zero
Grant, you have an amazing talent for teaching mathematics. I've watched many of your videos and have learned a ton. Brady, I've learned a ton from all of your numberphile videos you've produced over the years too. I have nothing but deep respect and admiration for you both. Thank you both for all you do for mathematics education for us common folks!
As a statistics student in university, this video blew my mind. That's because there is a relationship between probability of something happening and all that higher dimension spheres, e, pi... just amazing. Thank you so much for the great content.
Thanks for the video! I thoroughly enjoyed it, and thought about the puzzle for a bit, and I thought of an interesting way to attack the original puzzle. First, replace the original square with a circle with the same area, and the problem remains unchanged. Also, scale down the diagram so that the possible range of dart throws is the unit circle, and the initial bullseye is a circle of radius sqrt(pi/4). Now consider a point P chosen randomly from the unit circle, and let R be its distance from the origin. We claim that R^2 is uniformly distributed in the range [0,1]. This is because for some k
31:28 "What is the probability that the sum of all those squares is less than 1" That is the most helpful analogy for understanding higher-dimensional spheres.
This was one of the most informative videos on multidimensional geometry, quantum physics and probability. This puzzle answers the question for me that the universe when described in probabilities does not imply a multidimensional space. Thanks for the understanding… very powerful!
I’m glad you chose zeta as the outro theme. Vincent rubinetti really did a great job composing the music of 3blue1brown. And Zeta represents that amazingly. I could go off on a long tangent on how much of the music is centered around curiosity and wonder but you just have to listen to it.
sooo umm it was definitely a long video for me, guess it's just not my type of a content (= I'm not very smart). I felt like a two year old watching some very skilled magician doing tricks and trying to understand what's even going on. Saw everything, can explain almost nothing :( The saddest part here is that I have a bachelor degree in chemistry
I'm just mindblown by the fact that the volume of spheres "picks up" some extra pies as they increase dimensions. It's so weird! Maybe there's an interesting reason for it that you could explain in a video!
While I definitely don't have an answer, I'd check out 3Blue1Brown's "Why is pi here? And why is it squared?" It has a really nice explanation why a pi squared can show up. Maybe that same chain of logic can start to be applied here.
I'm not sure if this is the right explanation, but IIRC, there are two independent axes of rotation in 4D space. Assuming n-space has floor(n/2) independent axes of rotation, then I think every other integral would be bounded by [0, 2π]...? It's been a long time since I did any calculus
If you look up the formula for the volume of a torus (donut) in 3D space, it also has a pi^2. Because you can kind of think of there being two circles going on in a torus, the radial one and the cross sectional one. And then its volume is the area of the middle circle swept around in another circle. And then yes, as Okuno above me said, in 4-d space you can also have these two independent circles (say two coming from the first two coordinates and two coming from the second two) and so in some sense it makes sense that when integrating and multiplying stacks of circles to get volumes you get two factors of pi.
@@Krekkertje If you don't need great precision, a compass is easily improvised as well. All you need is a pin to center the arm, which could even be made of the same cardboard as his straightedge. Poke a hole through the arm at the distance you are scribing, and place your pen/pencil/marker tip through it. A string to tie from pin to writing implement is more accurate than the cardboard, but not quite as common in most offices. The string is particularly useful in geometry though, as it can also be used to measure pi. =]
@@Cuuniyevo You need a pin, and to poke a hole big enough for a pencil to fit through. Far easier to just take nearly any object and use it as a straightedge.
Super awesome video. Thought that without the animations, it would take me out of my zone (in my mind). But this puzzle had so many pieces, and I followed fully. I only regret that I would not be able to connect all those dots on my own, if I were presented the same question. But I'm practicing thanks to your help!
I really like how, if you include the radius of the circle into the calculations, the final result is instead e^(pi·R²/4) = e^(A/4) where A is the area of the initial circle. Even more naturally, A/4 is the ratio of the areas of the initial circle to the initial square. So the final result was e^(probability of an initial hit).
That’s actually a nice generalization, because now you can ask the harder question of if it was nonuniformly distributed or if it was radially symmetrical
This man is the best teacher i've found on youtube. He is an inspiration. He makes me excited to watch a conceptual math video. Much like numberphile does
4:54 Well, probably not rotationally symmetric; probably more elliptical than circular. You have to counter gravity in one plane and not the other, and the mechanics of your arm probably affect the result. But a rotationally symmetric gaussian distribution would likely indeed be more correct than a uniform distribution, yes.
As a 3blue1brown patron I've been waiting for this video. :) I've tried to solve the problem on my own for weeks and failed, so I couldn't be more excited right now!
You are right, I allways feel like the idiot who doesn't get it when I watch their video's. Still I love to watch them, in the hope it will make me a bit smarter, but I allways end up confirming to myself to be an idiot
Tbh, 3blue1brown is a much better channel. Numberphile has a bad habit of oversimplifiying problems (such as in the infamous -1/12 video), where 3blue1brown shows how mathematicians actually discover new math.
