How to see a sphere in 4D
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- Опубліковано 25 вер 2024
- Today we will explore the world of higher-dimensional geometry from simple lines in the first dimension to the intriguing concept of a 4D sphere.
We start our journey with the basics of 1D lines, 2D circles, and 3D spheres, gradually building up to the elusive 4th dimensions. Through intuitive explanations and geometric proofs, we cover all the areas in multi-dimensional spaces!
Key Highlights:
Visualizing and calculating the properties of shapes in different dimensions.
A deep dive into the volume calculation of spheres from 3D to 4D.
Understanding the Gaussian integral and its connection to higher-dimensional volumes.
Insights into the gamma function and its application in extending factorial functions to complex numbers.
This video is perfect for math enthusiasts, students, and anyone curious about the world of higher dimensions. Get ready to expand your mathematical horizons and explore the unseen geometry of the universe.
Useful Links:
Mathematics Stack Exchange Discussion: [math.stackexch...]
#higherdimensions #geometry #mathematics #educational #SphereVolume #5DGeometry #MathVisualization #GaussianIntegral
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replace the a in math with e and the channel has an insane glow up
changing anything to e in math results in crazy things!
@tr0m
*Glow _down._
@@qwertzuioppel
Yo, good point!
It has to be delibarate
meth
this video explained in one sentence: We show that 1+1 is intuitively 2, then we use a simple miracle whose explanation is beyond the scope of this video and wow: finished!
Its left as an exercise for the reader
As a 4 dimensional being I see this as an absolute win.
actually, the 4th dimension doesn't exist. its purely a creation from the mind if you know what I mean
@ That’s what an under cover fellow 4D being would say.
it does exist an you can't tell me otherwise@
fr imagine not believing in 4d bro
like bro imagine being in 3d
I wish the connection between the Gaussian pulse and the surface area of a n-dimensional sphere was explored more
Good video, but there are small things you could improve to engage less knowledgeable people in the audience. For the integral that defines the gamma function, you could show why the integral is related to factorials by doing some integration by parts and showing how the factorial gets built up. From there you can just use (n-1)! directly, and only then say that since the integral doesn't care about having non-integer inputs, it actually extends the factorial and is known as the Gamma function. That makes it softer on people who know less about special functions. Just some feedback, but other than that I enjoyed the video
A thoroughly entertaining video. I realized after 10 seconds that while I understood the meanings of the individual words you used I had no clue whatsoever as to what you were actually talking about. Couple that with formulas that looked like they should have been etched in a sword made it even more enjoyable. You obviously know your subject matter. But for me, this video was akin to watching the 'Dream sequence' in 2001 while listening to Shakespeare in Mongolian.
Bravo!
I enjoyed it very much. Although I really do not have much understanding of proofs and those kinds of mathematical concepts, I appreciated the explanation of the formula for areas of circles and sphere of all dimensions.
Also, by not having “much understanding” I actually mean I have next to no understanding. 😁
It doesn’t need to make sense, it’s just so satisfying to watch math mathing.
This video is the equivalent of a Minecraft video where the person does some building off camera.
Yeah fr
1:44 This proves A = rC/2, however circumference C is often found as a derivative of area A with respect to radius.
Assume we do not know that C is linear with respect to r.
C = f(r) = dA/dr = C/2 + r/2 dC/dr --> C = r dC/dr --> C/r = dC/dr But we only know that C = f(r) not that C = pr for some constant p
d^2 C / (dr)^2 = -C/r^2 + 1/r (dC/dr) = -C/r^2 + 1/r (C/r) = -C/r^2 + C/r^2 = 0
Now we know that dC/dr is constant, because the second derivative is 0. Hence we can say C = pr, p = C/r = dC/dr. And pi would be defined as p.
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA WHAT IS THAT
UOOOOHHHH 😭😭😭😭😭😭💙
@@Blurgertdon't be scared.. you're next.
isnt d^2 C / (dr)^2 = -C/r^2 + 1/r (dC/dr) = C/r^2 - 1/r (C/r) the way to go?
