i think the reason 4d doesn't have a name is mainly because by the time you go past 3d you're usually generalizing to any dimensionality. It is pretty cool to see that these relationships aren't unique to the 2d case
A fun thing is that these (shapes with as many equidistant points as a space allows) follow a pascal's triangle themselves (offset by 1) when expanding to further dimensions. The d-dimensional shape will contain $\choose{d+1}{b+1}$ b-dimensional shapes.
A short disclaimer about some of the maths of the 4d shape. I believe that a hyperpyramid is a general 4d pyramid although a pentacharon is specifically a 4d tetrahedron which is what we need. Secondly I am aware that for small powers of the expansion of the quadnomial, substituting the faces for pascal's pyramid and the edges with pascal's triangle does work but similar to with the pyramid, after a certain point we begin to move into the centre of the face. In the same way numbers in the tetrahedron start filling the middle but this becomes very hard to draw and harder for people to follow so I didn't mention it. The main point of showcasing the quadnomial is more the proof of concept and that something like that really does work. Also I will personally guarantee that I am replying to ALL comments and questions. Feel free to ask for any help :). Hope you enjoyed.
It seems like you've reinvented Pascal's simplices, but without the connection to multinomial coefficients. It's a well known generalization of Pascal's triangle, but still impressive that you've been able to do this on your own. I would advise you to look it up on Wikipedia, there are articles on the aforementioned topics (however small the one on Pascal's simplices may be), they may be of great interest to you.
@ValkyRiver Hey thanks for sharing this. I discovered this was possible although I decided not to mention it as it seemed that the video was long enough and it looked quite complicated to explain quickly. I appreciate you creating this to me as I'm sure some people will find this useful.
@@vladislavanikin3398 Hi, I actually have read about and derived the multinomial coefficients. A couple people have mentioned this and this was actually going to be part of the video although last minute I removed it. I actually did animate and voice it over so I have the video and I'll likely upload it in a week or two but it got quite algebraic and the point of this video was the idea. Thanks though :)
Maybe you (or somebody) could write an algorithm for a generalized n-dimensional hypertertahedron and let the computer do all the grunt work. That would indeed be interesting if it hasn't already be achieved by somebody...
Playing around with Pascal's pyramid for a bit, I realised that each layer is just part of Pascal's triangle multiplied by the bottom row. Foe example, the 5th layer is: 1 = 1 x 1 4 4 = 1 1 x 4 6 12 6 = 1 2 1 x 6 4 12 12 4 = 1 3 3 1 x 4 1 4 6 4 1 = 1 4 6 4 1 x 1 I would expect that this extends to higher dmensions as well. Cool!
I seriously haven't seen this before. This is really cool. Regardless, generating this in higher dimensions but like this video, more than anything it's cool it works. Honestly nice find - I haven't seen this idea before.
I wrote a thesis that included this. Loved your illustrations of the concepts. But I generalized the shapes' names as "Pascal's n-Simplex". It's fun to look at common identities on Pascal's triangle and see the more generalized ones in nD.
The binomial is describing how to partition a set into 2 parts: the 'x' part, and the 'y' part. The exponents describe how many element such a part will have. The coefficients give the number of ways to do so for a particular partitioning. In other words, it iterates the subsets of a powerset. The trinomial is describing how to partition a set into 3 parts: the 'x' part, the 'y' part, and the 'z' part. The exponents describe how many elements such a part will be. The coefficients give the number of ways to do so for a particular partitioning. This reasoning extends to infinity, with n-nomials being partitionings of a set into n many parts, the exponents being how many elements are in a specific part, and the coefficients the numbers of ways such a specific partitioning into n parts can be be done. In other words, you found a way to numerate the partitions of sets.
Really interesting video. I'm surprised that the trinomial distribution doesn't show up more often since it seems quite intuitive and has some quite real applications.
Hi! Yep it's something I also did think about but I think the main issue is that the 3d aspect does make it unnecessarily hard to interpret but I do think multinomial distributions are used algebraically atleast. The binomial is also fundamental and shows up in stats absolutely everywhere like in Poisson and Normal distributions so it is used far more often.
Oh yeah lol I never actually noticed that. Good catch. I'm currently slightly busy but I'll try to sort that out soon. I'm glad you enjoyed the video :).
one other really interesting thing you can see with pascal's pyramid is that each layer is like pascal's triangle multiplied by itself (you can see at 10:52, it's like a 5 layer pascal's triangle with each row multiplied by 1,4,6,4,1, this pattern continues with further layers of the pyramid. You can even see this showing up with the pentacharon with each tetrahedron of it being just a version of pascal's pyramid with each layer multiplied
@xicad1533 I'm sure you noticed this but we use a base of the equilateral triangle and then extend it to a certain number of dimensions which is why a hexagonal or square base won't work but im sure you realised that.
@@viks3864 You can follow the same principle of just adding the 4 numbers above for a square base and it works just not the same, hexagonal doesn't really work
Fantastic… once again a video I’m compelled to call brilliant. I enjoyed helping out dwww :) U deffo gave me some inspiration to make my own one some time :O. I need to finish watching this now! Can’t wait.
I have no idea why this comment got held for inspection. RUclips blocked this comment but I managed to let it through?? Anyways this is the legend himself who did the trinomial and binomial disributions. Incredible man. Anyways glad you enjoyed aiyush :)
We can generalize all triangles, tetrahedra, etc. with the term n-simplex. The regular n-simplex is simply the polytope in n dimensions that has n+1 equally-distant vertices. A 0-simplex is a dimensionless point. That's not a vertex, as we do not count the self as a facet. A 1-simplex is a line segment, it has 2 vertices. A 2-simplex is an equilateral triangle, it has 3 vertices and 3 edges.. A 3-simplex is a tetrahedron with 4 vertices, 6 edges and 4 faces. A 4-simplex is simply that polytope in 4 dimensions that has 5 vertices, 10 edges, 10 faces and 5 cells. It's often called a 5-cell for that reason.
the general multinomial coefficient is fun. it's the form (n choose k1,k2,...,km) = n!/(k1! * k2! * ... * km!), where km = n-k1-k2-...-k_(m-1). from that you can get the binomial coefficients by taking n choose k1,k2 where k2 = n-k1.
Yep it's something which other people did mention, and it was something I considered adding although the video was getting too long and the point of this video was the idea. I may upload it as a separate video but thanks for the feedback.
Really enjoyable video - well done. I love the sense of inevitability that you generate as you increase the dimensions and we find that previous results are incorporated. FYI, the generally accepted term for 4D solids of this type is "polychoron" (plural "polychora"), so your 4d shape is the regular pentachoron (which is the simplex in 4 dimensions). Anyone who wants to get their heads further into this stuff should get hold of a copy of Regular Polytopes by Harold Coxeter (Dover). Will be checking out your other vids - keep up the good work!
Hey, thank you so much for the kind words - I really appreciate it. I have received some similar comments regarding the naming of multidimensional objects on this same idea and have read up on it now but thanks since honestly I didn't really know about some of this stuff before. I'm glad you enjoyed :)
I was literally thinking of this exact problem and how this could be generalized to n-nomials raised to any power. This video was absolutely well-made and super clear throughout, well done!
I didn't expect so many people were interested in this topic. I wrote my thesis on this and it's amazing how many things you can do with this patter. There is actually a proof that there exists an object with any dimension that contains every multinomial of the same dimension and below that, since each of pascal's object contains the objects of lower dimension. Also another really cool thing is that you can find the multinacci numbers if you cut the pacal's object with an object of 1 dimension lower. Keep up the great visuals and have a pleasant day.