Same! I felt like a two year old watching some very skilled magician doing tricks and trying to understand what's even going on. Saw everything, can explain almost nothing :( The saddest part here is that I have a bachelor degree in chemistry
Really appreciate the call out that while geometric interpretations are useful, they're best not taken as only relevant for people living in a strange world experiencing that many spatial dimensions. Any collection of related measurements can and are reasonably considered "dimensions." Just a table with a few extra columns. Very practical. Nothing timey-wimey.
it's so cool that even though it is impossible for us to conceptualize or create or simulate upper dimensions, math already gives us properties of how those dimensions behave. It's like you're locked out of a house and you have no way of interacting with the insides of the house, yet through some big brainness you can figure out some things about what's in the house
Sophomore engineering student here! We just started getting into things where you translate physical units into other "domains" like the phasor domain in electrical engineering, various domains in differential equations, and Mohr's circle in mechanical engineering, and it really shows why all of these transforms are useful. It's simply way easier to understand stress transformations when you explain them graphical using Mohr's circle rather than their physical derivation.
I think it may seem counter intuitive to think that a 100 dimension circle takes up so little of the 100 dimension cube but if you step back and analyze what is visually happening when you increase a cube's dimensions you can see it is adding more and more "corners" and when you see that you realize the smooth circle that encompasses all of the cube except the corners you can see why as you get higher dimensions your "ball" takes up a smaller and smaller portion of the "cube". It may be wrong to think of increasing dimensions as increasing"corners" but that is one of the things that happens as you increase your dimension. I think maybe visualizing it that way would help make the fact that a 100d "sphere" takes up so little of a 100d "cube" a little more natural feeling
30:24, I'm afraid there's a small mistake here. The volume of a ball in 100D in 2.e-40, but the _proportion_ it covers from a cube is actually way smaller, since you're dividing by 2^100.
Some alien somewhere: "Oh no, I dropped my 100-dimensional ball into a 100-dimensional bucket, and spilled 10^-40 ml in the process! Gonna take me at least a Planck-time to clean this mess up..."
Depends. Is the boundary part of the circle or not. Which depends on wether the circle is a closed or an open set. In general sports, the boundary is included though.
@@AdeonWriter well, there is no such convention, you generally have to specify if you're considering an open or a closed disk (a circle is the boundary itself, much like a sphere is the boundary of a ball). This was never done in the video just because it was irrelevant to the problem, you get the same result in either case.
@@RodelIturalde Hitting the line would make it so that all the X's and Y's squared equal exactly 1. The formula states they must equal less than 1. So hitting the line count's as a miss.(timestamp 13:13)
Interesting puzzle, great video, love you both! I am not a math major nor any sort of academic worth the name, but I have a logical and mathematical mind, so this sort of content fascinates me. I've seen this video several times over the years, and each time I learn something new. Today, I think I learned an intuitive way to understand "probability zero" and why it's not the same as "impossible". My old preconceived notion was that the opposite of infinity is infinitesimal. And maybe it is, but I don't think that way anymore. Of course, infinity is conceptually the biggest possible thing, so infinitesimal should be the smallest possible thing. Mathematically, some infinities are bigger than others, but that's a later topic. If we confine our domain to the interval [0, 1] as we would when discussing probability, how might we define the smallest possible thing? One idea I heard about some time ago is "0*" (zero-star), a number so close to 0 that it is "confused with" 0 but not actually equal to it. We might think of this as 0.000...1, with infinitely many zeros between the decimal point and the final 1. We have seen something like this before at the other end of our probability range. What is 0.999...9 with infinitely many nines? That's 1, as was nicely explained in another Numberphile video. In part, it's because there is _always_ some real number between any two other real numbers (infinitely many, actually). If there are no numbers between 0.999...9 and 1, then 0.999...9 = 1. Similarly, if 0.000...1 exists, there _must_ be a number between it and 0. If so, then it's not the closest number to 0. If not, then the number can't exist in our domain/range. This leads to a contradiction. If 0.000...1 exists, then it must be equal to 0. Thus, a probability of 1/∞ = 0. Not "close to", not "confused with", but actually 0. The only caveat is that if you take the inverse, you can't say 1/0. You'd have to say 1/(1/∞) = ∞/1 = ∞. I'm not saying that this reasoning is correct or rigorous, but it's one way to make sense of it in my own head, so I thought I'd share it.
I'm a second year grad student with a BSc in Physics and Applied Mathematics, and the math that appears in stuff like this still blows my mind. Math really is magical. (Also, studying physics, I'm glad to see a Taylor series not chuncated at the second order.)
Many years ago I was working in the supermarket industry. We characterised shoppers by their propensity to shop in different areas: meat, dairy, fresh veg, canned goods etc, giving us a set of n scores. We used these as coords in n dimensional grocery space. This "space" was really easy to conceive and navigate. That was the only time I've ever felt comfortable with n>3 dimensions. Until I watched this video, that is. Thanks.
Great video! In fact Grant was quite close to proving that V(2n)=pi^n/n! Look at a version of the puzzle were the darts hit uniformly in the circle rather than the confining square. What is the probability of surviving at least n rounds? The ratio between the 2 versions must be (pi/4)^n Since for a random point X in a circle |X|^2 ~ U(0,1), the riddle mentioned briefly at 28:10 shows that the answer to the new version is 1/n!
For a realistic case, we can assume a Gaussian distribution instead of a uniform distribution. It would be interesting to get the expected value of such a case then it might give a value greater than 2.19 and I guess it would be around 4 to 5, in that way your number of throws might match the expected value.
uuuuhhhh... but a gaussian distribution would have nonzero probability everywhere. The probability of your dart missing the board and hitting the moon would be positive. Maybe a truncated gaussian, truncated at the original dartboard or "dart-square"?
