So why do the volumes of all even dimensional unit spheres sum to e^π?
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This is a follow up to my video on Euler's number ( • e (Euler's Number) is ... ). Here you'll see why all unit spheres in even dimensions have volumes that sum to e^π
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A corollary of this is that as you increase the dimension, the volume of a unit sphere tends towards zero, which is hard to visualise.
Tim Leach is there some sort of intuition you know of for this? maybe another formula/proof that also uses this?
ApplepieFTW the idea is that higher dimensional spheres are “spiky”. Already, between 2D and 3D spheres. Search this up :)
@@ApplepieFTW So... A sphere in a metric space is the set of all points that have fixed distance from a fixed point (its centre). To measure the distance between two points you use a metric (a distance function). People are usually familiar with the 2-norm known as the Euclidean Norm, which is (most likely) defined on n-dimensional real spaces. The Euclidean Norm is the one derived from Pythagorean theorem. So to measure the distance between two points you measure the norm of the vector that joins those two points. The Euclidean Norm of a vector is the square root of the sum of the squares of each component of the vector. By increasing the dimension of the space you are increasing the number of components of the vector. So if you fix the radius of the sphere, in this case 1 you have to have that the sum of all the squares of the components adds up to 1. This means that the more components you have, the less their value can be. It's like the sphere starts lacking outer points.
That's a great explanation!
@@ApplepieFTW Another explanation of this can also be found via calculus and limits, specifically in the ordering of dominating functions. First, take a to be some arbitrary constant and n to be our variable. At the limit as n goes to infinity (basically, as our variable assumes larger and larger AND LARGER values, just add more zeroes... more zeroes... MORE zeroes... Yeah still not big enough just keep going!!!), the following functions dominate one another in this order of precedence (without getting too fancy because I could sit here all day coming up with extras to slip into this list):
a (constant function) < n (linear function) < n^a (polynomial function) < a^n (exponential function) < n! (factorial; the gamma function also sits here) < n^n (tetration! The famous x^x function is this~)
If we take the example of the video, and only look at even-dimensional spheres just to keep the math simple (odd-dimensional sphere volumes will just slot in neatly in between their respective even-dimensional neighbors, so this simplification works), we wind up with this equation:
lim_(n->inf) (pi^n/n!), or the limit, as n goes to infinity, of pi raised to the power of n, over n factorial.
We end up with an exponential function over a factorial function.
When you have a limit of two functions where one function dominates the other as the variable goes to infinity, you end up with one of two situations:
# Let f(x) and g(x) be two functions such that g(x) > f(x) for arbitrarily large values of x approaching infinity...
1. Lim_(x->inf) f(x)/g(x) = 0
2. Lim_(x->inf) g(x)/f(x) = infinity
In the first case, it's basically like taking our arbitrary constant a and dividing it by n as n gets progressively bigger and bigger and bigger. In the latter case, it's like we take n and divide it by a, and no matter what it's just gonna race linearly off to infinity anyways.
We can conclude that the limit of pi^n/n! at infinity is zero because the factorial function (our g(x) in this case) dominates the exponential function (our f(x) in this case) at the limit of infinity. And therefore, we can conclude that as our spheres become arbitrarily n-dimensional for ever increasing values of n, their volumes become smaller and smaller and ultimately approach zero.
“...full circle.” Pun intended?
It's amazing that this converges. This also implies that after a certain point, the volume starts to decrease.
Also this is being really sloppy with the units...
Yeah it peaks at 5 dimensions and then the volume decreases.
really sloppy is an understatement ;) Espectially since since 4d volume contains infinite 3d volume and 3d volume contains infinite 2d volume. 2d volume (known as area) contains infinite 1d volume (known as length) and 1d volume contains infinite 0d volume (known as a point). Yes, it's just mathematics and all are just numbers. Though when you look at it from the actual geometric perspective we essentially add meters to square meters to cube meters ... (or units to square units to cube units ...)
That said there are other quite fascinating things about higher dimensional spheres. See Numberphile on packing higher dimensional spheres:
czcams.com/video/mceaM2_zQd8/video.html
What do you mean by being "sloppy with the units"? What are "units"?