That's honestly so cool. I was obviously joking in the video when i said I was the first to think of this but honestly it's interesting to know that people have researched this more. Is there any chance you can link me your thesis if it's online - I would love to have a read. If the comment gets blocked because there is a link, I can manually unblock it later. Have a good day.
@@viks3864 it's sadly not online and even more sadly it's written in german. I can send you a copy if you want. The equations should still be understanable though and I can answer questions.
@coolio9713 Oh that's unfortunate but its still really cool. Also yeah I'll take you up on your offer and I'll try my best to read through it since it sounds really interesting and I'll lyk if I have any questions. Thanks
I had good understanding of Pascal's triangle before watching this video but had no idea that its logic could be extended to higer dimensions. U sir just earned a new subscriber. Great video
It was very likely I was not the first as let's be honest, I'm 17 and it's 2023 - someone probably thought of it first. What was interesting though was that there were virtually no papers discussing it although trust me, I'm more than aware that I'm not the first.
@@viks3864I was also around 17 when I discovered the concept (I have a blog on it somewhere). You'd be amazed how much fun you can have independently discovering things others somewhere already very probably discovered - and Google's search engine and Arxiv isn't extensive enough to show you everything 😅 So I doubt your name (or nickname) vik will stick to the naming of higher dimensions _unless_ you get a very large following and become very trusted amongst the community of mathematicians who are obsessed with naming things (europeans and americans in general). Which means, if you're really determined to have it named that way, you're gonna need to keep doing math videos and math in general for a very very long time, playing the long game Nonetheless your video is really nice :)
I explored them in high school as well, but i never found the connection between foiling and the triangles, i was mostly interested in the recursive nature of the structures
Hi! I'm excited to see this video, and to find that I'm not the only one to figure this out! Several months ago, I actually explored the same concept. Over a year before, I derived an equation that generated Pascal's Triangle by myself, and so I decided to go on a mission to do the same thing with the three dimensional equivalent of Pascal's Triangle. I eventually succeeded, and came to the realization that a four dimensional equivalent exists. After a long time, I managed to derive an expandable equation that generated any 'n' dimensional equivalent. What really blows my mind, is that the equation self-references itself! The equation that describes any 'n' dimensional equivalent can be generated by only using similar copies of the original equation, with only the variables changed. It blew my mind when I first found this all out. Pascal's Triangle is effectively just a small 2 dimensional sliver, of an infinite dimensional object! Really changed the way I look at Pascal's Triangle.
I actually did hint at this towards the end and honestly it is such an elegant idea. I think it's so cool and I'm glad to hear you discovered this yourself too. I'm glad you enjoyed the video :D
@Domstopher would you mind sharing what you found, I'm very interested in finding a closed form for generating those coefficients, this has a lot of useful applications for such kind of different problems
@@viks3864 1 is the first 1 1 1 is the second and 1 2 1 2 1 2is the third but I don't get why there isn't a 3 in the middle as all ones "push" (for lack of a better word on the middle which should make a three. I know it wouldn't work with evaluating (a + b +c)^x but it would look so much nicer.
Such a nice video! Thank you for talking about this complex yet fun topic. Although I can´t really say I agree with the last section. There is nothing inherently wrong about it. It just seems really inconvinient to continue using a tetrahedron or a 4d shape. You could have just used a square pyramid and for the next expansion a pentagonal one and so on.
Quite nice video! Good luck with the new channel. Please have a look at your audio recording configs, 'cos its volume is quite low, making it hard to hear.
Interesting geometrization, but its much easier to use the tri or quad nomial formulas. (N!/(d_x!d_y!d_z!)) For trinomial just also divide by the degree of w for the quad nomial
Actually while I was making this video I actually stumbled upon the multinomial coefficient and was going to include it in the video. I actually animated and voiced it over. Last minute, I removed it all since I felt the point of the video was how multiple dimensions can be used visually and how cool it looks. Seriously though, good catch, I might actually upload that part as a separate video.
Erm if I'm honest I thought it just sounded funnier lol. If this name is ever used in significant fields of maths, it will be named pascal's hyperpyramid and then following on into higher dimensions. I just named it this since originally I did try to derive this myself, and it sounds more unique but obviously if this ever became important, it wouldn't be named after me.
I think you may have made a slight error at 9:21 where you wrote 2z^2 where it should say 1z^2. You also badly need a new mic ;) other than that, great video! Extrapolating from specific examples to more general cases is truly the soul of mathematics, and you've done a beautiful job of that here.
I have a few questions that I would like to ask. Could this be generalized to any dimension? If so, could there be a general formula for polynomial expansions? I would like to add that I love your presentation of the pascal’s pyramid and 4-d shape. I would name it the Vicks-Pascal hyper pyramid because it uses ideas both from you and Pascal! Although, the video could improve with some audio balancing.
Hey thanks for the feedback. I'm in the process of trying to get a microphone so don't worry, hopefully the next video will be better. Also it can be generalised algebraically to any dimension and likely be the focus of my next video.
Funny joke. I'd actually die trying to figure this out visually. If you really are interested, there are some quite elegant ways to do this algebraically. It makes doing any number of terms to any power very easy. If you'd like I can send it to you but I actually plan on making a video on this somewhat soon. Glad you enjoyed though :D
Yep its called the multinomial coefficient. I was planning on make a follow up video on it but I have been really busy lately but feel free to look into it online. If you need any help understanding it let me know.
There is a much more beautiful underlying approach that my teacher explained to me even when I was very young (before I even learned algebra), and it was a revelation for me. It shows the beauty of mathematics and the power of abstraction. We were studying combinatorics, and in this case, he explained an intuitive way to think about combinations, and he didn't use a single bit of algebraic notation, but rather real examples. It makes a lot of sense that the two results of expanding n-nomial expressions and n-dimensional simplexes yield the same results. In fact, both of them involve choosing k_1, k_2, ..., k_m elements from n objects for each element. As an example, take a binomial expression: (x+y)^3. For the coefficient of the term x^2 * y, you would be choosing every single combination that involves 2 of the element "x" and 1 of the element "y" from 3 total choices. To visualize this, you can expand the expression into (x+y)(x+y)(x+y). See to execute the multiplication, you are simply choosing 2 of the "x" to multiply and 1 of "y", resulting it the combination 3 choose (2, 1). As for the simplex, think of it this way: rather than thinking about adding the numbers from top to down, imagine you are walking along a path on the simplex. In fact, you will notice that each number represents a point in space in this world: therefore, you can express the position of this number using coordinates, which are precisely your choices of elements! As an example, take the Pascal's triangle: you can imagine the point 3 choose (2, 1) as being 2 steps to the left and 1 step to the right. As you can see, it is entirely equivalent to the problem above! Now let's derive the formula for a general multinomial coefficient in n dimensions. Let's take a concrete example first, it's always easier to deal with. Let's multiply (x+y+z)^6 and find the coefficient of x^2 * y * z^3. This is equivalent to 4 choose (2,1,1). But it is also equivalent to finding all combinations the elements x,x,y,z,z,z! So, how would we approach this? Let's try an easier problem first, with the case that all elements are distinct. During 6 steps, choose an element from the set {1,2,3,4,5,6} and remove it. If all elements are distinct, you would have 6 choices in the first step, 5 in the second, and so on, resulting in 6*5*4*3*2*1= 6! permutations. Now we know that we overcounted many cases. So we can now tackle our original problem: how much did we overcount, exactly? Well let's consider our example of multiplication: (x+y+z)^6. Let's say that we take the case where we chose z at indices i_1, i_2, i_3. Notice that in the earlier case, we counted these indices as being distinct: however, we don't care about the order, we just want to have 3 z's in the multiplication. So we condense all of these cases into a single one of z,z,z in the end. Therefore we simply divide by the number of permutations of 3 elements here! And the same applies for each of the other variables. So in total we have n! divided by k_i! for an index i ranging the total number of variables: n choose k_1, ..., k_m = n!/(k_1*...*k_m).