I hadn't read the title all the way through, so I made it about 4 minutes in before I stopped and went, "wait, that voice is familiar! That's 3blue1brown!" Something about the "you'll see at the end why we formatted it this way" sentiment is almost like a signature
24:25 I get your idea of hitting the bullseye being 0 but that would just be if the tip of the dart would be infinitely small as well because the area of where the dart hits includes the exact point of the bullseye. But theoretically you are right.
Thanks for having me on, this was a blast!
A 2n-dimensional ball could have exploded right next to me and I wouldn't have noticed, I was so engrossed in the video. :-)
Also, let's all acknowledge the real delightful collaboration at play here, which is that between pi and e.
Omg! This is pure logic with pure smartness incorporated in it... This literally blew my mind... My pupils still remain dilated..
when will you do the essence of probability and statistics?
"And these blast points? Only imperial storm troopers are that precise. The probability is zero."
One benefit of inviting 3blue1brown is that he does the animations himself.
He’s a very good guest!!!
A guest of rigour
@@numberphile this one felt like a collab more than an interview, I liked the video none the less. I hope this will give 3blue1brown even more exposure and maybe help out a few people who may not have the best of teachers, to learn calculus and linear algebra the right way.
@@numberphile Why does he say probability is zero for rational points? That's wrong. Please correct this.
There are an infinite number of rational points. The probability of choosing any single rational number therefore is 0.
Consistency is only a virtue if you’re not a screw up.
-Grant Sanderson, 2019
Savage advice. Also true.
True
Arvasu Kulkarni I cant remember when I saw it, but there was a demotivational poster with that as the line, and someone who had split an arrow...at the very edge of the target.
@@AlisterCountel yes! that takes me like 15 years back
Is Grant Sanderson the happiest person on the planet?
"Consistency is only a virtue if you're not a screw up."
Grant Sanderson - 2019
Surprisingly profound.
What does he mean by that tho
@@bentrash7885 In layman's terms: Doing things in a consistent way is only beneficial if you are consistently doing them well.
There is a saying that goes something like "Consistency is key." I assume it is a play on that.
screw up here, can confirm
Consistency is only a virtue if you're not a screw up. Well, it's possible but its probability is zero.
Grant : Has unnecessarily expensive and fancy compass
Also Grant: Uses random piece of cardboard as a "straight" edge
Well, you do need a pretty decent compass to draw large circles. Though a piece of string would be more in keeping with the "straight" edge.
He used all the money to buy that fancy compass... Duh!
XD
??.
So, now the most calm and relaxing voice on youtube has also a face
and a handsome one
@@cyansea2370 back up. Somebody already put a ring on it
He's actually shown it a lot before this
Calming voice used well. Literally made me stay and listen to his lectures than other lecturers do.
There is Bob Ross on TH-cam and *he* has the calmest voice.
“That’s not why mathematicians necessarily care about higher dimensions”
That statement and concept just blew my mind slightly.
Which minute mark does he say it?
Romanski right at the end. I’ll find it...
Romanski start from 26:20
@@Fogmeister Thank you!!
There are as many reasons to study math as there are mathematicians.
I felt so weird watching this video, because 3blue1brown is talking but, with his face
that was me in his first Q&A... "did you hire an actor or something"
I thought he was a π
How did you imagine him talking before?
One could say he switched form talking from a Pi to talking from a piehole..?
I'll show myself out
To paraphrase Bill Bailey: "Ain't you that guy from TH-cam? What are you doing talkin' round like normal?"
Everybody gangster till the animated personified pi shows up.
Alan Deutsch when you jump from 3D to 4D, it becomes gangster * π^2
This is probably the best demonstration of a useful application of higher dimensional math that I've ever seen. Compressing a series of 2D coordinates into a single nD coordinate to get the probability of the whole set. Awesome!
What is its practical use?
"2d coordinates that can be compressed (normed) into a single coordinate" is an all but complete operational definition for the field of complex numbers (the formal name for this would be the "canonical real structure" if im not mistaken). its no exaggeration to say every STEM field finds its own wildly different favorite use of complex numbers.
but more than that, the entire subject of algebra, as you may or may not know, has a pretty large hole in it that isn't actually possible to close using anything other than these same 2d numbers. so in a way, they really are the very thing ensuring any and all reasoning you do using your school textbook math will be logically rigorous; that it has the consistency to carry useful meaning by default in whatever practical context you can give it.
"I'm gonna try to make this a worse shot"
*hits the middle*
3Blue1Brown Suffering from Success
Overburdened with brilliance.
Task failed sucessfully
A victim of his success.
Sounds like an error message in Windows.
He beat the 0% odd of hitting exactly the same spot as before
I feel like we enter Grant's brain every time his animation appears
This is so trueee
We are!
Josh Bone lol
my parents:
"are you ever gonna get a girlfriend?"
me:
"it's possible, it's just probability zero."
But, as Grant said, "probability is zero, don't worry about it", LOL
That makes me a loser.
This is excellent. That could be a meme. It has potential on Reddit
So is that..."zero" potential?
Just say you're already in an unhealthy relationship with the number e.
I’ve had an unhealthy relationship with e since the late 80s.
Was Bob Holness your dealer?
I don't get it
I wasn’t sure whether I should make a pun about the vertical position of the very first dart thrown. Heck it, y-naught
Hahaha!
While some viewers may read the title and ask “Why?”
Numberphile says “y naught!”
Two videos we now need:
1. A Healthy Relationship with E: What the factorial sum really means
2. Calculating the volume of spheres in higher dimensions
Essence of tailor series.
check out grant's essence of taylor series video, and his e^πi video! they'll give you an intuitive understanding of e like none other
Calculating the volume of spheres in higher dimensions shouldn't be too hard, right?
