You'll be pleased to find out (I hope) that the next video is already halfway finished. We are in lockdown again here in Melbourne and as a consequence I've got a bit more time to spend on Mathologer. COVID is not all bad :) Update: I just decided to run an experiment. Went with a descriptive title and thumbnail for a day and a half and now switched to a more clickbaity title and thumbnail. Will be interesting what happens (if anything :)
I still strongly support the Victorian Government and Health team. The other day I heard someone calling them "tyrants", but I think he has no clue. Real tyrants like Bolsonaro let their people die, simply because they consider them inferior to themselves.
You have a gift for showing us what math is really about. It's pure amazement, wonder, curiosity and entertainment. Thank you for capturing the essence of math!
"And mathematicians wonder why people think they're weird." My mother was a singer, actress, secretary, homemaker, and social butterfly. NOT a math person (that was my dad). One day, I was trying to explain a math problem to her and I needed to pick a small number to use in a simple example. So I said "How about 1?", and she starts laughing. "Why 1? It's so small! Why don't you pick a REAL number?" 😄
I'd tell her that in a sense the only Real number is 1 as all other numbers are derived solely from 1. Even the transcendentals are derived from 1, by some arbitrary methods. 2 is nothing but the successor of 1, and 0 is hence nothing but the precursor to 1. And the operations we've acquired from that is + and - as 1 + 1 = 2 and 1 - 1= 0. To derive × and ÷ one has to do more steps, but one can derive them as well, and you can do this to fit an arbitrary amount of operations.
@@livedandletdie Wasn't the basis for numbers the empty set, which we denoted as 0. And then 1 is the successor of 0, etc... It has been a few years ago so I might remember wrong,
The video ended smoothly and the resding our minds at the boring part with a bitter sweet picture of a cat made the end a refreshing end for the video with that music making it happy and giving us a refeshed experience.
15:52 "There's no hidden trickery" I'm not complaining, but I'd say that counting faces is quite tricky, as it relies on topology. To see when we're removing a face and when we're not, is visually obvious and yet non-trivial. And also one needs to be careful not to disconnect the network (as you said, "starting from the outside" should guarantee this).
Very good point. In fact, when you have a close look at what I do in the proof, you may come to the conclusion that the first post-network step of adding diagonals is not necessary at all for the proof to work. Just prune away and you eventually arrive at a single polygon to which V=E applies, and so the V-E+F=F=2 (the inside and the outside of this polygon). But the reason why we are inserting the diagonals is to get more control over what is happening in the proof. For example, it's easier to argue that we can always prune so that the network does not split in two if we are dealing with a network composed of triangles, rather than a completely general one.
Amazing stuff as usual!! Thanks Mathologer :) Especially the spinning projections in 29:06 completely blew my brain up. I think it's because we're so used to interpreting overlapping lines on a 2D plane (on the page) as faux-3D objects, that interpreting them as just what they are (lines) when they move around basically short-circuits my brain.. once again, well done Burkardt :D
Great video! Lots of other RUclipsrs might have stopped after giving one explanation but you really went the extra mile with the animations and multiple proofs. Thank you for sharing this with us!
Love your stuff and how much fun you seem to have presenting it....I get lost pretty easily because I'm old but enjoy the journey....anxious to see what you have up your towel next
My God that was mindblowing, this channel has me obsessed!!! Everytime I watch a Mathologder video, I can't wait to explain it to everyone I know (though they aren't math nerds like me :D)
The entire video I was wondering what the 2 in the (x+2) formula was really referring to. Sooo satisfying to see the recurrence equation at 25:00, makes so much sense!
Me: *takes out ring, proposes* GF: *says yes, crying* Me: *starts talking about the number of vertices on the diamond of the ring* GF: *takes off ring*
Really loved this video thanks. Pascal's triangle is the gift that just keeps on giving. Although strictly speaking the rule here is: " *Twice* the number above left plus the number above right"
If instead of n-cubes, you look at n-tetrahedra, the number of vertices, edges, faces, etc. exactly match Pascal's triangle (with the last 1 in each row removed).
@@WarmongerGandhi is that (x+1)^n ? i think so, hey look i see how the binomial expansion correlates to the pascals triangle and NOW even the higher dimensional triangles.
I was really worried about the kitten trapped inside the hypercube, but then beard-man appeared and used his Shadow-Squish Super Power and saved the day! What a great story!
