I have seen the fact that the volume of balls picks up a pi every even dimension, but I never really understood why this only should happen every two dimensions. Seeing it written this way, where it always "gets half a pi" in each dimension, just that it happens to cancel out with the coefficient in the odd dimensions, is really cool!
I would term my discovery of deriving half factorial in high school has been one of the best feelings. Notice though it was discovered earlier, I did it independently myself and it felt like my own. But I was approaching it differently. My interest was not the half factorial but I wanted to know volume of 4d sphere. My mom was slicing potatoes and it gave me an idea. A 3d sphere is just slices of circles of radius from zero to R and back to zero. So I tried integrating the areas together and I was able to get volume of 3d sphere using area of 2d sphere(circle). Now for 4d sphere it is just integral of volume of several varying radii 3d sphere the same slicing idea! Using that I derived for 4 d. Then 5d and then 6d and so on. Then I realized a curious pattern for every even dimension the denominator was going like 1, 2, 6 ... Immediately I could see the pattern that it will be 24 or 4! For the 8d sphere. I checked it and magically I did get 24. Now I was able to get a general formula for n dimensions, and then I was like why not substitute n as 3 or 1... The truly magical thing happened ✓ pi came out of no where and I was able to derive half factorial without knowing gamma function! I was completely astounded and was in cloud 9. I never expected factorial and nd spheres were connected. I didn't sleep that entire night. Rarely one gets such joy, and that day was truly a memorable one. So that is my story of the gamma half!
You dont need n-balls. All I did was find the areas of a quarter of a simple 2 dimensional superellipse of the form x^(1/n)+y^(1/n)=1. You find that in terms of y that is y=(1-x^(1/n))^n. Integrating from 0 to 1 of that equation for an integer n will just give you (n!n!)/(2n)! That is literally it.
You must be careful. I agree this is an absolutely awesome feeling and an amazing find! But again, you must be careful. You didn't derive the value of the half factorial, you defined it. The reasons mathematicians prefer using the gamma function instead of writing factorial is that they mostly agree that it only works on the set of naturals. To reiterate, you found an amazing pattern and one version following the laws of the factorial, but you didn't derive it for half values, you defined it. Just something to remember
@@pedrosso0 Yes there is a difference between deriving and defining, but what is it? For example one finds, that i = exp(iπ/2) through some way. Would he be defining or deriving this fact? Just asking.
@@Qrudi234 Well I don't know all the ways of finding this value but if we mean by taylor series then the process would probably go like this: You know e^x = 1+x+x^2/2!+... And you play around with going beyond the rules and find e^i = -1 or even e^(i x) = cosx+ isinx And then you realize that hmmm what if we *define* e^x to be its taylor series analytical expansion and thus make that true? And then you can find that e^(i pi/2) So the cycle goes: derive define derive define... The process of mathematics really.
Pretty cool. I had the same idea a bit ago, and it turns out you can use this logic to actually define the factorial of ANY rational number without using the Gamma function. All you do is redo all this but instead of using n-spheres in the L2 metric, you do n-spheres in the Lp metric which will give you factorials like 1/p!. The recursion is more complicated for Lp spheres, but it works out. In general, if \pi_p is the volume of the Lp sphere embedded in p-dimensions, v_{n,p} the volume of the Lp sphere embedded in n-dimensions, then (n/p)! = \pi_p^{1/n} / v_{n,p}. And this reproduces exactly the formula for 1/2! using the L2 norm, but works for any rational number. Challenge: Prove it is well defined (eg, (6/9)!=(2/3)!)
I have come back home from the pub and so am 'tipsy' (that is, not drunk but also not sober) I am an engineer not a mathematician, but your video came into my feed because I watch Mathologer etc. The job of a video game (say Portal2) is to make you feel 'clever'. They spend a lot of time engineering things so that this is the outcome. After a few minutes of watching your video I realised that I'm not your target audience. I am not a mathematician. Quickly you got over my head. And yet I've watched videos on Lie Groups or Rieman Hypothesis and understood it (then quickly forgotten it). Having explained my positoin.. I want to offer unsolicited advice.... You need to slow down, and spend more time between each 'leap' if you want people like me (not mathematicians, but interested) to follow your content. Otherwise I have to say, I like your delivery, and the editing of your content etc. Thank you for making these videos, youtube should be used for education. When I'm sober, maybe I'll understand more? :)
Just as an empty sum is zero , an empty product is one! This is much closer to a truth than to a convention. It can be unified by saying : in a group the empty product is the neutral element.