Wolf Elkan For number 2, dr Peyam did a video on that
@@trickytreyperfected1482 it's kinda the same idea as with triple integrals, except it gets higher, you're still integrating independent variables nonetheless the bounds do depend on those variables, unless you go to higher dimensional spherical coordinates system, which is much easier integration is but might be harder to derive and understand intuitively
One day I’m gonna prove Grant wrong and find the real number line out in nature.
Said like a true engineer, sir!
I personally wouldn't expect the real number line to be so natural
@@tyzonemusic Made my day. Thank you!
Mathbro!!!
You might only find the natural number line. And certain constants of nature. Sad to say but reality is often disappointing.
I have a suspicion Grant chose this specific puzzle to flex his exceptional dart-throwing skills
I use arch btw
Lightning never strikes the same place twice
3B1B: *Nay, but my darts shall!*
it's possible, but the probability is zero.
It's possible, but it's probability is Zero
He's quite the "dartist" at this game
fun fact: lightning actually strikes multiple strikes during discharge
I messed up the like count :D
19:53 you mean.. the Grant finale? 😏
EYY
lol
Most people would see the subject of this video and say "why?"
Numberphile and 3Blue1Brown see it and say "yₒ".
That's beautiful
Exactly my thought when they said that
Damn breh you fuccen killin it out here!
I loved that quote when Ted Kennedy said it, and I love it even more here. :D
This took me over 5 minutes to get and I'm glad I did
"Infinity War was the biggest crossover in history"
Brady Haran: Hold my Brown paper
3 blue 1 brown paper
And im over here thinking that its possible, but probability 0
No that would be the mathvengers: eulergame on papa flammy's channel
Gabriel Kellar i
It's possible for me to know something about Infinity War, but because I don't the probability is 0.
I used to think that this crossover ever happening was impossible
But now I've realised it's just probability zero
But now that it's already happened, what's the probability of it happening again?
@@Why_It At least as much as grant hitting the same spot twice on a dartboard trying to do worse, that is to say 0, or in other words, definitely !
@@Why_It exactly as it was before
@@Soken50 Definitely! aka Definitely factorial. So it approaches 1
@@AlabasterJazz But I put a space to avoid confusion.
I am not a mathematician, most of the stuff on numberphile goes over my head but it is just SO satisfying to watch.
For example, I have no idea what e is, I didn't understand anything past explaining what the game is, but it is so damn satisfying to watch, it's like wizardry
The level of passion for mathematics Grant has is overwhelming. He inspires to quit everything, not just my job.. and just bury myself in my books.
When to try to hit the dart so badly that it's actually a bullseye. What an amazing Parker Shot that was...
A "Parker Miss"
:D
"I'm gonna try to make this a worse shot"
*proceeds to shoot exactly the same spot Merida style
Well, shooting the exact spot is possible; but its probability is 0 :)
That was the worst outcome possible.
You are the reference king.
imagine if darts were shot using a minibow
666 likes
The way I rationalized the ratio of the volumes of a hypercube and n-ball is with the corners. You keep adding more and more corners that the ball can't reach.
And the corners have more and more volume as the dimensions got higher. Just compare the corners in the Square/Circle to the ones int Cube/Sphere.
In 4D already you can put a whole hypersphere of half the radius on each of the corners.
nD-Hyper-Parker-Cube s much closer to nD-Sphere
Also the corners get farther apart. A corner of an 100-D cube is sqrt(100)=10 units distant from the centre (assuming that all sides are 2 units long).
One visualisation of this I like uses a projection into 2 dimensions.
One can cut a 3D cube with an xy plane rotated 45 degrees around the x axis. This cut gets you a rectangle of dimensions 2 * 2.82. This plane also cuts out a circle of radius 1 from a sphere or radius 1. If you draw it, you'd notice unused space -- there's a margin of 0.41 units on each side, and an even bigger margin in the corners.
A similar plane cut can be performed in 100 dimensions. A cut along orthogonal vectors a=(1,0,0,0 ...) and b=(0,1,1,1 ...) will result in a rectangle of size of 2 * 19.9 units -- lots of unused space there.
We get even more slack if we cut along (1,1,1,1 ...) and (1,-1,1,-1 ...) -- we get a square of size 14.1 * 14.1 units.
Using a compass on a glass table with no protection below it. That mad lad
:-)
pretty sure most metals can't scratch glass unless it's something like hardened steel
This kind of content really does make me realize my enjoyment for mathematics. When I’m able to break the form of rigid, applied math and enjoy some almost philosophical branches of mathematics, it makes me want to pursue the subject despite my previously held distaste for the topic.
I can't believe Grant has been handsome this whole time.
Kyle Wright his smile gives me life
Too cute for animations lol
@@jhuny Animations cannot contain his adorability. He only decided on pi to represent him because he's a cutiepie
@@trickytreyperfected1482 his cuteness makes me positively irrational 🤪
Right?!?
"I'm gonna try to make this a worse shot."
Grant Sanderson fails the way Matt Parker succeeds.
Of course I kid. Both of those kids put out amazing content. I always look forward to a new standupmaths or 3B1B video.
I'd love to see a collab with them. I'd be happy to have Matt Parker a just an audience member present in the room.
@@LeoStaley Oh, I bet a collab would be amazing. Matt's dry humor and wit with Grant's mellow genius... I'd watch that.