A shape is convex if, given any two of its points the line segment connecting the two points is fully contained in the shape. This definition of convex works in all dimensions :)
You do need a notion of "inside" of the shape, which is uncontroversial but does rely on some other theorems. Every simple closed (hyper)surface embedded in R^n partitions the space into three connected components: the surface itself, a bounded component called the interior, and an unbounded component called the exterior. This is a consequence of the Jordan-Brouwer separation theorem. So then we can say that a polytope is convex if it is simple and every line segment connecting two endpoints in its interior lies entirely in the interior (i.e. every point in the line segment is in the interior of the polytope).
I studied maths for six years after my high school degree and, still, I learned so much in this video! Thank you Mathologer for all the wanders you bring us. :)
@@ainsworth501 I got the degree called Baccalauréat which is what you get the year you turn 18 in France. I don't know what the equivalent is in other countries.
Tristan's proof is exactly multiplying by x+2. Wonderful. I wonder if there's a link between these generating functions and the genus of the figure they define.
I thought of a 3d version of your 3d polyhedron formula proof before it was mentioned. I'm slightly proud that it's an actual proof and not just something I initially thought *could* work.
Wow!! I derived this amazing formula years ago but it never occurred it could be obtained by a generating polynomial. :O Thank you as always to the Mathologer team! Fun facts: the dimension, m of the most numerous bits of an n-cube converges to n/3 as n increases. This can be shown by setting the derivative of 2^(n-m)*(n choose m) wrt to m equal to zero, while using the derivative of f(m)^g(m) and Sterling’s approximation. For a large n, we get a bell shaped curve presumably due to the DeMoivre-Laplace Theorem.
This video has made me understand the visual used to describe a tesseract even if it isn't what a 4th dimensional object would truly look like. Thank you!
Thank you. Your work is reaching into the future. My students LOVE watching your videos even if they just grasp the very edges of what you're talking about. They continue to think about your videos long after they have watched them. Thank you to you and all your team. Please keep up these great works!
I enjoyed your presentation. I like to get my hands on models. Here is a suggestion for another way to gain insight into 4-D figures. The "shadow pictures" that you presented, such as at time 26:30, are like "perspective" art because sizes are distorted as well as angles. I have found some fun in building "isometric" 4-D shadows where all edges are of equal length. For example, for the hyper cube, connect soda straws together with string. Use four colors of straws, eight straws of each color. Build one cube using three colors, one color for each axis. The cube is floppy, of course. Now build a second cube with the same colors in the same directions and with one vertex "inside" the first cube. Finally, use the fourth color of straws to join corresponding vertices of the two cubes. The resulting figure can be manipulated to show many different views of the hypercube. As long as you keep all the straws approximately parallel, every arrangement is a good representation. All the components of the hypercube are present, but the angles can not all be ninety degrees because this is a projection.
What a delight to watch your video's! Being a math teacher myself, I cannot help but notice the similarities in how we teach. Especially the animated (sometimes hand wavy ;) ) proofs are sublime. Most educational math videos on RUclips sure lack proofs and just summarize/explain statements. Hats off to you dear Sir! Hopefully you'll keep on educating us all!
I'm one of your student in the class of the nature and beauty of mathematics at monash university, i really love your teaching style and i review your videos from RUclips channel quite often. Thanks a lot for showing me how beautiful that math can be.
2:43 you enlightened me. This is probably the first time I can truly visualise an hypercube. It has as faces ... 8 cubes in 8 parallel 3d universes, and they are crossing 2 by 2 on 3d cubes, and 3 by 3 on lines ...This formula is soo visual !!!
Oh my god, just yesterday I was wondering exactly about that: how many "elements" (vertices, edges, ...) do n-dimensional cubes have! And now you made a video on it!
Fantastic! I have had so much fun over the last couple years telling my students, "maths is broken". I often show your videos to help create that bigger picture feeling about math.
Lieber Professor Polster, ich habe noch nicht das Video bis zum Hälfte gesehen, aber schon hat das Video ein LIKE von mir verdient. Herzlichen Glückwünschen.
I reckon the Iron Man title is more enticing than the +/- title. I had not cheched the video before, seemed such a dense, intimidating subject. Now it's like discovering an easter egg of the MCU, seems worth enduring the hard maths somehow.
Very cool (as usual). It would be interesting to see this concept transforming from the discrete to the continuous by comparing/contrasting hypercubes with hyperspheres.
If mathematics is one side of the video, the music is the other. Thank you for both astonishing mathematic topic and making me discover so great music.
Another great video! Thank you. BTW that rotating hypercube at the end is a torus whose surface rotates around the poloidal axis while remaining fixed on the toroidal axis. An interesting video would show rotations about both axes simultaneously.
I just love this channel and the way things are shown, and I also really like the shirts, this one from Space Invaders is really cool, especially because I'm from the oldies and I love this game!!! congratulations for this beautiful educational channel!!!