That's actually a really nice approach to the situation and I have read about multinomial coefficient and simplex and it was actually going to be part of the video. Unfortunately last minute I realised the video was getting too long and decided to cut it. I'm glad you found an approach you liked and I will try my best to make a follow up to this video eventually on that topic :D (my upload schedule is beyond horrible lol)
After a lot of thought and cool revelations, I figured out that for any number of terms in (a0+a1…)^n, you can can find the coefficient of the term with a0^p0*a1^p1… where the p’s add to n with the formula n!/(p0!*p1!…), which is apparently general knowledge in the wider math world. The resulting formula is apparently called a “multinomial”.
I'm so glad you derived it yourself since I did the exact same thing while making this video and it's so satisfying. The formula is so intuitive and elegant and I'm so glad you found it interesting too even though you were not the first. Good job!
The values of the 3d pyramid are Pa choose b and c}. Which is the number of ways to choose {b} objects for one group and {c} objects for another group from {a} total objects. a choose b and c = (a choose (b+c)) * ((b+c) choose c) Similarly, the values of the 4d hyper-pyramid are {a choose b and c and d}. Which is the number of ways to choose {b} objects for one group and {c} objects for a 2nd group, and {d} objects for a 3rd group, from {a} total objects. a choose b and c and d = (a choose (b+c+d)) * ((b+c+d) choose (c+d)) * ((c+d) choose d)
The math becomes harder too. I like to find the coefficients using permutations. So for an N-dimensional Pascal Pyramid: the Coefficient of the term [i, j, k, …] can be found using N! / a product of factorials of differences. I don’t have the formula memorized, but that’s a solid hint.
I have been wondering this since i learned about the pascal's triangle. I am extremely thankful that you made this video and that i found it. I actually kind of made the pascal triangle before learning it when i was studying probability (im not sure what's the name in english but in my language its something along the lines of "combinatory analysis"). I ended up in the pascal's triangle when i was writting down a grid with the number of posssible solutions for A+B+C+D...=Z ({A;B;C;D;E...Z} being naturals) The grid was number of icognites (NI), with values of Z. any given coordinate was the number of possibilities for that given NI and Z. So for example Z=10 NI=2 would be 66 possibilities, because X+Y=10 has 66 solutions (and its 12! / (2!.10!) not going to explain but i think you can understand how that turns to factorial pretty quickly) Anyways the point is that grid ended up being almost exactly the pascal's triangle. excpet kind of a pascal's square.
Possibly, the word you're searching for is "combinatorics." From Wolfram MathWorld: Combinatorics is the branch of mathematics studying the enumeration, combination, and permutation of sets of elements and the mathematical relations that characterize their properties.
In general, the coefficient of the term w^k * x^l * y^m * z^n equals (k+l+m+n)! / (k! l! m! n!) which generalizes straightforwardly to any number of variables in a multidimensional "Pascal's simplex".
I stumbled on the trinomial distribution trying to find a space-filling-curve on the RGB cube to get all colors from black (#000000) to white (#FFFFFF) changing at most one value in one color in each transition and also passing through each plane R+G+B = constant at a time. I couldn't find a solution that I liked but the first part of the solution had some connection with trinomials
Hey sorry I like to keep my code private although if you need help with learning manim, consider joining the discord or using the online reference manual. Feel free to ask me any questions too
wow, what a coincidence. I also gave a presentation on independently discovering multinomial expansion ~august of last year! it's almost eerie how similar what we did was (but the animations you made are way way nicer than my lame google slides presentation). I think it's a little disingenuous to say the tetranomial visualization is original though. There were probably hundreds (maybe thousands!) that had studied these sorts of things but they may have been seen as too trivial to publish. For me, this idea was the gift that kept on giving, you can use to to to prove some crazy stuff like n-bonacci identities and euler's polyhedral formula for multiple dimensions. I have no idea what else is in store but I definitely agree that it only occurs by asking that daring "what if?"
13:07 that's called a simplex. edit: nvm i'm wrong simplex is the simplest polytope in n dimensions so that's a 4d simplex, and the triangle is a 2d simplex
unfortunately, you're not the first. this is a natural thing to think of, so many people have thought of it and just didn't bother giving the 4d case a name, since they usually went on to consider the general multinomial coefficients. but it's still very nice to see someone interested. there are multiple fun things to do with simplices ("pyramids" generalized to all amounts of dimensions) and polynomials - specifically, polynomials where every term has the same sum of exponents of variables. i think these polynomials are called "homogenous" or something. for example, simplices give a very nice visual representation of some inequalities, such as the algebraic-geometric inequality. it was useful in olympiads a while ago, but when everyone started using it (or similar things), the problems got a bit more difficult. anyway, good luck on your journey through math. have fun!
That's actually quite interesting honestly. Also I almost knew for a fact that someone had thought of this first but I did think naming it what I did would be slightly funnier and more memorable although I am more than aware that someone else probably invented and used this way before me and decided not to name it. Thanks though :)
So yeah.... the obvious conjecture is that you can do this with a d-simplex and d variable polynomials. It should be not too hard to prove that this works always with some linear algebra.
Yep this has been pointed out to me although the point of the video is to consider the style of thinking and the fact that it works. Multinomial coefficient formulas are quite simple and elegant but I really like the way in which this method works.
You mentioned that it seemed like the fact that you see Pascal’s triangle in each of the edges of the pyramid doesn’t serve much of a purpose? However it seems to me like it’s very fundamental. It’s almost like Pascal’s triangle is a subset of Pascal’s Pyramid. Kinda like how a triangle is one of the faces of a pyramid. So Pascal’s pyramid would be whatever the threedimensional name for a “face” would be to a 4D hyperpyramid. It was also funny to think this but in theory, Pascal’s… line? Makes sense with the line of thinking I’m trying to show here. The sides of Pascal’s triangle (which I’d consider the 1D equivalent to a 2D face) are just 1, 1, 1, to infinity. And if you were to take the monomial a^n its monomial coefficient would be 1 for all powers of n. This video was awesome I wish you success with your channel
I completely agree and I thought it was really cool but the video was starting to get really long so I had to cut out some parts but yeah I do see what you mean and I think its really interesting.
Along with multinomial coefficients already mentioned in several comments, I will suggest to you to search and study the 'Restricted Occupancy Theory _ A Generalization of Pascal's Triangle' of J.E. Freund and written in 1956. His recursive formula for n-dimensional Pascal's generalized triangle, gives the method of summing coefficients of previous stages. Good read! :)
@@viks3864 It has nothing to do with the video, I was talking about the mind game called 'The Game' which the video's section name instantly reminded me of thus causing me to lose
oooh nooo TnT I mean awesome video dude, I tried to solve this problem once, I were trying to figure out a formula for a generic coeficient in a generic dimentional tree... To start I triend to find the generic coeficient in the pyramid and I already failed there, I thought you would touch on the subject and maybe had a solution for the coeficient or just showing it's impossible to have an algebraic formula for it.
Hi, I'm glad you enjoyed! There actually is a multinomial coefficient and I plan on making a video for the derivation for it soon and I think the solution is quite nice.