Lol Grant is the opposite of Matt
The ending was so fascinating. Talking about higher dimensions without even being in higher dimensions
Its easy to talk about 2 dimensions, while our world is 3D, so why not 4D? or 6D? or 100D?
@@pietervannes4476 Graham's number D! 😁
@@IceMetalPunk tree(Graham's number)D
Grant is great at talking about these things in a mind bendy way that I don't even think about before. I think that's why I really enjoy his stuff.
I've read a number of Egan's books, and considering some of the bizarre things that goes on in them, I can totally believe he'd cook something like this up in his spare time. Furthermore, I'd just like to say that 3Blue1Brown continues to have the most pleasing math sounds around.
Love this guy and love this channel. A few years ago, when I needed something a bit more mathy than Vsause, but my teenage brain couldn’t quite make it through a Numberfile video, here came Grant, explaining the beauty of math like no teacher or textbook I had had up to that point ever could. And probably ever since as well.
Hey Vsauce, Michael here!
So, obvi I knew Grant was smarter than me because I watch his math videos to be smarter, but I never imagined that he was also more attractive than me and this is really shaking my world view rn
Who says he is smarter than you? The mind isn't a fixed muscle. It can always develop and grow to be smarter.
@@constracted7331 It's one thing to believe you can get better which is entirely true, but it's another thing to think that with hard-work, and being bounded by the constraints of reality and time, you can become as talented as anyone, this is simply not the case as depressing as it may seem.
@@TheBatch62 Obvi u new wat ay mint sew y b pehdaentik?
@@constracted7331 not sure but pretty sure he's somewhere 23-29 years old yet he's a fking mathematician from those very clever solutions to hard math problems. And what I know that I think during that age is still a time on achieving PhD on mathematics yet this guy has seemed to become a maths professor.
Edin Zenon How old are you?
"A healthy relationship with e?" Clearly, Grant has never been to a rave.
AHHAAHAHAHAHAHAH nice one
dont get it
AttoBlaze e = ecstasy
I was gonna like your comment but dont wanna ruin 666 likes
best comment... ever
When you spend too much on the editing software, so you get a cardboard ruler.
I realize this was a joke, but I can't not state that 3b1b wrote it himself (python) and even published it.
Can’t not
Double negative eh...
@@thetophatgentleman4634 Well, yes. Deliberately, I wouldn't think anything else would have made a lot of sense.
Straight edge 😋
@@thecakeredux Surely you mean that nothing else would not have made a lot of sense?
The small volume of hyperspheres can also be understood in a Geometric way with sections:
think of the 2d and 3d case:
If you slice the 3d box in half and look at the section you will see exactly the 2d case, with the same proportion of ball and empty space.
But every other parallel slice will have less ball and way more empty space! So it's kinda obvious that the volume of the 3d ball is less, it would have to be a cylinder to be equal!
This extends to higher dimension by taking higher-1 dimensional slices.
Of course the algebraic phrasing is more precise.
Also some may think that this slice thing is cheating since you can't visualize that either: well, topologists and geometers are not superhumans and have developed lots of tools such as this to think about higher dimensions, and I think that all those tools deserve to be regarded as Geometrical thinking.
Hey! I know this is really late, but I tried the simpler version of the puzzle (x1 + x2 + x3 + ... < 1) and I got sqrt(e) as the answer. Is that correct?
@@khandanish2004 Hi! I think you made a mistake, it looks to me like you've considered the volume of the cross polytope in 2n-dimension to be 2^n/n!, while it is 2^2n / (2n)!.
Remember, in the original puzzle we only end up working in even dimension.
(A cross polytope is the n-dimension version of the unit ocrahedron, it's the counterpart to the hyperspheres when computing the probability |x1|+|x2|+...
@@99Megaluca99 My intuition was even simpler: Look at the relation between a square and a circle thus: You get the circle by smoothing out the corners of the square. If you approach higher dimensions you get 2 to the power of dimension verteces (=corners), therefore we should have exponentially more corners to cut as wonder into higher dimensions.
Okay, maybe not simpler, but an intuition nontheless. This ultimately shows as well, when you think about how you produce one higher-dimension ball: You spin it around an axle that is perpendicular to the dimensional set that contains our ball.
⁰Q'-x
"Because why not" is the definition of why math is so fun. The rules are clear, the reasoning rock solid, but the freedom to play with the assumptions is unmatched. No physics to restrain you. No "right" way to approach a problem.
Another fun one is "without loss of generality". I can't count how many times these words stopped me for quite a few minutes while I tried to understand why and make sure it really is... The revelation was usually fun. Not for obvious cases of course, but some authors can make the word "clearly" work VERY hard.
I get goosebumps when I see notification of videos like this
Bless you for having notifications on. 🔔
@@numberphile bless me too😉
@@numberphile I'm just happy to see Numberfile cares enough about it's subs to read the comments.
@@numberphile I hope you cab clarify why the probability of hitting the center is zero, because I don't think that's right.
@@leif1075 there are infinitely many points that can be hit, and the center is just one of them. it would be more accurate to say that the probability approaches zero
"This is the healthy way to think of e to the x" we need more math teachers like him ;;
Grant, you have an amazing talent for teaching mathematics. I've watched many of your videos and have learned a ton. Brady, I've learned a ton from all of your numberphile videos you've produced over the years too. I have nothing but deep respect and admiration for you both. Thank you both for all you do for mathematics education for us common folks!