How come C=r dC/dr
Hi, I'm Aryd from Indonesia.
I have a question for u, sir.
I found the n-dimensional Area of Sphere is really fascinating, but when I checked the coefficients for the r^n, it starts to decline after n approx 5.257 and converges to zero as n increases. Does it means our universe is actually shrinking after the 5th dimension, or is it possible that sphere doesn't exist in such a very high dimension?
The title suggests a gentle, intuitive visualization technique or at most some simple geometrical proof... The content is anything but! Nice proof, but false advertising haha-I almost didn't click because I thought it was going to be yet another "imagine a balloon getting big and then getting small again" video
I loved this video, and I'd love to see more like it in the future. I only wish I understood more than a quarter of it...
❤an interesting approach. Thank you very much publisher
Nothing in this video is about visualisation as is implied by the title and the intro.
What a coincidence that the Gaussian bell makes an appearance here... but wait, there are no coincidence in math so there must be a reason!... I get that you might not want to get into the details, but it could've been an opportunity to pose that question to the audience (why are these two things related?)... Also, it would have been nice to talk about how your formula also works for the three dimensional sphere (there you could have talked a bit more about the gamma function)... Cool none the less!
at the end i would've liked to see the equation applied to the 3rd dimension, as on first glance it doesnt seem to work.
It does work on n = 3 because of the Gamma(n/2) term, due to the recursive nature of gamma function, Gamma(x+1) = x•Gamma(x). So Gamma(3/2) is just 1/2•Gamma(1/2). And Gamma(1/2) is just sqrt(pi) which would just cancel a power off the sqrt(pi³) term above into sqrt(pi²) = pi
Once you realize that the gamma function at any half-integer argument = some rational number * sqrt(π) then the pieces fall into place. I agree though; I'd have liked to see that discussed as well, as it also explains the pattern where you only have π in the formula at n=2 and n=3 but you suddenly get π² at n=4.
On a related note, I felt like there also was a bit of a missed opportunity at the end to discuss the observation that this formula has a local maximum in n--that is, for a given R, there is some dimension n which maximizes the volume, after which the volume converges toward zero. For example, for a unit n-sphere (R=1), the maximal n-volume occurs at n=5... and at n=13, the n-volume even drops below 1.0! There's some interesting underlying intuition there: in higher dimensions, a sphere becomes vanishingly small vs. its circumscribed cube, as space becomes increasingly "stretched out" vs. the n=3 space we are accustomed to.
That's so cool!!@@indnwkybrd
I didn't know that, thanks!@@mosztzsch7007
You haven’t defined the volume for a 4D sphere though
2D sphere has an area, 3D has a volume. It is not immediately clear what “volume” would mean in 4D
its more like hyper volume cause their is L^4
The title: “How to see a sphere in 4D”
The video: “Here’s the equation for a Hypersphere’s volume, the visualization is up to you.”
Yep, “how to”. Not “show”
at 7:00, the gamma function has input z instead of n
No visualization is shown in this video. It's just a mathematical derivation of the volume. Also the text-to-speech isn't great.
Trying to understand a 4D sphere, in a 3D world, on a 2D screen, with my 1D brain.
trying to understand a 4d sphere in a 3d world on a 2d screen displaying a comment thats been copied so many times it wont even fit in 4 dimensions
😂
Definitely deserves more subscribers. Very nice voice and good explaining (I think)
(Not the biggest math guy XD)
I think it's AI generated lol
worst AI voice translation around. just narrate yourself. otherwise, pay an english speaker to do so.
I have no clue what I just watched but seems like it makes sense
How would this formula be applied when you go above the 4th dimension? It seems like the formula is limiting to the 4th dimension and below. Correct me if I'm wrong though
Dude, the title is misleading. There was no attempt whatsoever to visualize a 4D ball.