Thank you for this, this answered and sufficiently explained questions about extra dimensions I didn’t quite know to ask yet but had visualized in my head all this time. Pretty beautiful
9:25 In a dodecahedron (literally: ”12 faces”), you’d have 20 vertices, 30 edges, and 12 faces. Your V=12, E=30, F=20 -list corresponds to icosahedron, the dual of the dodecahedron; just flip the numbers for vertices and faces, and there you go. Also; setting x = 0, in the (x+2)^n -formula, doesn’t wipe *_EVERYTHING_* out: The left-hand-sides become (0+2)^n = 2^n; while the right-hand-sides wipe out; making the equations false. Besides that, however, great video 👍🏻.
I was the one who suggested this in the comment section of the last video - but I am still impressed by the number of connections you've made that I'd never thought of.
Yes, glad you made that comment :) If had a couple of very nice bits and pieces fall into place that had been waiting for just the right moment to come together :)
Always excited for a new Mathologer video! I especially enjoyed the final animation (not counting the closing credits!) of the spinning 4d cube. It helped to conceptualise the otherwise imperceptible nature of the hypercube. Hope you and family are well and flourishing in the current lockdown. Otherwise come back to South Australia where the total # of covid cases have jiggled between 2 and 4, over the past few weeks! 😎
I remember going through a full and rigorous proof of the euler characteristic formula in graph theory, and all I recall was it being quite a doozy! Enjoyed your mathologerized version very much.
thank you. i actually wondered the same question many years ago and ended up using similar techniques to figure out how many m-dim 'objects' in a n-dim hypercube. and then i proceeded onto simplexes as well as cross-polytopes. re-inventing the wheels, i know. but the feeling of figuring out all of those things by myself is still one of my most precious moments.
It would have been nice to have had this resource when I was a child. My mind could have handled it. But now is all gibberish. Thanks for trying to make this simple and accessible for people.
Thank you for uploading such beautiful videos on mathematics sir, it really helps to understand the beauty of studing such a fascinating subject which is considered dull otherwise.(by many)
at the end when you started rotating the shapes, the projected 3D shape looks 3d in the shadow because we are viewing it on a 2d screen, that really made the 4D projection click for me. Great work!
I'm pleased to say I actually have heard of some of the things in this video before! I'm just a simple humanities person so I get excited when not everything in a maths video is completely new to me. :)
So the "2" in "x+2" came from the fact that an edge/vertex is defined by two points and a line/edge between them. Is there a mathematical entity where you would use "x+3" or any arbitrary "x+N"?
Since there is no connected graph with one edge and more than two vertices, x+3 would correspond to two vertices connected by an edge, and a third unconnected vertex. If you "square" that, you get one square, two unconnected lines, and one unconnected point. It probably makes more sense if you think of it as ((x+2)+1), where ((x+2)+1)^2 = (x+2)^2 + 2*(x+2) + 1
The tetrahedron family is almost x+1. To make this family we start with one vertex in dimension 0. To go from dimension n to n+1 we add a new vertex in the higher dimension and connect that vertex to all existing vertices. This creates a new m+1 dimensional object for each m dimension object as well as having one new vertex and all the existing objects. So P_{n+1}(x) = 1 + P_n(x) + xP_n(x). This has the solution P_n(x) = ((x+1)^(n+1) - 1) / x. That is, we expand x+1, remove the final 1 and shift everything down a dimension. 0 dimensional = 1 1 dimensional = 1 + 2 2 dimensional = 1 + 3 + 3 3 dimensional = 1 + 4 + 6 + 4 etc.
@@jay_13875 So an x+1 would correspond to 1 point connected with an edge, like, a curve? If the edge has to be straight, would this work in non euclidean spaces?