Really interesting derivation. I just felt the graphics were a bit too fast. Text appeared for a fraction of a second, and the well built graphics just moved too quickly to follow with the comments and the developed equations hammering my brain as well!
@@mathkiwi Don't overestimate your audience, and just make it comfortable for them. If you have ever taught, you'll know that most (really most!) people (I'm not talking about von Neumann) need more time to just see and then digest what they see. And avoid making them feel rushed.
Same I had to play it at half speed in order to read everything and pause over and over again in order for what you were saying to sink in. Really good video overall though, thanks!
I really like this way of deriving the volume of balls/spheres. I did it in class using the usual multidimensional integral stuff and when I saw that you were doing this I immediately realized what your definition was going to be, but this is a much more intuitive way of looking at things.
Nice video. There's a typo at 11:11 onwards, B_1 should be 2R. Additionally a factor of 2 is missing from the right between 11:41 and 11:46. As the transitions between equations were so fast, looking at everything by pausing/rewinding took longer than if I were to write out everything myself. Maybe use slower transitions or multiple lines of proof at once if you really want to show each step of algebraic manipulation. The rendering and graphics were great.
"The others will define themselves according to the recursions of alio......" at 3:32. Hard to understand--please speak slower and clearer or write this down.
good video but i find a little problem: your pronunciation is a little difficult for me to understand, and that happens to be a problem because there are not subtitles. :(
12:00 I don't get why he was able to define that expression as 1. Like, what argument did he use? I don't see an argument to force both N-odd and N-even expressions to be the same. Why they are the same?
That's really a very amazing interpretation between topology and factorials And the question about why the surface is the derivative of the sphere was always a question i thought about and couldn't get it If u might mention some sources for why it's so
This video is full of "Here's how it works in one case, so it works in all of them", the only time where this wasn't entirely the case was with the hypercube proof of why the nth ball's hypervolume is proportional to that of r^n. The very premise of the video is at the end "Okay let's just define it to be like this", paraphrasing of course We set out to find (1/2)! without using the gamma function, not to define it. Which if we consider what determines a factorial that's completely impossible because you have to define it. You should, like in the video of the youtuber "lines that connect" state clearly that you intend to define (1/2)!, and then state the things you decree shall hold for a factorial. You did do the latter with saying that n! = n * (n-1)! but you only really said that as a law which is perfectly true for the natural numbers but you didn't make it clear that "Hey I'm deciding that this shall hold for all valid inputs of this new factorial I'm defining". You also very quickly decided "okay let's just define it to make this make sense", you mentioned defining but you just kind of handwaved it.
The formulas are shifting and transforming so fast, with so many skipped steps. At 11:15, I really didn’t recognize that as a factorial expansion; it was missing the 1, the only other terms were 3 and 2, and the “k is even” disclaimer technically is enough info but my brain couldn’t process it within seconds. There’s not enough setup to see how our recursive factorial formula fits into the n-is-odd case; the division by (1/2)! seems random. The motivation for making these formulas match wasn’t explicitly stated. Afterwards, steps are skipped in the already-too-fast formula transformations, starting at 11:42. The -1 exponent is resolved at the same time as moving pi into the numerator. I didn’t understand the transformation at 11:44 until I wrote it down; the entire formula appears to change, but really only variable names are changing. I still don’t understand what the 2+k transformation is at 11:46. And I don’t understand how you went to -1/2 at 12:21. The pacing was great until the final meat of the argument. You just need to sloooooow down.
Very cool! The last time I saw formulae for the surface area and volume of N-balls, it used the floor function and the "double factorial" (which is more like a "semi-factorial" than a "double factorial", because mathematicians are bad at naming things) so that the argument to the factorial remain integers. Also, the Gamma function is _not_ *the* extension of the factorial to the Real (or rather Complex) numbers. Gamma is the extension of _(n + 1)!_ to the Real/Complex numbers. The _Pi_ function is the real extension of n!, and in fact, Π(-½) = (½τ)^½, just like the half-integer factorial defined here.