Fortunately, mathematics is free of such pointlessly restraining concepts ;)
@@reddmst mathematics has all kinds of units. i.e. units of distance, area, volume, 4D volume, etc. This treats them all the same
@@pendragon7600 In mathematics, there is no concept of "units" as you describe it. Perhaps you're confusing mathematics with physics, where it does indeed matter whether you're measuring distance or volume?
If you're confused about how a zero dimensional sphere has non-zero volume, you need to remember that in mathematics, "sphere" refers to the SURFACE of a ball. So an ordinary basketball would actually be a 2-sphere, even though it exists in 3D space. Similarly, a 0-sphere is actually a line segment, whose "volume" is it's length.
This video actually deals with even dimensional unit balls. An ordinary basket ball is a 2-sphere but its inside is a 3-ball, and its volume 4/3*pi^3 does not appear in the sum. Simirarly, a 0-sphere is actually just a pair of points and a 1-ball is a line segment, but then it is odd dimensional and of length 2 : it does not appear in our sum. The real question is why does a 0-ball has non zero volume 🤔
A zero dimension sphere HAS a 0 volume. This video confuses volume with the Lebesgue measure, which is not the same thing. In effect, the Lebesgue measure of a 0-dimensional sphere is its length.
@@alex95sang52 No, you are wrong. In higher geometry, it is standard terminology to refer to the n-dimensional Lebesgue measure of an n-dimensional set as simply its volume. This usage of terminology is inconsistent with how the terminology is used in the mathematics you are taught in schools, but it is, nonetheless, entirely correct.
@@marinricros9555 A 0-ball has nonzero 0-volume, because the 0-volume is just the number of points, which for a 0-ball, is trivially, exactly 1. In the context of the video, "volume" refers to n-volume, not the 3-dimensional Lebesgue measure. When mathematicians talk about the volume of an n-dimensional object, they are talking about n-volume. This is just standard terminology.
I have a guess of why e shows up here that can sort of be intuitively understood. Lets look at the definition of e:
lim{n->inf} (1+(1/n))^n
The equation we are taking the limit of is technically the volume of an n-dimensional hypercube with edges of length 1+(1/n). Right in the definition for e there's a fairly easily understood connection to volumes in higher dimension.
You are right!!
If you start your series at the second term, you get
∞
Σ V(2k) = e^π - 1
k=1
which is really cool because it's *almost* Euler's Identity. (Were V(n) is the volume of the n-dimensional sphere).
Finding out Zach Star has an educational channel is like the equivalent of finding out your funny uncle is your college professor.
Fantastic video, not only was this a nice, simple proof of something pretty cool. But your explanation of the gamma function was probably one of the best I've ever heard in terms of simplicity without losing too much truth about the function.
Wow. I wish I knew more about this stuff. Its so interesting.
Knew this stuffs at my Calc 2 book! infinite series like Taylor and mclaurin series are formally introduced here.
@@buhatmarccarlson4596 oh yeah. applications of derivatives. those theorems were superior forms(or derived forms) from langrange's mean value theorem. i almost forgot.
0:48 those functions look a heck of a lot like the spectrum of light given by temperature X by blackbody radiation
Probably because it's the roughly the same equation
You are 1 smart dude!! Wiens curve!!
Very cool and interesting stuff. At first i thought this would be complex but you made it easy to understand
Dope stuff, cool to see everything come together with the maclaurin series
Your explanation is understandable... Thanks... Loved it
Not sure why I'm watching this as a 16 year old British boy who should be studying for his GCSE history exams right now, but great video! I just about managed to not have no idea what you were talking about 👍 I see this as an absolute win
Hey, how were your GCSEs?
@@weltschmerzistofthaufig2440 Lmao they were pretty good thanks. Ended up getting a 7 in history which I was pretty happy with. Maths and science have always been my favourite though, so now I'm at uni studying computer science and electronic engineering.
@@billyusher4907 Oh, that's awesome! I'm glad you found a fulfilling passion. All the best at your university!
"The problem is all inside your head, she said to me / The answer is easy if you take it logically..." - Paul Simon
It's simple when you put the formula for n-capacity of an n-ball of radius r in its most elegant form:
V(r,n) = [(πr²)^(n/2)] / (n/2)!