@@FunkyDexter You do not need need any particular spatial embedding. Euler formula works in topological structures, not strictly geometric polytopes. And in topology, everything is rubbery and infinitely elastic. For example, for a sphere the Euler characteristic is still 2, as it is for a cube, although it has neither faces not edges. You simply pump air into a cube till it blows up into a sphere; QED. In other words, it this extended sense, it's not a property of a polytope in R^3, but rather property of any planar graph living in a 2-sphere. Remember that then Burkard projected a cube on a flat sheet, the top face corresponded to the infinite "everything outside" the planar graph. But if you project (strictly, declare equivalence of) all points on the sheet of paper to be a single point, the "specialness" of this projection disappears. You can deform the sheet so it becomes a sphere (in topology everything is infinitely "elastic"; only tears and creases are prohibited), this top face's projection is no longer "special", as there is no longer "outside" of the graph, the outside is also encircled by it, as any "inside" projection is. The problem gains more welcome symmetry tho. You can try to draw the graph with a marker on an inflated balloon to see how really more symmetric it looks. Then take another balloon, and draw approximately equally spaced vertices on it (imagine a cube inside the balloon to get some precision), and connect them with 12 edges on the surface of the balloon. See that the two graphs are actually same, only their embedding ("layout") is slightly different. You can drag all vertices to the new positions, and temporary bend edges so that the graph stays planar (you may "straighten" them later, remember, everything is elastic). What you'll get is a shadow of a cube sitting inside the balloon and a light source in its dead center. Much more symmetric embedding than on a sheet of paper! Note that the graph is still planar, as the surface of the balloon is still topologically flat. But for a 2-torus, the characteristic is 0: this weakly corresponds to the (much stronger) condition that only convex polytopes qualify for the Euler characteristic of 2. This is a reason why the famous "3 homes must be connected each to to 3 utilities" problem cannot be solved on a plane (or surface of a sphere), but can on the surface of a 2-torus: The Euler characteristic _is that of a graph, not a polytope!_ Someone (3blue1brown?) ordered mugs with the same homes and utilities on it, and gave it to other RUclipsrs. The solution exists, but necessarily involves drawing some utility lines over the handle. The handle is a key: a glass without a handle is still topologically same as sphere, or same as a sheet of paper with "everything else" belonging to one face, so no drawing through the inside of the glass will help, it would be still impossible. “A topologist does not know the difference between a coffee mug and a doughnut” is a very true joke! :)
A wonderful video. Thank you for sharing it. The only additional bit that I would request is a few lines that might give some intuition about why (X+2) is the the magic formula for this. (Why not (X+1) or (2X+2) etc.)
Just wanted to say this is one of the most interesting and entertaining channel I'm following. Each video brings out my curiosity and a smile on my face. Thank you! And as for this specific video I happen to have a copy of the book "Euler's gem" on my nightstand, a real spoiler 😆
Just finished watching your quadratic reciprocity video......what a treat. I am a little amateur in mathematics..... would look forward to your video on permutations as you had mentioned there ...it would definitely help in appreciating the complete beauty of the proof (not sure if it's published already)...also sorry for posting unrelated topics to this video.... just wanted to post on an active thread.
Ah, nerts. Still over my head. Videos like this make me WANT to learn more advanced math, though! Thank you for sharing your insights, and thank you even more for sharing your unmistakable and infectious love of the subject! We’re so very fortunate to have people like you, OC Tutor, 3blue1brown, etc who create amazing content which inspires curiosity and imparts knowledge (for free, at that) to anyone who seeks it. Imagine what Euler, Gauss, or Newton could have done with so powerful a means of communication!
These geometry videos are always so enjoyable :) Even when the subject is something I thought I understood reasonably well in uni, the videos always help to make it more intuitive instead of just being an algebraic truth. I really hope one day I'll get to see mathologerised versions of Lie groups/algebra. Like it's one thing to sit down and work out that SU(2) and SO(3) algebras are isomorphic but there *ought* to be a way that that's visual right? Given how visual SU(2) and SO(3) are...
What's really nice here, that maybe wasn't stressed very hard in the video, is how the geometrical process of adding a dimension (points become segments, segments become squares, squares become cells) is modeled perfectly by the algebraic process of multiplying by x (thus increasing the exponent). This is a great example of how polynomials are their own kind of object, beyond just a functional relationship between numbers. Once we see the algebraic consequence of the geometrical process, it means we can manipulate algebra symbols and expect that to tell us about geometry. The fact that mathematicians do this is not obvious and deserves to pointed out explicitly.