The isomorphy is in a different category: It's just a volume-preserving bijection (except for some points / the edges of the cylinder but they don't have a volume in this dimension)
This is so cool. It breaks my brain in a wonderful way. Need to go through the math myself a bit slower to track it all, but either way, it's beautiful the way it all came together.
I worked out the formula for any number of dimensions all by myself and found it was a different one for an even number of dimensions than an odd number of dimensions.
Well, I'm going to have to watch this again, but I have to say that this connection between the surface and volumes of n-spheres is delightful. I've seen the formula for the n-volume of an n-sphere decades ago, but this is certainly an amusing update. Regarding the n-1 surface of an n volume I can also encourage the readers to check out differential forms and the exterior calculus.
I enjoyed the video. My one bit of feedback is that it was hard to parse everything you were saying when you spoke so quickly. That's a stylistic choice that you're free to make your own decisions about, but I think that particularly for a topic like math, having a slower speaking speed than usual and/or more pauses would help. This is especially true when leaving out steps. No matter how trivial the bit of algebra or calculus or recursion, it can still take a couple seconds for the viewer to fully understand one or more omitted steps, and if you transition immediately to the next thing, the viewer will miss parts of what you're saying, and then get more confused, or need to pause or rewind.
Great video indeed. Yet a few ( I hope constructive remarks) . R1) A bit fast altogether add a breathing second from time time . R2) Several time you show one line , make a change ( in it) , comment the result. It is better to keep two lines ine a row. Thank you any way.
I don’t see why factorials of negatives have to be undefined. If we just go the other way with the multiplication we let get that (-1)! = -1 x 0!. And we’d then get that it’s -1. (-2)! would then be 2, (-3)! would be -6, and so on.
Who says the 0-ball doesn’t have as its boundary the (-1)-sphere? Just say that the (-1)-sphere is the empty set. The (-1)-sphere is the set of vectors in R^0 (the 0-dimensional vector space over the real numbers) such that the norm of the vector is equal to 1. Namely, the empty set.
With a better definition of factorial, a separate rule for 0! is unnecessary. Why not _n! = 1 times all the positive integers less than or equal to n_ ? From that definition alone, 0! must =1, because there are no positive integers less than or equal to 0.
Why is it that we can just define that constant to be 1, to have the even and odd formulas be equal? That seemed hand-wavey to me and I didn't understand why we could just do that. I'd really appreciate an explanation!
Loved the video, although a have a few constructive suggestions. Some times, the narration can be a bit harder to follow or too fast. I understand since I’m not a native English speaker, but as 31b1 does, subtitles and captions for important points could be very helpful. Slowing down how fast you speak could also help, since it makes it easier to understand. I look forward to more videos, and hope this helps!
I really loved this. It provided whole new insights into Bn and Sn for me. On a lighter note, we now have the perfect answer to the people arguing about whether to use Pi or Tau. The answer is neither. Replace Pi with (-1/2)! ^2
I think it is not true or can at least not be simply assumed that the volume of the n-ball is proportional to the n-cube as it will take up less and less space inside the n-cube.
5:45 A homothecy of ratio k on R^n is a linear transformation of determinant k^n. The rest of the proof is paperwork and sadly depends on one’s definition of volume and surface. I don’t want to introduce n-lebesgue measures, nor differential form pullbacks here 🙊
What the video shows that if you define (1/2)! as proposed, you get a unified formula for the volume of n-spheres that does not require a distinction between odd and even dimensions. It is a definition guided by heuristics - not a proof. When exploring maths you’d next want to check if this definition makes sense and exhibits the properties you’d expect. In this case, things work out, but in general this does not need to be the case. It is e.g. not a priory clear that the volumes of odd and even dimensional spheres would be related in a neat way that is compatible with other ways of generalising the factorial (e.g. via complex analysis). While it does turn out to be the case here, the video does nit provide a proof. I didn’t understand this to be its ambition, but making this more explicit would be a good way to increase the intellectual honesty level of this video. :)
does it means that math obey law of nature?? Oh no so hypothetically that, some domain of math requires reality? This is against our common undestanding that math is un-tangled with reality.. By including pi in this formula, we can say that math is not consistent? Because it depends on the value of pi.. which in our spescific reality is 3.14
What is "reality"? In what reality pi is not about 3.14? I heard that pi is 4 in one of United States (it seems, Indiana). But in Euklid geometry pi (relation of the length of circle to its diameter) is ~3.1415926. And there is a proof that relation of the round's surfase to its radius is the same. In Riman geometry the length of circle is less, in Lobachevsky geometry the length of circle is more than pi diameters, but they both approximate to pi when the diameter approximates to zero. And pi as itself may be calculated without geometry model, despite it is some sophisticated (many calculations, too slow).