Let n = 2k, and sum over k, from 0 to ∞, and you have the Taylor Series for eˣ, where x = πr².
So when r = 1, ∑₀ººV(1,n) = ∑₀ººπᵏ/k! = e^π.
And you're right; this *is* pretty remarkable!
Fred
Damn!! Your IQ must be 420! Minus 250 for the Paul simon quote= 170....a better quote...."and so I record "the annals of Rome" not from desire, but because God destined me to suffer the curse!" Tacitus
what suprised me more is how the volume start decreasing with the dimension's increase! very interesting video
I was just reading the title at the thumbnail and was like ".......... what?"
Wow! The first CZcams Channel that's over my head!
Awesome!
The first? You haven't searched much yet, haven't you? Not to criticize him in any way, but holy ** are there loads of channels to blow your mind out there
Check out the 100 x 100 Rubik's cube!!
And e^pi ~ g which makes sense cuz earth is a sphere.
Nice video! Thanks!
Great video!
Thanks. I needed that.
Somehow i simultaneously understood yet didn't understand it.
That's nothing a little observing can't fix
I think you understood you just need to let it sink, digest... play with it a little, maybe if you see different sources you will get more details, have fun
Just observe it, and you will find out if you actually understand it or not... 😀
nicely done
I wouldn't have understood jackshit from this video just 4 years ago, and look at me now I can predict what you're gonna do next im honestly shocked how did i get here, great video!
So the sum of all volumes of even dimensional unit spheres to the power of i equals -1... neat
Good math to wake up to!🙌gm from CA.
Although I understand skipping it for brevity, I think you should have clarified that you mean n-dimensional volume as volume is usually just three dimensional. Also not mentioning anything about the odd number or all number series left me wanting more. Despite all this, super cool video!!
Thanks! I am working on a video now where the subject is essentially 'cool things about every dimension', and I will go into more detail on the stuff this video skipped over since I've been getting a lot of questions in the comments.
"We have finally come full circle."
I see what you did there xD
Very good!
As soon as I saw the gamma function I was like yayyyy Taylor series!
Thank you Brady
The circumference of a circle --> 2πr
The area of a circle is the anti-derivative of it's circumference --> πr^2
The surface area of a sphere --> 4πr^2
or four times the area of a circle with the same r
The volume of a sphere is the anti-derivative of it's surface area --> 4/3πr^3
A nice way to remember circle and sphere formulae.
This holds true for every dimension sphere
I have algebraically proved this in undergrad but never understood it. Thanks for this!
great video thanks a lot
Great video! Hug.
Can you make a video comparing aeronautical and aerospace engineering??
This is one of those things where I can follow you as it's being explained, but could in no way reproduce the explanation. Lots of things are like that.
Please make a video on William Lowell Putnam Mathematical Competition
Please do more videos on statistics and probability!
Good job
this is fucking awesome, thank you
Keep the r - that way you get e^{πr^2}!
why neglect r^n? otherwise, you would get e^(pi*r^2), which is more informative/nicer.
each term of the sum would contain r^n
Because he’s doing it for unit spheres r=1
@@jonathanallan5007 no because he only takes the even ones so it will always be r^(2n)
@@jonathanallan5007 and by doing so, he loses information. There is absolutely 0 reason to set R to 1.
@@thatyougoon1785 indeed, n will always be even (you could call it say 2m), but it is a different n in each term of the sum (i.e. ... + pi^m/m!*r^(2m) + ...) and the limit of that sum wont be e^pi*(r^2). EDIT - ah, do you mean e^(pi*r^2)?
The fact that it's a finite sum is mind blowing, even for someone like me who is used to this sort of math.
So if the sum of even-dimension balls is e^(1/2 tau), is the sum of all multiple-of-three-dimension balls e^(1/3 tau)? And so on for multiples of 4, 5, etc?
I know I'm 2 years late, but no, with multiples of 4 it converges to 11.5919... and e^pi/2 is 4.8104.
Ok, so we saw that by grabbing the formula from wikipedia and plugging in values, we are now looking at the exponential formula.. great, but where's the explanation?
I was hoping for some bigger picture.