As soon as you pointed out that the number of vertices, edges, faces, etc. of an n-cube sum to 3^n, I had a total "aha moment", where I thought, "of course they do, just as a Rubik's cube has 1 mini-cube for each vertex, edge, face, and (hypothetically) an interior mini-cube for the 1 cell to make 27". And then I paused the video and worked out a visual proof sketch (of the (x+2)^n coefficients counting the elements of an n-cube). I haven't kept records, but maybe once in 4 videos or so, something early will give me an "aha, I see where he's going" moment and that's always fun. And usually my aha moment pans out later in the video. But this time you never went in my direction, so I'll present my proof concept (EDIT: Well, Tristan's proof is the same concept, but it treats the vertices and edges directly as algebraic objects instead of using n-volumes like my concept below. I thought my idea was missed because of the Euler tangent.) If you have a 1+x+1 line segment partitioned into segments of length 1, x, and 1, you can raise it the nth power and get an n-cube with n-volume (1+x+1)^n that is partitioned into 3^n n-cuboids. Each vertex is adjacent to exactly one n-cuboid with n-volume 1^n*x^0=1. Each edge is adjacent to two vertex-cuboids, but also exactly one n-cuboid with n-volume 1^(n-1)*x^1=x. Each face is adjacent to vertex and edge n-cuboids, but also exactly one additional n-cuboid with n-volume x^2. And so forth. (And finally there is one with n-volume x^n in the center.) It is clearer to start by illustrating for n=1,2,3: imgur.com/a/VIajEwu (1+x+1)^2 creates a square with 9 pieces, 4 of area 1 at each vertex, 4 of area x along each edge, and 1 of area x^2 in the center. (1+x+1)^3 creates a cube with 27 pieces, 8 of volume 1 at each vertex, 12 of volume x along each edge, 6 of volume x^2 in the middle of each face, and 1 of volume x^3 in the center. Now, I'm not an expert at turning visual proof sketches into proofs, but I think it's very pretty.
Well, that was exiting. Thank you for providing new videos. This one reminded me at another aspect of Eulers formula: Graph duality demonstrated e.g. by "Euler's Formula and Graph Duality" (3Blue1Brown). We'd love you to talk more (as you indicated) about 'meta-cubes'...
Great video as always! Inspired by the coordinate-proof from the video, here's a proof of "an n-dim cube consists of 3^n bits and pieces": Consider any bit/piece, its vertices form a subset W of the set V = {vertices of the n-dim cube}. Now focus on the m-th (1
You'll be pleased to find out (I hope) that the next video is already halfway finished. We are in lockdown again here in Melbourne and as a consequence I've got a bit more time to spend on Mathologer. COVID is not all bad :)
Update: I just decided to run an experiment. Went with a descriptive title and thumbnail for a day and a half and now switched to a more clickbaity title and thumbnail. Will be interesting what happens (if anything :)
Love❤❤❤ u sir.Stay safe.
Pls make video on Collatz Conjecture.
Do we get a hint of what the next video is about? :)
I still strongly support the Victorian Government and Health team. The other day I heard someone calling them "tyrants", but I think he has no clue. Real tyrants like Bolsonaro let their people die, simply because they consider them inferior to themselves.
Covid tyranny is all bad though!
@@WillToWinvlog Are you violating the rules? You are just prolonging it, you doof!
German mathematician: "Here's another kitten, in a cube. Very cute. Feeling revived?"
Quantum mechanics students: "NO ERWIN, PLEASE, NOT AGAIN!"
QM students : "Iron Man, please don't, we know your Erwin in disguise."
Kitten killing lessons were my favorite at math classes actually
@@sitter2207 ZAP THEM lol
love cats so a kitten is always good
@@francisgrizzlysmit4715 Same here. 😻
You have a gift for showing us what math is really about. It's pure amazement, wonder, curiosity and entertainment. Thank you for capturing the essence of math!
This!
I couldn't agree more, all I feel is amazement watching this
What are you sending to him that is so amazing, wonderful, curious, and entertaining?
Meanwhile school: a + b
As he hails satan with 6's.
Ah yes, my favorite mathematicians, iron man and towel man!
:)
Don't forget to bring a towel...
Every mathematician should be as well prepared for galaxy hitchhiking as Euler was.
@@Robert_McGarry_Poems 42
they have a fight
triangle wins
"And mathematicians wonder why people think they're weird."
My mother was a singer, actress, secretary, homemaker, and social butterfly. NOT a math person (that was my dad).
One day, I was trying to explain a math problem to her and I needed to pick a small number to use in a simple example. So I said "How about 1?", and she starts laughing. "Why 1? It's so small! Why don't you pick a REAL number?" 😄
:)
At least she did not say something more complex.
Ok, honey. 1.0.
I'd tell her that in a sense the only Real number is 1 as all other numbers are derived solely from 1. Even the transcendentals are derived from 1, by some arbitrary methods.
2 is nothing but the successor of 1, and 0 is hence nothing but the precursor to 1. And the operations we've acquired from that is + and - as 1 + 1 = 2 and 1 - 1= 0.
To derive × and ÷ one has to do more steps, but one can derive them as well, and you can do this to fit an arbitrary amount of operations.
@@livedandletdie Wasn't the basis for numbers the empty set, which we denoted as 0. And then 1 is the successor of 0, etc...
It has been a few years ago so I might remember wrong,
The video ended smoothly and the resding our minds at the boring part with a bitter sweet picture of a cat made the end a refreshing end for the video with that music making it happy and giving us a refeshed experience.