Other people actually asked this question and I use manim. It's the community version and is something which 3Blue1Brown developed. There are a lot online guides on how to use it and it runs off python. I personally recommend using visual studio code too. Please feel free to contact me if you need help with learning manim :).
Nice video! Have a look at this: en.wikipedia.org/wiki/Multinomial_theorem At some point, my brain doesn't think about the n-D planes, volumes, etc... and simply about the elegant summation condition in the main definition of the multinomial theorem: Under the summation sign: For m=2: k_1 + k_2 = n For m=3: k_1 + k_2 + k_3 = n ... and so on, where k_i are dummy variables. The solution of all k_i above, forms the geometry you are explaining, given that any k_i variable must be >= 0. This is really all of the combinations of numbers that sum to N -> which is the power that you're expanding for. I really do think about it this way, it helps move away from the geometry which is harder to visualize, but of course, equally valid.
I actually did end up researching the multionomial theorem when making this but ultimately didn't include it in the final video as it got quite complex in an already long video. Thank you though for the explanation as I'm sure others may find this useful!
@SSJProgramming By the way, if you link comments in RUclips comments, it automoderates them and doesn't let it appear for any other users until I manually approve it. I don't really mind but I thought I'd just let you know in case you help anyone else.
Neat! At the end you could have shown all of the cross sections for each given volume, that could have been visualized using your earlier code! Neat concepts though, great work
@@MrConverse Yep animating this took two weeks, voicing it over took a day so there are some moments where I make some random mistakes but thanks for mentioning it and honestly good job for noticing since most people are asleep by that point lol.
@@viks3864I’m not even sure you should label it a mistake. A tetrahedron is three-sided pyramid, yes? So you are not wrong, although most of us think of a four-sided pyramid when we hear the word pyramid. I think it’s more of a precision error. ;-)
Hello, I have solved arbitrary polynomials raised to arbitrary power in general, I'm not sure how I can share the formula to you but if you interested and have an idea how I can send it to you, we can make it work
Sorry bork, I know my videos continue not to intellectually challenge you and I sincerely apologise. The next video will hopefully be good enough for you.
Hey, I am actually unsure of how the simplex works completely. I believe it is a part of graph theory and is actually in the maths course I'm learning although we dont utilise it this year. Some other people actually mentioned simplex to me and how this method is apparently quite similar to it, although it is just a coincidence. There is a decent chance everything I just said is waffle.
@@viks3864simplex just means "like triangle, tetrahedron, pentachoron, etc." without specifying a dimension. These come up all over math, for instance in "simplicial complexes" which are basically shapes built out of triangles and tetrahedron, etc.
I just read about him, seems like he did quite a lot and its a little unfortunate that the more popular name for the triangle is named after pascal but its interesting nonetheless.
lol even tartaglia wasn't even close to the first person to come up with the triangle. generally, most people consider al karaji of persia (~1029 ad) to have been the first to come up with binomial theorem (because he was able to generalize it) but there are records of ancient indian mathematicians that studied the combinatorics of the coefficients ~200 BC. I wouldn't be surprised if it were discovered earlier either. in general, I think that we should do away with mathematicians naming things after themselves and keep pascal's original name: the arithmetic triangle.
This has been a wonderful visualization of the Multinomial Theorem! I was impressed and enjoyed your geometric reasoning. In fact, it's reminiscent of topics in Discrete Differential Geometry in your use of Simplicial Complexes. en.wikipedia.org/wiki/Multinomial_theorem
i think the reason 4d doesn't have a name is mainly because by the time you go past 3d you're usually generalizing to any dimensionality.
It is pretty cool to see that these relationships aren't unique to the 2d case
It does have a name, it's just uncommonly used. It's called a pentatope. It's made of 5 tetrahedra fused together in 4D space.
A fun thing is that these (shapes with as many equidistant points as a space allows) follow a pascal's triangle themselves (offset by 1) when expanding to further dimensions. The d-dimensional shape will contain $\choose{d+1}{b+1}$ b-dimensional shapes.
A short disclaimer about some of the maths of the 4d shape.
I believe that a hyperpyramid is a general 4d pyramid although a pentacharon is specifically a 4d tetrahedron which is what we need.
Secondly I am aware that for small powers of the expansion of the quadnomial, substituting the faces for pascal's pyramid and the edges with pascal's triangle does work but similar to with the pyramid, after a certain point we begin to move into the centre of the face. In the same way numbers in the tetrahedron start filling the middle but this becomes very hard to draw and harder for people to follow so I didn't mention it.
The main point of showcasing the quadnomial is more the proof of concept and that something like that really does work.
Also I will personally guarantee that I am replying to ALL comments and questions. Feel free to ask for any help :). Hope you enjoyed.
Here are the first four cross-sections of Viks’ pentachoron using the cross-section trick twice:
1
1
1
1 1
1
2
2 2
1
2 2
1 2 1
1
3
3 3
3
6 6
3 6 3
1
3 3
3 6 3
1 3 3 1
It seems like you've reinvented Pascal's simplices, but without the connection to multinomial coefficients. It's a well known generalization of Pascal's triangle, but still impressive that you've been able to do this on your own. I would advise you to look it up on Wikipedia, there are articles on the aforementioned topics (however small the one on Pascal's simplices may be), they may be of great interest to you.
@ValkyRiver Hey thanks for sharing this. I discovered this was possible although I decided not to mention it as it seemed that the video was long enough and it looked quite complicated to explain quickly. I appreciate you creating this to me as I'm sure some people will find this useful.
@@vladislavanikin3398 Hi, I actually have read about and derived the multinomial coefficients. A couple people have mentioned this and this was actually going to be part of the video although last minute I removed it. I actually did animate and voice it over so I have the video and I'll likely upload it in a week or two but it got quite algebraic and the point of this video was the idea. Thanks though :)
Maybe you (or somebody) could write an algorithm for a generalized n-dimensional hypertertahedron and let the computer do all the grunt work. That would indeed be interesting if it hasn't already be achieved by somebody...
Playing around with Pascal's pyramid for a bit, I realised that each layer is just part of Pascal's triangle multiplied by the bottom row. Foe example, the 5th layer is:
1 = 1 x 1
4 4 = 1 1 x 4
6 12 6 = 1 2 1 x 6
4 12 12 4 = 1 3 3 1 x 4
1 4 6 4 1 = 1 4 6 4 1 x 1
I would expect that this extends to higher dmensions as well. Cool!
I seriously haven't seen this before. This is really cool. Regardless, generating this in higher dimensions but like this video, more than anything it's cool it works. Honestly nice find - I haven't seen this idea before.
This is reminiscent of the fact (x+y+z)^n=((x+y)+z)^n
Thanks I was hoping to find a solution for how to get the coefficients now its easy to calculate (x+y+z)^n for small n :)
A very nice and clear explanation!
Thanks lad :)
I'm literally "wow"-ing out loud; I can't say that for any of the other some videos I've seen so far. Awesome job!!!
Hey honestly I'm so glad you enjoyed it. It took a while to make but I am quite happy with how it turned out. Have a great day :)
I wrote a thesis that included this. Loved your illustrations of the concepts. But I generalized the shapes' names as "Pascal's n-Simplex".
It's fun to look at common identities on Pascal's triangle and see the more generalized ones in nD.
The binomial is describing how to partition a set into 2 parts: the 'x' part, and the 'y' part. The exponents describe how many element such a part will have. The coefficients give the number of ways to do so for a particular partitioning. In other words, it iterates the subsets of a powerset.
The trinomial is describing how to partition a set into 3 parts: the 'x' part, the 'y' part, and the 'z' part. The exponents describe how many elements such a part will be. The coefficients give the number of ways to do so for a particular partitioning.