You need to have some understanding of derivatives for that. If you do, watch this video: czcams.com/video/3d6DsjIBzJ4/video.html
If you don't, watch the whole series. You won't regret it.
What have you majored in?
Wonder how many other people will be showing up here after digging deeper into the Animation vs Math video
I wonder if there is any nice result for the odd dimensions
math fellas the video starts at ~3:20 the rest is like an introduction to the measure of the n-ball + basic gamma function properties
Ohh myy lawwddd its beautiful
OMG, how is it that CZcams has not suggested this channel to me yet ???
Glad they finally did!
@@zachstar Never late than never ! But now I have to catch up before the next surprise exam !
What program you use to graph those functions?
5:18 *full sphere
What if we sum up the odd ones too?
We can perceive it's actually not the sum of volumes, but sum of space variable of cycle dimensions. But It's not a clickbait because another title would be impossible! Lol
If you believe in bigger than 3 dimensions that is ok, other way is to interpret as holographic principle where information about volume, information is imprinted on surface area. So full circle unit radius area integral is A=-2iln(i)=pie, and from here we get Gelfond constant e^pie=(-1)^-i=2,314... or approximate 20+pie, or approximate (60!)^(1/60).
*Great video, but I was wondering WHY it approaches e^pi*
check out 3b1b's video on taylor series
I mean... the guy just proved to you why they did. If by "why" you mean an intuitive explanation rather than mathematical one, I don't know and there is a chance that there isn't one. It would be like asking why is pi=3.141... We can prove mathematically that it is like that, but no one could tell you why it couldn't be equal to something else if our world started anew. It's just that in this universe we inhabit the ratio between a circle's circumference and radius is equal to 3.141...
Maybe pi could have been any number but just so happened to be this one, maybe it can only have this one particular value. Maybe it even depends on some meta-conditions we do not know of that shape the way maths work in a particular universe. We do not know these things and as such, discussing them makes little sense
Whenever you see euler's number somewhere, usually the answer is just "because e".
Errenium well our logic is based off of our creation. We make sense of our world and have learned to interpret it. If the world was different, perhaps our reasoning would be different. Although using logic to describe a fundamentally different logic is quite tricky.
@@Assault_Butter_Knife your IQ must be around 500!pi was explained to me as a cheap farmer wants a fence to cover as much area for his animals for the least amount of linear fencing and money...
Go through the book 'Consider Dimension and Replace Pi'...you will come to know the correct formulas of Nth-Dimensional sphere...
thank you for the recommendation
My knowledge dose not pass cal2 but I was able too follow along, scary!!!
Yeah ur knowledge of writing “dose” not pass grade 2 either, spooky
So I was interested in the total sum, and apparently, it's... 46? Just 46? Meaning the sum of the odd spheres is 46-e^pi? Wow. Just wow.
They should update the answer number in hitchhikers guide to the galaxy...
Funfact!
Mindblown
and now I'm wondering what the sum of the volumes of the odd unit spheres is equal to.
what about the sum of the odd dimensional spheres? does it converge
It has to converge, because V_{2n+1}(1) is trapped between V_{2n}(1) and V_{2n+2}(1) for large enough n, both of which converge. You need to know what Γ(n+½) is to do the sum, but you end up with e^π × erf(√π). What? The √π is fine, but what's that erf thing? Well, it's a function defined by an integral, and erf(√π) is pretty close to 1 (it's about 0.9878). Credit: I cheated and used Wolfram Alpha.
Can anyone tell me what graphing programme he uses?
Euler was a beast, those who cry Gauss can keep crying
Brilliant
what is this? Advance algebra? Is there a name for these crazy looking equations? Where do you start if you want to be able to understand all of this? Thanks
How did you figured out it was getting closer to e^pi instead of another exponential function?
He didn’t need to guess and see that it was close to e^pi. He just saw that it ended up being the same as plugging pi into the maclaurin series for e^x
Maclaurin series...I'm having PTSD flashbacks to Calc 3...
You should change the channel name hahahaha. Love the vids tho. Keep up with the good work
Is the gamma function an analytic continuation of the factorial function to the reals?
Sort of... It can be used that way but the gamma function is a subfield not a function as the projection to the reals is not unique. Convergence is conditional - rationals v irrationals... its COMPLEX :)
Can you do videos about Engineering Technology?