15:52 "There's no hidden trickery"
I'm not complaining, but I'd say that counting faces is quite tricky, as it relies on topology. To see when we're removing a face and when we're not, is visually obvious and yet non-trivial. And also one needs to be careful not to disconnect the network (as you said, "starting from the outside" should guarantee this).
Very good point. In fact, when you have a close look at what I do in the proof, you may come to the conclusion that the first post-network step of adding diagonals is not necessary at all for the proof to work. Just prune away and you eventually arrive at a single polygon to which V=E applies, and so the V-E+F=F=2 (the inside and the outside of this polygon). But the reason why we are inserting the diagonals is to get more control over what is happening in the proof. For example, it's easier to argue that we can always prune so that the network does not split in two if we are dealing with a network composed of triangles, rather than a completely general one.
@@Mathologer Thanks for the answer! I didn't think about it, but indeed, the exposition as it is already helps in bridging the gap.
this is the most beautiful video I have ever seen and felt
Another video of Mathologising beauty. The 4D cube rotating in space was a delight.
Yes. And it sort of hints at the 2-nested-tori nature of the hypersphere.
Fred
@ss It was only the shadow and not the real one. ^^
Plus it’s only a 2D projection of a 3D shadow of a 4d object ! 😄
Amazing stuff as usual!! Thanks Mathologer :)
Especially the spinning projections in 29:06 completely blew my brain up.
I think it's because we're so used to interpreting overlapping lines on a 2D plane (on the page) as faux-3D objects, that interpreting them as just what they are (lines) when they move around basically short-circuits my brain.. once again, well done Burkardt :D
It’s amazing how algebra and geometry can be connected by such a pretty formula.
And the derivation using recurrence is simple and… simply stunning.
Great video! Lots of other RUclipsrs might have stopped after giving one explanation but you really went the extra mile with the animations and multiple proofs. Thank you for sharing this with us!
Love your stuff and how much fun you seem to have presenting it....I get lost pretty easily because I'm old but enjoy the journey....anxious to see what you have up your towel next
My God that was mindblowing, this channel has me obsessed!!! Everytime I watch a Mathologder video, I can't wait to explain it to everyone I know (though they aren't math nerds like me :D)
The entire video I was wondering what the 2 in the (x+2) formula was really referring to. Sooo satisfying to see the recurrence equation at 25:00, makes so much sense!
Thank you, Mr. Mathologer. You explain the geometric meaning of mathematical formula precisely. We are happy to see more.
Me: *takes out ring, proposes*
GF: *says yes, crying*
Me: *starts talking about the number of vertices on the diamond of the ring*
GF: *takes off ring*
”But first, we need to talk about walls, floors and ceilings, for 12 hours.”
Finally a math topic I’ve never heard about! Thank you Mathologer, you’re great
Really loved this video thanks. Pascal's triangle is the gift that just keeps on giving.
Although strictly speaking the rule here is: " *Twice* the number above left plus the number above right"
If instead of n-cubes, you look at n-tetrahedra, the number of vertices, edges, faces, etc. exactly match Pascal's triangle (with the last 1 in each row removed).
@@WarmongerGandhi is that (x+1)^n ? i think so, hey look i see how the binomial expansion correlates to the pascals triangle and NOW even the higher dimensional triangles.
My favorite channel strikes again! I've been going through the Mathologer backlog, waiting patiently
The animation at the end is a thing of beauty. It lets me intuitively understand what it means. Thank you.
I was really worried about the kitten trapped inside the hypercube, but then beard-man appeared and used his Shadow-Squish Super Power and saved the day! What a great story!
I think it was a hyper-kitten, but then all kittens are hyper. Puppies, too.
I love your approach to math. You take such complicated topics and make them so intuitive and easy to understand conceptually. I love you :D
10:12 just out of curiosity, how do we differentiate between higher dimension convex and concave polyhedra??
A shape is convex if, given any two of its points the line segment connecting the two points is fully contained in the shape. This definition of convex works in all dimensions :)
@@Mathologer Well, assuming being on the boundary counts as being “inside” the shape. :)
@@Mathologer what an elegant definition! So simple, yet bulletproof.
Although technically, Euler's polyhedron formula also works perfectly for non-convex (concave) polyhedra, as long as they don't have any holes.
You do need a notion of "inside" of the shape, which is uncontroversial but does rely on some other theorems. Every simple closed (hyper)surface embedded in R^n partitions the space into three connected components: the surface itself, a bounded component called the interior, and an unbounded component called the exterior. This is a consequence of the Jordan-Brouwer separation theorem. So then we can say that a polytope is convex if it is simple and every line segment connecting two endpoints in its interior lies entirely in the interior (i.e. every point in the line segment is in the interior of the polytope).