This reasoning extends to infinity, with n-nomials being partitionings of a set into n many parts, the exponents being how many elements are in a specific part, and the coefficients the numbers of ways such a specific partitioning into n parts can be be done.
In other words, you found a way to numerate the partitions of sets.
Really interesting video. I'm surprised that the trinomial distribution doesn't show up more often since it seems quite intuitive and has some quite real applications.
Hi! Yep it's something I also did think about but I think the main issue is that the 3d aspect does make it unnecessarily hard to interpret but I do think multinomial distributions are used algebraically atleast. The binomial is also fundamental and shows up in stats absolutely everywhere like in Poisson and Normal distributions so it is used far more often.
At 9:21 in the video, "1z²" morphs into "2z²" in the lower left triangle. Despite this minor typo, it's still a great video!
Oh yeah lol I never actually noticed that. Good catch. I'm currently slightly busy but I'll try to sort that out soon.
I'm glad you enjoyed the video :).
I saw that too and was hoping i want the only one
@@johncorn7905 tbf nice spot, fair enough
A fantastic video! A wonderful way to both introduce Pascal's triangle, and to see what happens when we expand upon it :)
Lol seriously this guy is genuinely incredible. Please sub to him - my vote for best entry for SoME3. Also thanks :)
Very cool video and it's interesting that this method can be extrapolated into multiple dimensions.
Wow thanks kind stranger. What an interesting and concise way to explain this video!!
I was messing around with something similar to this recently and seeing this made me so excited, thank you for making this
also did some stuff with square/hexagonal base but never got very far with it, not sure how itd apply to this
one other really interesting thing you can see with pascal's pyramid is that each layer is like pascal's triangle multiplied by itself (you can see at 10:52, it's like a 5 layer pascal's triangle with each row multiplied by 1,4,6,4,1, this pattern continues with further layers of the pyramid. You can even see this showing up with the pentacharon with each tetrahedron of it being just a version of pascal's pyramid with each layer multiplied
I'm really glad you enjoyed! Pleasure to help out.
@xicad1533 I'm sure you noticed this but we use a base of the equilateral triangle and then extend it to a certain number of dimensions which is why a hexagonal or square base won't work but im sure you realised that.
@@viks3864 You can follow the same principle of just adding the 4 numbers above for a square base and it works just not the same, hexagonal doesn't really work
Fantastic… once again a video I’m compelled to call brilliant. I enjoyed helping out dwww :) U deffo gave me some inspiration to make my own one some time :O. I need to finish watching this now! Can’t wait.
I have no idea why this comment got held for inspection. RUclips blocked this comment but I managed to let it through?? Anyways this is the legend himself who did the trinomial and binomial disributions. Incredible man. Anyways glad you enjoyed aiyush :)
We can generalize all triangles, tetrahedra, etc. with the term n-simplex. The regular n-simplex is simply the polytope in n dimensions that has n+1 equally-distant vertices. A 0-simplex is a dimensionless point. That's not a vertex, as we do not count the self as a facet. A 1-simplex is a line segment, it has 2 vertices. A 2-simplex is an equilateral triangle, it has 3 vertices and 3 edges.. A 3-simplex is a tetrahedron with 4 vertices, 6 edges and 4 faces. A 4-simplex is simply that polytope in 4 dimensions that has 5 vertices, 10 edges, 10 faces and 5 cells. It's often called a 5-cell for that reason.
I’m a big fan! Love your videos
Thanks!
Job well done !
Glad you liked it :)
at 8:44 the pattern can be visualized, as the bottom rows are rows in the side of the pyramid, which follow the law used to create pascal's triangle.
the general multinomial coefficient is fun. it's the form (n choose k1,k2,...,km) = n!/(k1! * k2! * ... * km!), where km = n-k1-k2-...-k_(m-1). from that you can get the binomial coefficients by taking n choose k1,k2 where k2 = n-k1.
Yep it's something which other people did mention, and it was something I considered adding although the video was getting too long and the point of this video was the idea. I may upload it as a separate video but thanks for the feedback.
Really enjoyable video - well done. I love the sense of inevitability that you generate as you increase the dimensions and we find that previous results are incorporated. FYI, the generally accepted term for 4D solids of this type is "polychoron" (plural "polychora"), so your 4d shape is the regular pentachoron (which is the simplex in 4 dimensions). Anyone who wants to get their heads further into this stuff should get hold of a copy of Regular Polytopes by Harold Coxeter (Dover). Will be checking out your other vids - keep up the good work!
Hey, thank you so much for the kind words - I really appreciate it. I have received some similar comments regarding the naming of multidimensional objects on this same idea and have read up on it now but thanks since honestly I didn't really know about some of this stuff before. I'm glad you enjoyed :)
wow what an amazing video. You’re taught me so much
Glad you enjoyed :)
I was literally thinking of this exact problem and how this could be generalized to n-nomials raised to any power. This video was absolutely well-made and super clear throughout, well done!
I didn't expect so many people were interested in this topic. I wrote my thesis on this and it's amazing how many things you can do with this patter. There is actually a proof that there exists an object with any dimension that contains every multinomial of the same dimension and below that, since each of pascal's object contains the objects of lower dimension. Also another really cool thing is that you can find the multinacci numbers if you cut the pacal's object with an object of 1 dimension lower.
Keep up the great visuals and have a pleasant day.
That's honestly so cool. I was obviously joking in the video when i said I was the first to think of this but honestly it's interesting to know that people have researched this more. Is there any chance you can link me your thesis if it's online - I would love to have a read. If the comment gets blocked because there is a link, I can manually unblock it later. Have a good day.
@@viks3864 it's sadly not online and even more sadly it's written in german. I can send you a copy if you want. The equations should still be understanable though and I can answer questions.
@coolio9713 Oh that's unfortunate but its still really cool. Also yeah I'll take you up on your offer and I'll try my best to read through it since it sounds really interesting and I'll lyk if I have any questions. Thanks
@@viks3864 sent you a mail to the email in your yt info tab
I had good understanding of Pascal's triangle before watching this video but had no idea that its logic could be extended to higer dimensions. U sir just earned a new subscriber. Great video
Hey seriously thanks :). It's always nice when someone says they liked video. I'm glad you enjoyed.
You are not the first person, I played around with the concept of higher dimensional pscals triangles in 11th and 12th grade, it was really fun!
It was very likely I was not the first as let's be honest, I'm 17 and it's 2023 - someone probably thought of it first. What was interesting though was that there were virtually no papers discussing it although trust me, I'm more than aware that I'm not the first.
@@viks3864I was also around 17 when I discovered the concept (I have a blog on it somewhere). You'd be amazed how much fun you can have independently discovering things others somewhere already very probably discovered - and Google's search engine and Arxiv isn't extensive enough to show you everything 😅
So I doubt your name (or nickname) vik will stick to the naming of higher dimensions _unless_ you get a very large following and become very trusted amongst the community of mathematicians who are obsessed with naming things (europeans and americans in general).
Which means, if you're really determined to have it named that way, you're gonna need to keep doing math videos and math in general for a very very long time, playing the long game
Nonetheless your video is really nice :)
Wow! Very impressed that you are only 17, are you studying a math course?