Which software you use for graphs
I'm just fourteen and all through the video I'm like
Yes, yes... Yeah, I'm totally understanding that......, thats obvious, ..
In my mind: whats a factorial?
Its the number multiplied with every number before it,
4! is 4 * 3 * 2 * 1 = 24
3! is 3 * 2 * 1 = 6
@@pentilex4338 oh thanks
its a very strong sqquence
@@Qaptyl lol thanks dude, but I'm 16 now and about to give my engineering entrance. And it includes a hell lot of calculus, so I'm well aware of factorials now :P
@@TheMartian11 yes but i just wanted to mention its very very strong
edit: joke went badly because i guess factorial isnt as strong as like tetration or smth
Making me think up the explanation myself isn't necessarily a bad thing, but it would have been helpful to specify n-dimensional volume and to explain why it gets smaller as you increase dimensions.
The best explanation I came up with is that if you look at the n-1 dimensional cross sections of an n-dimensional sphere, the volume of a center cut will be the same as that of the n-1 dimensional sphere, but as you move outwards, the cross section spheres shrink at a much faster rate the more dimensions are involved, since they have a term of r^(n-1).
Yeah that would've been good to include but I really did want this to be a quick video that just showed the math cause the intuition is an entirely different thing (you should look up sphere packing if you like this kind of stuff). Someone else in another comment gave a great explanation and basically they said if you think about a circle of radius 1 inscribed in a square (of side length 2), the circle will take up 78.5% of the square. If you put a sphere of radius 1 inside a cube of side length 2, then the sphere will take up 52.3% of it. That percentage keeps decreasing cause the object is curving in higher and higher dimensions and as the dimensions approach infinity, the sphere will take up 0% of the higher dimensional cube.
DOPE
I would like to know what is the volume of a circle.
The area.
anyone who wants to know more about why this works should look up 3b1b's series on the essence of calculus, specifically his videos on derivatives and taylor series, the math is much less intimidating than you may think
damn mind blown
What is thing You used For Graphs?
It's desmos, you can see it in the frame in the bottom left.
What program are you using ?
How about the sum of the all odd dimensions?
Is there also a geometrical interpretation, based on sphere pictures?
In terms of the sum I would guess no because we are adding things of different units (m^2 + m^4+...). But there is the interpretation that as the dimension of a sphere approaches infinity, its volume approaches zero.
@@zachstar Cool answer, many thanks.
I love this kind of stuff.. suggest me a course/degree
Shivam Sharma number theory
Yes I understand and it is very good. But it means very little. Because you are adding up volumes in different dimensions, it means nothing. It is like adding the area of a square and the volume of its corresponding sphere. Or like adding the length of a line and the area of a square. They are under different perspectives.
Shouldn't this mean e^pi is transcendental since you need infinitely many subtractions to make it zero?
Re Trend e^π is transcendental but not for that reason. There are plenty of convergent infinite series that are algebraic (eg. 1/2 + 1/4 + 1/8 + 1/16 +... =1)
I like how you basically zoomed into pi at 4:52, was that intended lol?
complete accident actually lol. Didn't notice until reading this comment.
Fascinating, but I have a question. This may have been asked already, but I take issue with your assertion that the term R^n can be ignored. It's true that for all unit spheres of any dimension, R=1 by definition, but unlike the quantities e, pi, or the gamma function, which are dimensionless, R has the dimension of length, and therefore the terms cannot be added as if they have no dimension. For example, you cannot add the area of a circle to the volume of a sphere, or to the hyper-volume of an over-sphere. Am I missing something?
He just added up the numerical values, which, in a way, can be interpreted as the volumes of some spheres. Don't think too hard about it, because it is just playing around with numbers.
I feel like we're confusing volume with Lebesgue measure...
it's easier to refer to it as 'volume' for the unitiated
Can u upload basics of lebesgue measure.. or suggest some link..
I'm not experienced in this subject matter, but it appears from a quick Google search that the Lebesgue measure is an extension of length/area/volume to any dimension, so in an intuitive sense it is, essentially, "volume" in other dimensions.
But, you didn't give me an intuition about why that happens.