I studied maths for six years after my high school degree and, still, I learned so much in this video!
Thank you Mathologer for all the wanders you bring us. :)
Wow! Which degree did you get at high school?
@@ainsworth501 I got the degree called Baccalauréat which is what you get the year you turn 18 in France. I don't know what the equivalent is in other countries.
Es un placer ver, escuchar y entender! Muy bien logrado Mathloger! 👍 Esperamos el próximo. 😀
woah, it's not often that i upvote a 30 minute video in the first minute, but that cube thing is just too cool!
Tristan's proof is exactly multiplying by x+2. Wonderful.
I wonder if there's a link between these generating functions and the genus of the figure they define.
I thought of a 3d version of your 3d polyhedron formula proof before it was mentioned. I'm slightly proud that it's an actual proof and not just something I initially thought *could* work.
Wow!! I derived this amazing formula years ago but it never occurred it could be obtained by a generating polynomial. :O Thank you as always to the Mathologer team!
Fun facts: the dimension, m of the most numerous bits of an n-cube converges to n/3 as n increases. This can be shown by setting the derivative of 2^(n-m)*(n choose m) wrt to m equal to zero, while using the derivative of f(m)^g(m) and Sterling’s approximation.
For a large n, we get a bell shaped curve presumably due to the DeMoivre-Laplace Theorem.
That was amazing. Wow! My mind is blown! The feelings & emotions I am experiencing is indescribable.
This video has made me understand the visual used to describe a tesseract even if it isn't what a 4th dimensional object would truly look like. Thank you!
Thank you. Your work is reaching into the future. My students LOVE watching your videos even if they just grasp the very edges of what you're talking about. They continue to think about your videos long after they have watched them. Thank you to you and all your team. Please keep up these great works!
Love the higher dimensional and geometry based videos!! Very inspiring and helpful!
I enjoyed your presentation. I like to get my hands on models. Here is a suggestion for another way to gain insight into 4-D figures.
The "shadow pictures" that you presented, such as at time 26:30, are like "perspective" art because sizes are distorted as well as angles. I have found some fun in building "isometric" 4-D shadows where all edges are of equal length. For example, for the hyper cube, connect soda straws together with string. Use four colors of straws, eight straws of each color. Build one cube using three colors, one color for each axis. The cube is floppy, of course. Now build a second cube with the same colors in the same directions and with one vertex "inside" the first cube. Finally, use the fourth color of straws to join corresponding vertices of the two cubes.
The resulting figure can be manipulated to show many different views of the hypercube. As long as you keep all the straws approximately parallel, every arrangement is a good representation. All the components of the hypercube are present, but the angles can not all be ninety degrees because this is a projection.
What a delight to watch your video's! Being a math teacher myself, I cannot help but notice the similarities in how we teach. Especially the animated (sometimes hand wavy ;) ) proofs are sublime. Most educational math videos on RUclips sure lack proofs and just summarize/explain statements. Hats off to you dear Sir! Hopefully you'll keep on educating us all!
Sounds like you're a cool teacher
I'm one of your student in the class of the nature and beauty of mathematics at monash university, i really love your teaching style and i review your videos from RUclips channel quite often. Thanks a lot for showing me how beautiful that math can be.
Great work as always. I hope you will show us the astonishing beauty of math for years
2:43 you enlightened me. This is probably the first time I can truly visualise an hypercube. It has as faces ... 8 cubes in 8 parallel 3d universes, and they are crossing 2 by 2 on 3d cubes, and 3 by 3 on lines ...This formula is soo visual !!!
WOW, very inspiring. Easy to understand. TOP animations. Thank you!
Oh my god, just yesterday I was wondering exactly about that: how many "elements" (vertices, edges, ...) do n-dimensional cubes have! And now you made a video on it!
Fantastic! I have had so much fun over the last couple years telling my students, "maths is broken". I often show your videos to help create that bigger picture feeling about math.
That animation of the rotating 3D and 4D cubes was very illuminating. Thank you for doing these videos.
God these videos are still great. You're still the best mathematician on youtube, in my opinion!
Im always fascinated by your discussion of proofs!
Lieber Professor Polster, ich habe noch nicht das Video bis zum Hälfte gesehen, aber schon hat das Video ein LIKE von mir verdient. Herzlichen Glückwünschen.
I love this guy! Keep 'em coming!
I reckon the Iron Man title is more enticing than the +/- title.
I had not cheched the video before, seemed such a dense, intimidating subject.
Now it's like discovering an easter egg of the MCU, seems worth enduring the hard maths somehow.