Would be very interesting if this can be published in some journal article
I explored them in high school as well, but i never found the connection between foiling and the triangles, i was mostly interested in the recursive nature of the structures
@johncorn7905 yeah tbf the concepts are pretty abstract but it's quite interesting
Hi! I'm excited to see this video, and to find that I'm not the only one to figure this out! Several months ago, I actually explored the same concept. Over a year before, I derived an equation that generated Pascal's Triangle by myself, and so I decided to go on a mission to do the same thing with the three dimensional equivalent of Pascal's Triangle. I eventually succeeded, and came to the realization that a four dimensional equivalent exists. After a long time, I managed to derive an expandable equation that generated any 'n' dimensional equivalent. What really blows my mind, is that the equation self-references itself! The equation that describes any 'n' dimensional equivalent can be generated by only using similar copies of the original equation, with only the variables changed. It blew my mind when I first found this all out. Pascal's Triangle is effectively just a small 2 dimensional sliver, of an infinite dimensional object! Really changed the way I look at Pascal's Triangle.
I actually did hint at this towards the end and honestly it is such an elegant idea. I think it's so cool and I'm glad to hear you discovered this yourself too. I'm glad you enjoyed the video :D
@Domstopher would you mind sharing what you found, I'm very interested in finding a closed form for generating those coefficients, this has a lot of useful applications for such kind of different problems
Im not sleeping where is the quadnomial distribution?
Great video. Why doesn’t the third stage of the pyramid have a 3?
Please my mind hurts from editing these videos lol. What do you mean by the third stage of the pyramid?
@@viks3864 1 is the first 1 1 1 is the second and 1 2 1 2 1 2is the third but I don't get why there isn't a 3 in the middle as all ones "push" (for lack of a better word on the middle which should make a three.
I know it wouldn't work with evaluating (a + b +c)^x but it would look so much nicer.
Nice video!
Thanks! Glad you enjoyed :)
Legend has it that this was conceived on the toilet when a certain someone was in year 2
Dhruv you handsome devil. Only cool people know this. Sub to this man if you read this comment
Nitpick: It's "polychoron" (choron basically meaning "room"), not "polycharon" (charon being the ferryman guiding souls to the underworld).
*I should have corrected "pentacharon". A pentachoron is a polychoron just like a pentagon is a polygon.
3:30 I'm sorry but I couldn't not see the x¹x⁰ where x¹y⁰ should be...
love it!
Hey thank you so much. I'm so glad you enjoyed it :)
Such a nice video! Thank you for talking about this complex yet fun topic.
Although I can´t really say I agree with the last section. There is nothing inherently wrong about it. It just seems really inconvinient to continue using a tetrahedron or a 4d shape. You could have just used a square pyramid and for the next expansion a pentagonal one and so on.
Quite nice video! Good luck with the new channel. Please have a look at your audio recording configs, 'cos its volume is quite low, making it hard to hear.
Yep I am working on it for the next one since I know the moc quality isn't great. Glad you enjoyed though.
Great video boss 🐐
Thanks boss man
Beautiful
Interesting geometrization, but its much easier to use the tri or quad nomial formulas. (N!/(d_x!d_y!d_z!)) For trinomial just also divide by the degree of w for the quad nomial
Actually while I was making this video I actually stumbled upon the multinomial coefficient and was going to include it in the video. I actually animated and voiced it over. Last minute, I removed it all since I felt the point of the video was how multiple dimensions can be used visually and how cool it looks. Seriously though, good catch, I might actually upload that part as a separate video.
I think there’s an importance in keeping the geometry as the focal interpretation, especially for practicing intuition
@@RickyMudYep exactly my thought process too. I will likely upload the algebraic aspect as a separate video but I completely agree with you.
You checked an important box on the crank list: naming something after yourself. Don't do that. Never do that!
Erm if I'm honest I thought it just sounded funnier lol. If this name is ever used in significant fields of maths, it will be named pascal's hyperpyramid and then following on into higher dimensions. I just named it this since originally I did try to derive this myself, and it sounds more unique but obviously if this ever became important, it wouldn't be named after me.
Stigler's law
I think you may have made a slight error at 9:21 where you wrote 2z^2 where it should say 1z^2. You also badly need a new mic ;) other than that, great video! Extrapolating from specific examples to more general cases is truly the soul of mathematics, and you've done a beautiful job of that here.
Yep the mic really does limit the quality and hopefully I'll be able to upgrade it soon. Thank you for the feedback though and I'm glad you enjoyed.
I have a few questions that I would like to ask. Could this be generalized to any dimension? If so, could there be a general formula for polynomial expansions? I would like to add that I love your presentation of the pascal’s pyramid and 4-d shape. I would name it the Vicks-Pascal hyper pyramid because it uses ideas both from you and Pascal! Although, the video could improve with some audio balancing.
Hey thanks for the feedback. I'm in the process of trying to get a microphone so don't worry, hopefully the next video will be better. Also it can be generalised algebraically to any dimension and likely be the focus of my next video.
diffrent degreas of perpendicular are equive to base function, this will work for any (N1, N2, Nn)power to any N...
cool video
can't wait for the pascal hexateron
Funny joke. I'd actually die trying to figure this out visually. If you really are interested, there are some quite elegant ways to do this algebraically. It makes doing any number of terms to any power very easy. If you'd like I can send it to you but I actually plan on making a video on this somewhat soon.
Glad you enjoyed though :D
This is really cool. I have one question: Is there a nice factorial based formula for each coefficient in the pyramid like there is for the triangle?
Yep its called the multinomial coefficient. I was planning on make a follow up video on it but I have been really busy lately but feel free to look into it online. If you need any help understanding it let me know.
There is a much more beautiful underlying approach that my teacher explained to me even when I was very young (before I even learned algebra), and it was a revelation for me. It shows the beauty of mathematics and the power of abstraction. We were studying combinatorics, and in this case, he explained an intuitive way to think about combinations, and he didn't use a single bit of algebraic notation, but rather real examples.
It makes a lot of sense that the two results of expanding n-nomial expressions and n-dimensional simplexes yield the same results. In fact, both of them involve choosing k_1, k_2, ..., k_m elements from n objects for each element.
As an example, take a binomial expression: (x+y)^3. For the coefficient of the term x^2 * y, you would be choosing every single combination that involves 2 of the element "x" and 1 of the element "y" from 3 total choices. To visualize this, you can expand the expression into (x+y)(x+y)(x+y). See to execute the multiplication, you are simply choosing 2 of the "x" to multiply and 1 of "y", resulting it the combination 3 choose (2, 1).
As for the simplex, think of it this way: rather than thinking about adding the numbers from top to down, imagine you are walking along a path on the simplex. In fact, you will notice that each number represents a point in space in this world: therefore, you can express the position of this number using coordinates, which are precisely your choices of elements! As an example, take the Pascal's triangle: you can imagine the point 3 choose (2, 1) as being 2 steps to the left and 1 step to the right. As you can see, it is entirely equivalent to the problem above!
Now let's derive the formula for a general multinomial coefficient in n dimensions. Let's take a concrete example first, it's always easier to deal with. Let's multiply (x+y+z)^6 and find the coefficient of x^2 * y * z^3. This is equivalent to 4 choose (2,1,1). But it is also equivalent to finding all combinations the elements x,x,y,z,z,z! So, how would we approach this? Let's try an easier problem first, with the case that all elements are distinct. During 6 steps, choose an element from the set {1,2,3,4,5,6} and remove it. If all elements are distinct, you would have 6 choices in the first step, 5 in the second, and so on, resulting in 6*5*4*3*2*1= 6! permutations.