As a statistics student in university, this video blew my mind. That's because there is a relationship between probability of something happening and all that higher dimension spheres, e, pi... just amazing. Thank you so much for the great content.
Boy, I love Grant. He's the kind of guy with whom you'd like to have long discussions about life.
2:04 "I'm going to try to make this a worse shot..."
Finding failure in victory more quickly than Myles Garrett.
Excellent comparison!
a Parker-shot would be amazing!
Suffering from success.
@@yourguard4 such a Parker Shot!
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Someone: You can't just "probability zero" your way out of every problem
3blue1brown: Observe, physicist.
Always remember to hydrate when doing high-impact mathematics.
Grant:It's possible to hit an exact bullseye, it is just probability zero.
*cue Mark Rober's auto-bullseye dart!*
Thanks for the video! I thoroughly enjoyed it, and thought about the puzzle for a bit, and I thought of an interesting way to attack the original puzzle.
First, replace the original square with a circle with the same area, and the problem remains unchanged. Also, scale down the diagram so that the possible range of dart throws is the unit circle, and the initial bullseye is a circle of radius sqrt(pi/4).
Now consider a point P chosen randomly from the unit circle, and let R be its distance from the origin. We claim that R^2 is uniformly distributed in the range [0,1]. This is because for some k
"so you are not quite twice as good as someone who has no skill whatsoever"
the burn :D
31:28 "What is the probability that the sum of all those squares is less than 1"
That is the most helpful analogy for understanding higher-dimensional spheres.
thought the same right there
This guy should start a channel of his own. He could be famous
he does, it's called 3blue1brown
Yea, he could teach some amazing lessons and help a lot of people with mathematic
@@stanley2696 Yes, he could use those impressive animation skills to aid visualisation of complicated topics.
@@Jehannum2000 I would love to see some introduction to Calculus by him. I have a feeling that he could do amazing geometric explanations
@@stanley2696 Linear algebra too.
This was one of the most informative videos on multidimensional geometry, quantum physics and probability. This puzzle answers the question for me that the universe when described in probabilities does not imply a multidimensional space. Thanks for the understanding… very powerful!
I’m glad you chose zeta as the outro theme. Vincent rubinetti really did a great job composing the music of 3blue1brown. And Zeta represents that amazingly. I could go off on a long tangent on how much of the music is centered around curiosity and wonder but you just have to listen to it.
Woah! I watched the whole video and it felt like 10 minutes but it is 32 minutes
Huh, you're right. I had to scroll up to check.
Long video with actual equations...got to love it!
Wow, I didn't even realize that's how long it was until reading your comment and checking - time flies when you're engaged!
sooo umm it was definitely a long video for me, guess it's just not my type of a content (= I'm not very smart). I felt like a two year old watching some very skilled magician doing tricks and trying to understand what's even going on. Saw everything, can explain almost nothing :(
The saddest part here is that I have a bachelor degree in chemistry
I watched on 2x and it felt like 2 hours, also 1 hour 56 minutes passed.
I'm just mindblown by the fact that the volume of spheres "picks up" some extra pies as they increase dimensions. It's so weird! Maybe there's an interesting reason for it that you could explain in a video!
While I definitely don't have an answer, I'd check out 3Blue1Brown's "Why is pi here? And why is it squared?" It has a really nice explanation why a pi squared can show up. Maybe that same chain of logic can start to be applied here.
It passes through bakeries and it gives its high numerators and denominators for pies.
I'm not sure if this is the right explanation, but IIRC, there are two independent axes of rotation in 4D space. Assuming n-space has floor(n/2) independent axes of rotation, then I think every other integral would be bounded by [0, 2π]...? It's been a long time since I did any calculus
@@okuno54 The correct explanation is the bakery I mentioned in my comment.
If you look up the formula for the volume of a torus (donut) in 3D space, it also has a pi^2. Because you can kind of think of there being two circles going on in a torus, the radial one and the cross sectional one. And then its volume is the area of the middle circle swept around in another circle. And then yes, as Okuno above me said, in 4-d space you can also have these two independent circles (say two coming from the first two coordinates and two coming from the second two) and so in some sense it makes sense that when integrating and multiplying stacks of circles to get volumes you get two factors of pi.
I've never seen Grant's face before. He's cute af
he's done a few interviews on his channel if you dig through his older videos!
He looks like an angel!
I know! Too bad he's married
@@jedkemekt2062 wym too bad? I'm sure she's a cutie too O.o
@@VectorNodes too bad for us :(
Honestly, this guy is probably a genius. He spouts and remembers all this complex math but almost makes me understand it...almost.
Pi Megami Tensei Nocturne: Featuring e from the Constant May Cry series.
mariosonicfan2010 😂😂😂
26:07
Brady: "you're not quite twice as good as someone who has absolutely no skill whatsoever"
me af
2 x 0 = 0
cries in multiples of zero
The man has a compass ready for the video but no rulers in sight lol
Rulers are easily improvised. Compasses are harder
@@Krekkertje If you don't need great precision, a compass is easily improvised as well. All you need is a pin to center the arm, which could even be made of the same cardboard as his straightedge. Poke a hole through the arm at the distance you are scribing, and place your pen/pencil/marker tip through it.
A string to tie from pin to writing implement is more accurate than the cardboard, but not quite as common in most offices. The string is particularly useful in geometry though, as it can also be used to measure pi. =]
@@Cuuniyevo You need a pin, and to poke a hole big enough for a pencil to fit through. Far easier to just take nearly any object and use it as a straightedge.