You forgot to mention what software did you use to make your animations. Great video.
its really not that hard to make a cross sectional visualization that at least matches the video title
I'll give that the gamma function is the most popular extension of the factorials, but I'd argue it's _not_ the most useful. _That_ honor goes to the pi function (no relation to the other pi... mostly). The difference between the gamma function and the pi function is that Π(n) = n! for all ℕatural numbers n, also meaning that Π(z) = Γ(z + 1). This has the nice bonus of simplifying the integral slightly by replacing the t^(z-1) with just t^z.
(n/2)Γ(n/2)? You mean Γ(n/2 + 1) = Π(n/2)?
I don’t see how this is a meaningfully different extension though - the content of any extension is in how it chooses to fill the gaps, while these two functions do not differ in this respect
@@joelbraun8584 It's a matter of pedagogy. They can do the exact same things, but which is easier to use and teach? I'd argue that the one that more closely aligns with the existing factorial and doesn't have a suspiciously artificial "-1" in its definition is simpler to understand, and as I showed at the bottom, it also yields a simpler formula at the end as a bonus.
Basically the same thing
this is almost like the pi vs tau debate
@@adrin181 With pi vs tau, I can understand why both would be used. With the gamma function though, the "z - 1" just feels so artificial and seemingly does nothing but distance the function from its discrete counterpart. While there are cases where pi gives a nicer result than tau, I've never seen a case outside of defining the digamma function where the gamma function gives a nicer result than the pi function.
The urge to become a math genius after watching this kind of videos is insane 😅
I think like the 1 d sphere have negligible breadth , 2d sphere have negligible height then 3d sphere must have some negligible 4d quantity . All we have to do is to find that quantity and put limit from zero to infinite to get a 4d sphere.
There was no proper visualization/equation explanation for the formulae, hard to understand
4D sphere is called glome, it is possible to move in three directions on the surface and not get any closer to the center of the glome
Well the question lies... Does it really change its shape or does it change its shape exclusively in other dimensional territories...
Also, there are 4D spherical coordinates as well as the standard 4D Cartesian/Minecraft, polar, and cylindrical coordinates. There are also toric coordinates, that uses tori, or circles with 2 radii, the most common cross-section of a torus is a donut. The formula for volume in 4D spherical coordinates are ⨌(р³ * sin²(φ) * sin(ψ))dрdφdψdθ, where:
w = р * cos(φ)
ρ = р * sin(φ)
z = р * sin(φ) * cos(ψ)
r = р * sin(φ) * sin(ψ)
x = р * sin(φ) * sin(ψ) * cos(θ)
y = р * sin(φ) * sin(ψ) * sin(θ)
using this, we can see what 4d looks like. Example: if we can put a 3d object in 2d ( drawing a sphere on paper ) then we can theoretically draw a 4d sphere. Any objections?
So to prove the true dimension of a foreign sphere we could attempt to measure it and see how it matches these values. However, possible not all dimensions contribute mass. And in fact most won’t. More likely forms instability governing harmonics. Static. Radiation. Hums. The whoville.
как работает формула (на 9.20) при n=3?
I feel incredibly dumb watching this video...
And I love it. It means I have something to learn still!
Hi
@@krio1267 oh hi!
this is the man teaching me 4d in a 3d world in my 2d screen in my 1d brain😅😅😅😂😂😂😂
I was with you Up to the third dimension
I think that there is no 1d because if you think that 1d is a line it have some width so that we can se it otherwise we cannot see it.😊
We cannot see it
why is the title how to SEE it in 4d? I don't see it in 4d at all, we just got the volume of it
Why are you using a text to speech program? It definitely doesn't sound like a normal voice (says intaegral instead of integral)
He really titled it “how to see a 4D sphere” instead of “how to find the volume of a 4D sphere” 💀
no 4d sphere for us ;-;
Volume of a 4D cube, or tesseract, is easy, just zeit (raise to 4th power, due to time sometimes being the 4th dimension, and Zeit is German for "time") the edge length.