Now we know that we overcounted many cases. So we can now tackle our original problem: how much did we overcount, exactly? Well let's consider our example of multiplication: (x+y+z)^6. Let's say that we take the case where we chose z at indices i_1, i_2, i_3. Notice that in the earlier case, we counted these indices as being distinct: however, we don't care about the order, we just want to have 3 z's in the multiplication. So we condense all of these cases into a single one of z,z,z in the end. Therefore we simply divide by the number of permutations of 3 elements here! And the same applies for each of the other variables. So in total we have n! divided by k_i! for an index i ranging the total number of variables: n choose k_1, ..., k_m = n!/(k_1*...*k_m).
That's actually a really nice approach to the situation and I have read about multinomial coefficient and simplex and it was actually going to be part of the video. Unfortunately last minute I realised the video was getting too long and decided to cut it. I'm glad you found an approach you liked and I will try my best to make a follow up to this video eventually on that topic :D (my upload schedule is beyond horrible lol)
After a lot of thought and cool revelations, I figured out that for any number of terms in (a0+a1…)^n, you can can find the coefficient of the term with a0^p0*a1^p1… where the p’s add to n with the formula n!/(p0!*p1!…), which is apparently general knowledge in the wider math world. The resulting formula is apparently called a “multinomial”.
Oh, I hadn’t read the comments yet. I still think it’s cool, cause I did it on my own.
I'm so glad you derived it yourself since I did the exact same thing while making this video and it's so satisfying. The formula is so intuitive and elegant and I'm so glad you found it interesting too even though you were not the first. Good job!
The values of the 3d pyramid are Pa choose b and c}. Which is the number of ways to choose {b} objects for one group and {c} objects for another group from {a} total objects.
a choose b and c = (a choose (b+c)) * ((b+c) choose c)
Similarly, the values of the 4d hyper-pyramid are {a choose b and c and d}. Which is the number of ways to choose {b} objects for one group and {c} objects for a 2nd group, and {d} objects for a 3rd group, from {a} total objects.
a choose b and c and d = (a choose (b+c+d)) * ((b+c+d) choose (c+d)) * ((c+d) choose d)
The math becomes harder too. I like to find the coefficients using permutations.
So for an N-dimensional Pascal Pyramid: the Coefficient of the term [i, j, k, …] can be found using N! / a product of factorials of differences. I don’t have the formula memorized, but that’s a solid hint.
I have been wondering this since i learned about the pascal's triangle. I am extremely thankful that you made this video and that i found it.
I actually kind of made the pascal triangle before learning it when i was studying probability (im not sure what's the name in english but in my language its something along the lines of "combinatory analysis").
I ended up in the pascal's triangle when i was writting down a grid with the number of posssible solutions for A+B+C+D...=Z ({A;B;C;D;E...Z} being naturals) The grid was number of icognites (NI), with values of Z. any given coordinate was the number of possibilities for that given NI and Z. So for example Z=10 NI=2 would be 66 possibilities, because X+Y=10 has 66 solutions (and its 12! / (2!.10!) not going to explain but i think you can understand how that turns to factorial pretty quickly)
Anyways the point is that grid ended up being almost exactly the pascal's triangle. excpet kind of a pascal's square.
Hey I'm so glad this helped out. It's interesting to see its other applications since I honestly didn't consider them :)
@@viks3864 honestly im not gonna watch the video yet because i wanna find out how it works in 3D by myself first lol. but your videos are really good
@buh3746 Hey thanks I really appreciate that. Good luck!
Possibly, the word you're searching for is "combinatorics." From Wolfram MathWorld:
Combinatorics is the branch of mathematics studying the enumeration, combination, and permutation of sets of elements and the mathematical relations that characterize their properties.
In general, the coefficient of the term
w^k * x^l * y^m * z^n
equals
(k+l+m+n)! / (k! l! m! n!)
which generalizes straightforwardly to any number of variables in a multidimensional "Pascal's simplex".
I stumbled on the trinomial distribution trying to find a space-filling-curve on the RGB cube to get all colors from black (#000000) to white (#FFFFFF) changing at most one value in one color in each transition and also passing through each plane R+G+B = constant at a time. I couldn't find a solution that I liked but the first part of the solution had some connection with trinomials
well done!
Hey thanks, appreciate it :D
Fabulous! “Viks’ Pentacharon” is a great name, too. Now, about that quadnomial distribution…. 😅
very nice, waiting for new videos such that one
Thanks, glad you enjoyed :)
This is great. Can you share the source code for this video with us?
Hey sorry I like to keep my code private although if you need help with learning manim, consider joining the discord or using the online reference manual. Feel free to ask me any questions too
The Unomial Infinigon, start with 1 at () aka (0,0,0... ect.) and add one dimension every step
wow, what a coincidence. I also gave a presentation on independently discovering multinomial expansion ~august of last year! it's almost eerie how similar what we did was (but the animations you made are way way nicer than my lame google slides presentation). I think it's a little disingenuous to say the tetranomial visualization is original though. There were probably hundreds (maybe thousands!) that had studied these sorts of things but they may have been seen as too trivial to publish. For me, this idea was the gift that kept on giving, you can use to to to prove some crazy stuff like n-bonacci identities and euler's polyhedral formula for multiple dimensions. I have no idea what else is in store but I definitely agree that it only occurs by asking that daring "what if?"
(my bad, didn't notice the other comments already bombarded you for the tetranomial stuff)
13:07 that's called a simplex.
edit: nvm i'm wrong simplex is the simplest polytope in n dimensions
so that's a 4d simplex, and the triangle is a 2d simplex
unfortunately, you're not the first. this is a natural thing to think of, so many people have thought of it and just didn't bother giving the 4d case a name, since they usually went on to consider the general multinomial coefficients.
but it's still very nice to see someone interested. there are multiple fun things to do with simplices ("pyramids" generalized to all amounts of dimensions) and polynomials - specifically, polynomials where every term has the same sum of exponents of variables. i think these polynomials are called "homogenous" or something.
for example, simplices give a very nice visual representation of some inequalities, such as the algebraic-geometric inequality. it was useful in olympiads a while ago, but when everyone started using it (or similar things), the problems got a bit more difficult.
anyway, good luck on your journey through math. have fun!
That's actually quite interesting honestly. Also I almost knew for a fact that someone had thought of this first but I did think naming it what I did would be slightly funnier and more memorable although I am more than aware that someone else probably invented and used this way before me and decided not to name it. Thanks though :)
To be fair, Pascal wasn't the first either but his name still got slapped onto the triangle. Maybe we can talk about Vik's simplices from now on?
@@RuthvenMurgatroyd That would be really cool lol but there are thousands of smarter and more worthy of the name.
So yeah.... the obvious conjecture is that you can do this with a d-simplex and d variable polynomials. It should be not too hard to prove that this works always with some linear algebra.
Yep this has been pointed out to me although the point of the video is to consider the style of thinking and the fact that it works. Multinomial coefficient formulas are quite simple and elegant but I really like the way in which this method works.
Ok i havent watched the video, but its 17:29 long and i am utterly amazed
What about the octagonomial?
Don't even joke
What about the Quadnomial distribution?
:(
smart guy, got nothing on dna origami but we can’t all be me
Freddie you cringe man you are doing med. Stay Small.
@@viks3864my cross sectional volume is bigger than urs
You mentioned that it seemed like the fact that you see Pascal’s triangle in each of the edges of the pyramid doesn’t serve much of a purpose? However it seems to me like it’s very fundamental. It’s almost like Pascal’s triangle is a subset of Pascal’s Pyramid. Kinda like how a triangle is one of the faces of a pyramid. So Pascal’s pyramid would be whatever the threedimensional name for a “face” would be to a 4D hyperpyramid.