@kevin M i always wear a bracelet and if you have two pencils its just as easy
As a mathematician, it's very important to always carry an emergency compass
"What if the dart is landing exactly on the line"
"The probability is zero, dont worry about it"
My absolutely favorite moment
The really nice bit is that the dart landed exactly on the same point... again. The probability is zero squared.
A real gem 😂
Whereas the probability of hitting the very centre of the bull while trying to miss by as much as possible is apparently 1.
@@dlevi67 I know this is probably a joke but I don't like reading it anyway
Super awesome video. Thought that without the animations, it would take me out of my zone (in my mind). But this puzzle had so many pieces, and I followed fully. I only regret that I would not be able to connect all those dots on my own, if I were presented the same question. But I'm practicing thanks to your help!
I really like how, if you include the radius of the circle into the calculations, the final result is instead e^(pi·R²/4) = e^(A/4) where A is the area of the initial circle.
Even more naturally, A/4 is the ratio of the areas of the initial circle to the initial square. So the final result was e^(probability of an initial hit).
That’s actually a nice generalization, because now you can ask the harder question of if it was nonuniformly distributed or if it was radially symmetrical
Wow, the two best maths TH-cam-channels at once?
This man is the best teacher i've found on youtube. He is an inspiration. He makes me excited to watch a conceptual math video. Much like numberphile does
[31:00] If I ever find the real number line while hiking through the woods, can I make a Numberphile episode please?
I would love to see it if you find it.
This is the best collaboration among the best 2 math channels. Looking forward for more of these!
4:54 Well, probably not rotationally symmetric; probably more elliptical than circular. You have to counter gravity in one plane and not the other, and the mechanics of your arm probably affect the result. But a rotationally symmetric gaussian distribution would likely indeed be more correct than a uniform distribution, yes.
You managed to make Grant trade in his CG animations for a dartboard, brown paper and markers. Brady you are too powerful.
As a 3blue1brown patron I've been waiting for this video. :)
I've tried to solve the problem on my own for weeks and failed, so I couldn't be more excited right now!
When the two channels that you don’t understand because of how smart they are combine
You are right, I allways feel like the idiot who doesn't get it when I watch their video's. Still I love to watch them, in the hope it will make me a bit smarter, but I allways end up confirming to myself to be an idiot
Tbh, 3blue1brown is a much better channel. Numberphile has a bad habit of oversimplifiying problems (such as in the infamous -1/12 video), where 3blue1brown shows how mathematicians actually discover new math.
Add PBS Spacetime to the mix and it will boil my brain.
Same! I felt like a two year old watching some very skilled magician doing tricks and trying to understand what's even going on. Saw everything, can explain almost nothing :(
The saddest part here is that I have a bachelor degree in chemistry
It’s always fun when e and pi synergize so beautifully, being the 2 most important transcendental constants.
Really appreciate the call out that while geometric interpretations are useful, they're best not taken as only relevant for people living in a strange world experiencing that many spatial dimensions. Any collection of related measurements can and are reasonably considered "dimensions." Just a table with a few extra columns. Very practical. Nothing timey-wimey.
Please tell me the collab doesn’t end with this one!
it's so cool that even though it is impossible for us to conceptualize or create or simulate upper dimensions, math already gives us properties of how those dimensions behave. It's like you're locked out of a house and you have no way of interacting with the insides of the house, yet through some big brainness you can figure out some things about what's in the house
Grant: "I'll call this x-naught"
Me: Why not
Maybe he doesn't want to
Ayayai.
I am embarrassed by how long that joke took me to get
Sophomore engineering student here! We just started getting into things where you translate physical units into other "domains" like the phasor domain in electrical engineering, various domains in differential equations, and Mohr's circle in mechanical engineering, and it really shows why all of these transforms are useful. It's simply way easier to understand stress transformations when you explain them graphical using Mohr's circle rather than their physical derivation.
I think it may seem counter intuitive to think that a 100 dimension circle takes up so little of the 100 dimension cube but if you step back and analyze what is visually happening when you increase a cube's dimensions you can see it is adding more and more "corners" and when you see that you realize the smooth circle that encompasses all of the cube except the corners you can see why as you get higher dimensions your "ball" takes up a smaller and smaller portion of the "cube". It may be wrong to think of increasing dimensions as increasing"corners" but that is one of the things that happens as you increase your dimension. I think maybe visualizing it that way would help make the fact that a 100d "sphere" takes up so little of a 100d "cube" a little more natural feeling
30:24, I'm afraid there's a small mistake here.
The volume of a ball in 100D in 2.e-40, but the _proportion_ it covers from a cube is actually way smaller, since you're dividing by 2^100.
He was merely estimating
Some alien somewhere:
"Oh no, I dropped my 100-dimensional ball into a 100-dimensional bucket, and spilled 10^-40 ml in the process! Gonna take me at least a Planck-time to clean this mess up..."
""What happens if you hit the edge"
Then you have to hit the single center point on the next round or you lose.
Depends.
Is the boundary part of the circle or not. Which depends on wether the circle is a closed or an open set.
In general sports, the boundary is included though.
@@RodelIturalde Geometrically, if you have a circle with radius 3, a point 3 units away on the plain is part of the circle, yes.
@@AdeonWriter well, there is no such convention, you generally have to specify if you're considering an open or a closed disk (a circle is the boundary itself, much like a sphere is the boundary of a ball). This was never done in the video just because it was irrelevant to the problem, you get the same result in either case.