What is "reality"? In what reality pi is not about 3.14? I heard that pi is 4 in one of United States (it seems, Indiana). But in Euklid geometry pi (relation of the length of circle to its diameter) is ~3.1415926. And there is a proof that relation of the round's surfase to its radius is the same. In Riman geometry the length of circle is less, in Lobachevsky geometry the length of circle is more than pi diameters, but they both approximate to pi when the diameter approximates to zero. And pi as itself may be calculated without geometry model, despite it is some sophisticated (many calculations, too slow).
We just "define it to be one...", and it works. Well, then you may define whatever you like, to make things mathematically work, but reality should be the judge though.
WOW! Amazing explanation! It is a very beatiful problem, and i would've ever thought, that it has anything to do with volumes and surface areas. Thank you!
As the video started, I was fully ready to be told the history of speedrunning of factorials or something like that. I am broken. (Whoever gets this reference please tell me I'm not alone)
So actually, the whole spheres, bodies and stuff do not have any connections to the "! operation", you (or somebody before you) just definited something, and (in this case) it works, as it is defined to do so.
6:38 Is this true for all solids or just Platonic Solids? You seem to be suggesting that surface area is the derivative of volume, which it is not in all cases. Is it not that you have developed a heuristic approach to the question based upon cubes and spheres as your instruments?
Interesting! this is so sick when u see something derived from somewhere else. For me it still was kinda fast, need to watch that a couple of more times to understand every step, but thx!
I'm courious about something, what is the general definition of surface area? I understand the volume (which is how many hypercubes fit in the object) but I'm clueless on how to define surface "area" for higher dimensions
@@gabitheancient7664 I believe a surface is by definition something 2-dimensional, something that locally looks like R². So the n-sphere is not really a "surface" unless n=2. Still, it seems standard terminology to speak of the "surface area" of the n-sphere. In any event, the n-sphere S_n looks locally like R^n, i.e. you can form it by bending (but not stretching) and patching together pieces you cut out of R^n. The surface area of S_n is the n-dimensional total volumne of the pieces you need. E.g. you can construct the 1-sphere by cutting out a piece of R¹ of length 2xpi, bending it into a circle and gluing the ends together.
This is secretly using the gamma function, as the higher dimensional sphere integral includes the gamma function inside. A correct proof without explicit use of gamma uses only the property that (x+1)! = (x+1) x!, and then taking the limit of (n+1/2)!/n! as n becomes large. Assuming that the factorial function doesn't oscillate wildly between large integer values, you recover the square root of the Wallis product representation for pi. I forget the details, I worked it out decades ago, but it works, and it doesn't require doing integrals, just taking limits and reasonable assumptions about the interpolation of n! between integer values.
It's not "secretly using the gamma function" because this line of argumentation does not rely on an existing formula for the higher dimensional sphere integral
Yes they have to be since the first formula claimed it works for all numbers N, so the formulas have to be the same The general formula is Bn+k = (n/2)!/((n+k)/2)! pi^(k/2) R^(k) Bn If Bn = pi^(n/2) R^(n) / (n/2)! Then Bn+k = pi^((n+k)/2) R^(n+k) / ((n+k)/2)! Which is the same formula, therefore it must be true for all numbers, including odd numbers
@@mathkiwi you could have saved us 10 minutes of our lives if you said that you were going to define it like that for convenience of a series of formulae.