Well there is the sphere packing problem... Where you have a 4 unit cube in n dimensions and you fill it with p unit spheres so that the spheres are always touching the cube and n other spheres after all it's a n dimensional problem. Well for n = 2 you get 4 spheres touching each other, and you could fit a sphere in the middle of the cube that touch the packing spheres and the sphere in the middle would have a radius of sqrt(2)-1.
Now for each dimension we can generalize this so that the center sphere that can fit between all of the packing spheres in the center to be sqrt(n)-1 So in 4 dimensions you could fit a whole unit sphere in the middle which would mean you could fit 17 spheres in there. However after the 4th dimension the sphere in the middle has a bigger radius than the packing spheres. And after the 9th dimension the inner sphere which is bounded by the packing spheres has a radius bigger than the unit distance of the center of the cube to the edge of the cube. Let's take dimension 16 for example, in which the distance from the center to the edge of the cube is 2 units, but the sphere in the middle of said cube has a radius of 3 units.
The radius of the packing spheres that line the inside of the cube is still 1=((Γ((n/2)+1)^(1/2))/sqrt(π))v^(1/n)
and the Volume is as we know as R is one V=(π^(n/2))/Γ((n/2)+1) so for let's say dimension 16 we know that the volume of the spheres is V= pi^8 / Γ(9) or roughly 0.24 units. the volume of the cube is however really interesting as well, as it's 4^n and as the dimension was 16 the volume of the cube was 4^16 or 4'294'967'296 units.
So the amount of packing spheres you could fill said cube with would be roughly 18 billion.
Here's a derivation for the sum of all dimensions, even and odd: www.johndcook.com/blog/2019/05/26/sums-of-volumes-of-spheres/
How does the volume decrease by adding a dimension?
The area (2 dimensional volume) of an unit circle is pi and the area on the 2 by 2 square the circle fits inside is 4. So the area of a circle is ~79 % of the square. That's because the circle is curved and has less area than the square. The volume of unit sphere is 4*pi/3 and the volume of the cube it fits in is 2 by 2 by 2 = 8. So, the volume of the sphere is only ~52 % of the cube. That's because the sphere is curved in an additional dimension compared to a circle and it loses volume even more! The more dimensions, the more volume is lost due to curved shape and that's why the volume aproaches zero when dimension aproaches infinity. Zero-dimensional circles are curved in zero dimensions (not at all) and are practically the same as zero-dimensional squares and that's why the zero-dimensional volume of a zero-dimensional unit-circle is 1.
@@pojuantsalo3475 To be fair, we also should factor in the fact that the volumes of said hypercubes grow exponentially (by a factor 2), but it boils down to the fact that we have a factorial in the denominator, and factorials grow faster than any exponential (after some time), in particular pi^n.
How come the volume of a 0-dimensional sphere (which is a dot or singularity) is 1 and not 0 ?
Ummer Farooq What do you mean by that, exactly?
Great video! But what's a 0 dimensional sphere??
A point
0, 1, and 2 dimensional ‘spheres’ aren’t really spheres. As already stated 0 dimension is a point and 1 dimension would be a line segment.
MajorPrep yeah I appreciate it. However for 2dimensional "sphere", we can imagine a 3d sphere sliced by a plane passing through the center to yield a circle. In 1dimension we can also imagine a 3d sphere intercepted by a straight line passing by the center to yield 2 points equidistant to the center. But I don't understand the 0 dimension. What could it be?
0 dimensions is what you get at the intersection of 2 lines.
MajorPrep yeah I see thank you!
eat the pie?
e to the ...
I‘ll see myself out
Careful! You're a hero!
Why woundln't you aline the dots at 3:58 ??? ._.
Yet another occasion where pi wins out over tau!
...except for that the derivation of the n dimensional hypersphere in Euclidean space involves 2pi and it was (yet another) coincidence that the 2's canceled out during calculation
If you add up *all* dimensions (not just the even ones), would you get e^tau? Because then Tau wins. 😂
Sir , how can I be good in maths ?
Practice that's it
Who are you ?
Rayeed Hasan 😂 you commented publicly, yet you don’t expect others to reply
@@localboys7449 , I appreciate his reply , I just wanted to know who he is