@@RodelIturalde Hitting the line would make it so that all the X's and Y's squared equal exactly 1. The formula states they must equal less than 1. So hitting the line count's as a miss.(timestamp 13:13)
Is everyone forgetting that you have zero probability of landing on the boundary anyways, since the distribution is assumed uniform?
30:22 reminds me of something I once read in a maths book: Spheres are pointy, and the higher you go in dimensions, the pointy-er they become.
Interesting puzzle, great video, love you both!
I am not a math major nor any sort of academic worth the name, but I have a logical and mathematical mind, so this sort of content fascinates me. I've seen this video several times over the years, and each time I learn something new.
Today, I think I learned an intuitive way to understand "probability zero" and why it's not the same as "impossible".
My old preconceived notion was that the opposite of infinity is infinitesimal. And maybe it is, but I don't think that way anymore. Of course, infinity is conceptually the biggest possible thing, so infinitesimal should be the smallest possible thing. Mathematically, some infinities are bigger than others, but that's a later topic.
If we confine our domain to the interval [0, 1] as we would when discussing probability, how might we define the smallest possible thing? One idea I heard about some time ago is "0*" (zero-star), a number so close to 0 that it is "confused with" 0 but not actually equal to it. We might think of this as 0.000...1, with infinitely many zeros between the decimal point and the final 1.
We have seen something like this before at the other end of our probability range. What is 0.999...9 with infinitely many nines? That's 1, as was nicely explained in another Numberphile video. In part, it's because there is _always_ some real number between any two other real numbers (infinitely many, actually). If there are no numbers between 0.999...9 and 1, then 0.999...9 = 1.
Similarly, if 0.000...1 exists, there _must_ be a number between it and 0. If so, then it's not the closest number to 0. If not, then the number can't exist in our domain/range. This leads to a contradiction. If 0.000...1 exists, then it must be equal to 0.
Thus, a probability of 1/∞ = 0. Not "close to", not "confused with", but actually 0. The only caveat is that if you take the inverse, you can't say 1/0. You'd have to say 1/(1/∞) = ∞/1 = ∞.
I'm not saying that this reasoning is correct or rigorous, but it's one way to make sense of it in my own head, so I thought I'd share it.
I'm a second year grad student with a BSc in Physics and Applied Mathematics, and the math that appears in stuff like this still blows my mind. Math really is magical. (Also, studying physics, I'm glad to see a Taylor series not chuncated at the second order.)
The animated Pi has a good-looking live-action avatar.
"Infinity War is the most ambitious crossover in history"
Mathematicians:"Hold my higher dimensional dart"
Disney: "Infinity War is the most ambitious crossover event in history."
Me:
Many years ago I was working in the supermarket industry. We characterised shoppers by their propensity to shop in different areas: meat, dairy, fresh veg, canned goods etc, giving us a set of n scores. We used these as coords in n dimensional grocery space. This "space" was really easy to conceive and navigate. That was the only time I've ever felt comfortable with n>3 dimensions. Until I watched this video, that is. Thanks.
Great video!
In fact Grant was quite close to proving that V(2n)=pi^n/n!
Look at a version of the puzzle were the darts hit uniformly in the circle rather than the confining square.
What is the probability of surviving at least n rounds?
The ratio between the 2 versions must be (pi/4)^n
Since for a random point X in a circle |X|^2 ~ U(0,1), the riddle mentioned briefly at 28:10 shows that the answer to the new version is 1/n!
What more could we ask for...
@Selarom Ogeid no I think 3Blue1Brown in real life on Numberphile is better...
For a realistic case, we can assume a Gaussian distribution instead of a uniform distribution. It would be interesting to get the expected value of such a case then it might give a value greater than 2.19 and I guess it would be around 4 to 5, in that way your number of throws might match the expected value.
uuuuhhhh... but a gaussian distribution would have nonzero probability everywhere. The probability of your dart missing the board and hitting the moon would be positive. Maybe a truncated gaussian, truncated at the original dartboard or "dart-square"?
Me: "you think darts is fun?"
Mathematician: "Yes! you know what happens when an unrealistically bad darts player plays a totally made-up darts game?"
Basically all of math in a nutshell lol
also all games are made up
31:22 HAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHA OMG I'm a child.
I hadn't read the title all the way through, so I made it about 4 minutes in before I stopped and went, "wait, that voice is familiar! That's 3blue1brown!"
Something about the "you'll see at the end why we formatted it this way" sentiment is almost like a signature
39:33 "Which I think is beautiful and clever" and so is Greg Egan's science fiction. Go on give yourself a treat.
This collab is my wet dream!!
Leave it to us Indians to post comments like this 😂
Now it’s your wet reality 🙂
I've never related to a comment more.
Conan O'Brien: Hol up lemme just hit this youth serum
...
3Blue1Brown: Your relationship to the number e is unhealthy human
OMG I was trying to remember who he reminded me of!
"I'm gonna try to make this a worse shot"
Proceeds to hit the _exact same spot_ as last time, to within a millimeter
Ikr 😂
hitting the exact same spot is probability 0 but it is possible lulz
*probability zero*
Whenever I see a Scientist's or Mathematician's bookshelf, I always try and see if we have the same books. Wonder if I'm the only one that does that.
24:25 I get your idea of hitting the bullseye being 0 but that would just be if the tip of the dart would be infinitely small as well because the area of where the dart hits includes the exact point of the bullseye. But theoretically you are right.