What is R and why with respect?
i came here to see a sphere in 4D where the hell is it
wow that was so elegant!
I feel smart when i see an number😊
Good video. Now do the Mystery of √ -1
lemme see how much of a genius you really are.
ok, pero... como se mira la esfera de la cuarta dimensión?
if n is always a whole number, why would you use the gamma function instead of the regular factorial?
We live in 3 spatial dimensions, so the surface area of a sphere is (r√π)³/(3/2)! = (r√π)³/(1.5!)
1.5! = ∫_0->∞ t^1.5 e^-t dt
the fourth dimension is merely the variability of the third dimension and therefore stands for the meter value of time in our physical construct of the universe. If you imagine a cube whose vertices are all expanded by one, you get a self-contained teatract, theoretically a cube that has itself as its core. this says that the fourth dimension is first the mutability of all things and can even cause atoms to exist multiple times even though they are all one and the same and have the same origin
nobody can see 4th dimension
we can observe it only in our imagination
but i'm sure hypersphere is lots larger because there's more space for it to exist
i like geometric but really bad at math numbers
Are you using an AI voice? It's slightly off but still quite realistic.
Crazy to think that we are actually watching a 4d representation on a 2d screen, which realistically is just 1 dimension of pixels at any given time that scan down really fast through time. But further more, it's actually 0 dimensional pixels quickly firing down a 1 dimensional line and then bumping down a notch 100s of times per second.
wow great knowledge . who wrote the script is amazing.
It's really interesting. I will work with my imagination to imagine 4 dimension in my brain. I guess it's could be done, but really challenging and useless...
So what does it look like?
Why leave so much explanation out? Like where do you suddenly get d from?
How about 5th dimension
that sphere at the start made me think there were smudges on my screen
The video is good, but the title seems misleading. I expected to see a video about how to visualize a 4D hypersphere, not to calculate the volume
crystal math more like crystal meth(ok just joking, love your videos)
Im your 999th subscriber🙂
underrated
Interesting stuff man
What is 2^53|~5
Bro, I don’t understand anything, so go all out, don’t say “is beyond the scope of this video”, explain everything
you did not answer how to see a sphere in 4d
Me:*confused unga bonga sound*
We can generalize further by going to different norm-balls. This video covers the 2-norm but we can also do the 1 norm up to the infinite norm balls.
Scientists need this bro
Maths neds this bro
the whole time I watched was me trying to comprehend what you were saying
Some random creature in 2D:
"How to see circle in 3D"
I feel like the establishment of all the equations after 3D was sort of arbitrary. Like “here’s this new equation”
1 zero=1 point
2 point=1 line
3 line=1 triangle
4 triangle=1 tetrahedron
5 tetrahedron= 1tetrateradron
the only 4d object that fits 3d
What is 100d
fun fact: its impossible to draw an 1D line in our 3D universe
@Aleph_Null_ so far no one has tried to debunk me lol
instructions unclear:my brain exploded
Also, your equation is only valid for even dimensions. There is no root pi in the equation for volume of a 3 sphere.
THE POWER OF TWWWOOOOOOOOOOOOOOOOO- 5:13
I don't have enough knowledge to understand this knowledge
Your math's blowing my mind.
Dude you destroyed our Bain 💀
and kids, this is how we learned about the glome, any questions?
What about n=3?🤔
Gamma of 3/2 is sqrt(pi)/2
1:19 kinda looks like a parallelogram to me but you do you
So you say that a line has a radius by definition? This is where I turn the video off.
alternative title: how to pass a math test in 9 minutes
My brain is melting.
This video proves the guys at crystal math are on crystal me-
Can someone tell me WHAT IS “R”
Radius maybe
love your channel name 😂
I read a paper on hyperspheres, but forgot which dimension has maximum volume.
N=5.
"the square root of pi" is the mathiest thing i have ever heard
So, where the sphere???
👽: damn, they found our hiding spot
Wow
very misleading title