Darts in Higher Dimensions (with 3blue1brown) - Numberphile
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- Опубліковано 22 чер 2024
- Grant Sanderson from 3Blue1Brown joins us to discuss an intriguing puzzle with a shrinking bullseye.
More links & stuff in full description below ↓↓↓
3Blue1Brown: / @3blue1brown
Grant Sanderson on the Numberphile podcast: • The Hope Diamond (with...
Greg Egan's tweet which started it all: / 1160461092973211648
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Grant did the animations for this one!
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Special thanks to our friend Jeff for the accommodation and filming space. - Наука та технологія
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."
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.
"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
??.
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?
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
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 UA-cam and *he* has the calmest voice.
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
I’ve had an unhealthy relationship with e since the late 80s.
Was Bob Holness your dealer?
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 UA-cam? What are you doing talkin' round like normal?"
"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.
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
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
"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
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.
19:53 you mean.. the Grant finale? 😏
EYY
lol
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!”
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
I feel like we enter Grant's brain every time his animation appears
This is so trueee
We are!
Josh Bone lol
Grant:It's possible to hit an exact bullseye, it is just probability zero.
*cue Mark Rober's auto-bullseye dart!*
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.
“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 have a suspicion Grant chose this specific puzzle to flex his exceptional dart-throwing skills
I use arch btw
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'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.
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 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?!?
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!
Why is Grant's smile the only thing that keeps me going?
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.
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
„Consistency is only a virtue when you‘re not a screw up“ 😂
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.
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 😋
@@felixmerz6229 Surely you mean that nothing else would not have made a lot of sense?
Someone: You can't just "probability zero" your way out of every problem
3blue1brown: Observe, physicist.
This is the best collaboration among the best 2 math channels. Looking forward for more of these!
This is insanely beautiful!
Thanks Numberphile and 3b1b! You've made my day!
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.
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
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.
Pi Megami Tensei Nocturne: Featuring e from the Constant May Cry series.
mariosonicfan2010 😂😂😂
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
"so you are not quite twice as good as someone who has no skill whatsoever"
the burn :D
Boy, I love Grant. He's the kind of guy with whom you'd like to have long discussions about life.
Great job guys. Fantastic puzzle, happy to see you two working together.
"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
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.
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!
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!
"This is the healthy way to think of e to the x" we need more math teachers like him ;;
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!
Error: task failed successfully!
"im not that bad" *casually throws a near bullseye barely looking*
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.
"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
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.
Honestly, this guy is probably a genius. He spouts and remembers all this complex math but almost makes me understand it...almost.
That was one of the better math videos I've seen in a while. Loved it.
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!
Always remember to hydrate when doing high-impact mathematics.
19:52 "Alright are you ready for the Grant finale?"
Because of my research in information theory, I already knew the formula for the high-dimensional volume, and sensed what is coming up before Grant showed us. It was still an amazing experience. Very cool puzzle with a very cool solution.
"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
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
This explanation was truly amazing. Thanks for the video. I really loved it!
It is always a treat to watch these videos.
Thanks for sharing
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
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
The perfect crossover doesn’t exi...
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.
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 :(
Wow, the two best maths UA-cam-channels at once?
39:33 "Which I think is beautiful and clever" and so is Greg Egan's science fiction. Go on give yourself a treat.
After all of these years of watching and following 3Blue1Brown it was Numberphile that finally integrated a face to his voice!
"Infinity War is the most ambitious crossover in history"
Mathematicians:"Hold my higher dimensional dart"
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!
It’s always fun when e and pi synergize so beautifully, being the 2 most important transcendental constants.
after rewatching this for the 4th time now... definitely one of the best NP videos ever
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.
Seeing his face feels like forbidden knowledge
"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.
Fascinating lesson, Grant, thank you for the enjoyable hour!
The animated Pi has a good-looking live-action avatar.
Disney: "Infinity War is the most ambitious crossover event in history."
Me:
Wow, great video! Fascinating how higher dimensional spaces and the Taylor series of e^x pop out of this puzzle!
One of the best puzzle I have ever heard of ! Bravo
You managed to make Grant trade in his CG animations for a dartboard, brown paper and markers. Brady you are too powerful.
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?
What a great problem and video! Ties so many concepts together neatly
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
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.
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.
This explanation really helped me visualize the concept of e and applications of higher dimensional math better than university ever had. Thanks so much for opening my mind 🤯
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
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
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.
What a great collaboration! Thanks both for a great video!
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