It was also funny to think this but in theory, Pascal’s… line? Makes sense with the line of thinking I’m trying to show here. The sides of Pascal’s triangle (which I’d consider the 1D equivalent to a 2D face) are just 1, 1, 1, to infinity. And if you were to take the monomial a^n its monomial coefficient would be 1 for all powers of n.
This video was awesome I wish you success with your channel
I completely agree and I thought it was really cool but the video was starting to get really long so I had to cut out some parts but yeah I do see what you mean and I think its really interesting.
Along with multinomial coefficients already mentioned in several comments, I will suggest to you to search and study the 'Restricted Occupancy Theory _ A Generalization of Pascal's Triangle' of J.E. Freund and written in 1956. His recursive formula for n-dimensional Pascal's generalized triangle, gives the method of summing coefficients of previous stages. Good read! :)
Hey thanks for the suggestion. I'll give it a read when I have a chance but it sounds pretty cool.
i love viks
I love you more ant
Go even general: there is a thing called the multinomial distribution
Amazing so informative
Thanks :)
0:00 I lost the game
Don't worry, people don't tend to pick up addition until they are 3 or 4.
I promise I'm actually not mean if you met me in real life.
@@viks3864 The game, where the objective is to not think about the game, if you do so you lose and have to announce it.
Yeah me too
@@Amechaniaa I'm sorry I'm not really following?
@@viks3864 It has nothing to do with the video, I was talking about the mind game called 'The Game' which the video's section name instantly reminded me of thus causing me to lose
oooh nooo TnT
I mean awesome video dude, I tried to solve this problem once, I were trying to figure out a formula for a generic coeficient in a generic dimentional tree...
To start I triend to find the generic coeficient in the pyramid and I already failed there, I thought you would touch on the subject and maybe had a solution for the coeficient or just showing it's impossible to have an algebraic formula for it.
Hi, I'm glad you enjoyed! There actually is a multinomial coefficient and I plan on making a video for the derivation for it soon and I think the solution is quite nice.
@@viks3864 can't wait man! I bet I was just thinking of it the wrong way.
What software do you use to animate?
Other people actually asked this question and I use manim. It's the community version and is something which 3Blue1Brown developed. There are a lot online guides on how to use it and it runs off python. I personally recommend using visual studio code too. Please feel free to contact me if you need help with learning manim :).
Nice video!
Have a look at this:
en.wikipedia.org/wiki/Multinomial_theorem
At some point, my brain doesn't think about the n-D planes, volumes, etc... and simply about the elegant summation condition in the main definition of the multinomial theorem:
Under the summation sign:
For m=2: k_1 + k_2 = n
For m=3: k_1 + k_2 + k_3 = n
...
and so on, where k_i are dummy variables.
The solution of all k_i above, forms the geometry you are explaining, given that any k_i variable must be >= 0.
This is really all of the combinations of numbers that sum to N -> which is the power that you're expanding for.
I really do think about it this way, it helps move away from the geometry which is harder to visualize, but of course, equally valid.
I actually did end up researching the multionomial theorem when making this but ultimately didn't include it in the final video as it got quite complex in an already long video. Thank you though for the explanation as I'm sure others may find this useful!
@@viks3864 Right on :)
Keep up the research, nothing is more satisfying than finding these nuggets of truth
@SSJProgramming By the way, if you link comments in RUclips comments, it automoderates them and doesn't let it appear for any other users until I manually approve it. I don't really mind but I thought I'd just let you know in case you help anyone else.
@@viks3864
Had no idea, thanks!!
So cool
Glad you enjoyed :). Feel free to ask if you have any questions.
0:00 not to be a hater, but mathematicians have a real knack for coming up with the least fun "games" imaginable
3:08 x⁰y⁰
I'll leave you in your despair and wish you a good day anyways my dear sir
Should be easily expandable to n-nomials.
9:38 2z^2😂
Neat! At the end you could have shown all of the cross sections for each given volume, that could have been visualized using your earlier code!
Neat concepts though, great work
That would have actually been a lot better lol. It would have been a nice change in hindsight - thanks for the feedback
You absolutely leng
I couldn't agree more jay
*6:25, tetrahedron.
There are tetrahedral numbers and pyramidal ones too. They are different. I’m sure you are already ware of that! Hope it helps.
@@MrConverse Yep animating this took two weeks, voicing it over took a day so there are some moments where I make some random mistakes but thanks for mentioning it and honestly good job for noticing since most people are asleep by that point lol.
@@viks3864I’m not even sure you should label it a mistake. A tetrahedron is three-sided pyramid, yes? So you are not wrong, although most of us think of a four-sided pyramid when we hear the word pyramid.
I think it’s more of a precision error. ;-)
3:16 Error: you put x^0 where it should've been y^0.
Oh yeah good spot
The generalizations do exist, as “Pascal’s n-simplex”, relating to the multi nominal theorem. Still, great video!
cool
Thanks :)
Hello, I have solved arbitrary polynomials raised to arbitrary power in general, I'm not sure how I can share the formula to you but if you interested and have an idea how I can send it to you, we can make it work
My email address is in the description and the about page of my channel. Feel free to send me any ideas there.
@viks3864 sure, see you soon!
Lengiof the video of 1729
But it is!
I agree, the title is sarcastic. Obviously the world is flat and the moon landing was fake.
Audio too soft!
Yeah, I noticed after I uploaded it but I'll sort it out for the next one. Thanks for the feedback tho
No quadnomial distribution?? Unsubbed.
Sorry bork, I know my videos continue not to intellectually challenge you and I sincerely apologise. The next video will hopefully be good enough for you.
@@viks3864I think he was joking about unsubscribing.
@@sebas31415 We can't joke about unsubscribing. Its too serious.
simplex ?
Hey, I am actually unsure of how the simplex works completely. I believe it is a part of graph theory and is actually in the maths course I'm learning although we dont utilise it this year. Some other people actually mentioned simplex to me and how this method is apparently quite similar to it, although it is just a coincidence.
There is a decent chance everything I just said is waffle.
@@viks3864simplex just means "like triangle, tetrahedron, pentachoron, etc." without specifying a dimension. These come up all over math, for instance in "simplicial complexes" which are basically shapes built out of triangles and tetrahedron, etc.
Tartaglia triangle
I actually didn't know there was a different name for pascal's triangle, that's actually pretty cool.
@@viks3864 in Italy, land of Tartaglia, we use his name, he lived 70yrs before Pascal. Have you ever heard of him?
I just read about him, seems like he did quite a lot and its a little unfortunate that the more popular name for the triangle is named after pascal but its interesting nonetheless.
lol even tartaglia wasn't even close to the first person to come up with the triangle. generally, most people consider al karaji of persia (~1029 ad) to have been the first to come up with binomial theorem (because he was able to generalize it) but there are records of ancient indian mathematicians that studied the combinatorics of the coefficients ~200 BC. I wouldn't be surprised if it were discovered earlier either. in general, I think that we should do away with mathematicians naming things after themselves and keep pascal's original name: the arithmetic triangle.
What is this, volume for ants?
yes so
*pentachoron
And also... *quadrinomial
Daddys first
Good job sujal I'm proud of you
@@viks3864you forgot timestamps i think
@@sninja332Don't worry it was just updating. Thanks though.
you sound like wilbur soot
Bri'ish person innit mate
314th like
Nice one. I'm surprised no one claimed 272nd like.
"promosm"
Wow I'm so famous even bots comment on my videos. What an honour
This has been a wonderful visualization of the Multinomial Theorem! I was impressed and enjoyed your geometric reasoning. In fact, it's reminiscent of topics in Discrete Differential Geometry in your use of Simplicial Complexes.
en.wikipedia.org/wiki/Multinomial_theorem