Very cool (as usual). It would be interesting to see this concept transforming from the discrete to the continuous by comparing/contrasting hypercubes with hyperspheres.
If mathematics is one side of the video, the music is the other. Thank you for both astonishing mathematic topic and making me discover so great music.
Just in case you are interested today's music is Floating Branch by Muted.
Another great video! Thank you.
BTW that rotating hypercube at the end is a torus whose surface rotates around the poloidal axis while remaining fixed on the toroidal axis. An interesting video would show rotations about both axes simultaneously.
I just love this channel and the way things are shown, and I also really like the shirts, this one from Space Invaders is really cool, especially because I'm from the oldies and I love this game!!! congratulations for this beautiful educational channel!!!
I would love to see a video on the Road Coloring Problem! (Great work with this one, by the way)
Never heard of that one. Very interesting concept. Also just had a look at the proof. Doable :)
Thank you for this, this answered and sufficiently explained questions about extra dimensions I didn’t quite know to ask yet but had visualized in my head all this time.
Pretty beautiful
papa flammy pog
Immer einer der ersten :)
Erinnere mich daran - wie war dein ursprünglicher Kanalname?
@@Mathologer Na aber natürlich :)
@@godfreypigott :^)
The most beautiful math video I have seen in the web ! Thank you ! Thank you ! Thank you ! 😀
The video was interesting as usual. And then you conjured Euler's formula out of thin air! Wow!!
Haven't even watched yet, but when YT showed me a brand NEW Mathologer vid, I immediately smiled.
Damn straight! : ) ... I was at my kids' competition, so couldn't watch immediately.... but a new Mathologer vid is the perfect cherry on top
9:25 In a dodecahedron (literally: ”12 faces”), you’d have 20 vertices, 30 edges, and 12 faces. Your V=12, E=30, F=20 -list corresponds to icosahedron, the dual of the dodecahedron; just flip the numbers for vertices and faces, and there you go. Also; setting x = 0, in the (x+2)^n -formula, doesn’t wipe *_EVERYTHING_* out: The left-hand-sides become (0+2)^n = 2^n; while the right-hand-sides wipe out; making the equations false. Besides that, however, great video 👍🏻.
I was the one who suggested this in the comment section of the last video - but I am still impressed by the number of connections you've made that I'd never thought of.
Yes, glad you made that comment :) If had a couple of very nice bits and pieces fall into place that had been waiting for just the right moment to come together :)
This is astounding, the tying together of something so prosaic as (x+2)^3 to a deep understanding of multidimensional cubes. Plus kittens.
What a nice surprise! I've been hoping you would release another video soon!
Beautiful stuff Mathologer!
No commercials - You are my hero.
One of the most beautiful videos i have ever seen
The music in this video is great, and also the video is great.
Today's music is Floating Branch by Muted
The video was on point! You've not lost your edge. Let's face it, the video was excellent.
Woot woot, tidying up my list of things to watch before the year is done!
Saw this video with the non marvel thumbnail a week ago and did a double take now, i love it!
Always excited for a new Mathologer video!
I especially enjoyed the final animation (not counting the closing credits!) of the spinning 4d cube. It helped to conceptualise the otherwise imperceptible nature of the hypercube.
Hope you and family are well and flourishing in the current lockdown. Otherwise come back to South Australia where the total # of covid cases have jiggled between 2 and 4, over the past few weeks! 😎
I remember going through a full and rigorous proof of the euler characteristic formula in graph theory, and all I recall was it being quite a doozy! Enjoyed your mathologerized version very much.
thank you. i actually wondered the same question many years ago and ended up using similar techniques to figure out how many m-dim 'objects' in a n-dim hypercube.
and then i proceeded onto simplexes as well as cross-polytopes.
re-inventing the wheels, i know. but the feeling of figuring out all of those things by myself is still one of my most precious moments.
ur vids are better than any netflix web series
Been Watching You Forever, as a Mathematician; You Are My Favorite One On The "Tube" ~
That's great :)
It would have been nice to have had this resource when I was a child. My mind could have handled it. But now is all gibberish.
Thanks for trying to make this simple and accessible for people.
Thank you for uploading such beautiful videos on mathematics sir, it really helps to understand the beauty of studing such a fascinating subject which is considered dull otherwise.(by many)
"I'd like to finish off the video" he says roughly half way through the video...
at the end when you started rotating the shapes, the projected 3D shape looks 3d in the shadow because we are viewing it on a 2d screen, that really made the 4D projection click for me. Great work!
I'm pleased to say I actually have heard of some of the things in this video before! I'm just a simple humanities person so I get excited when not everything in a maths video is completely new to me. :)
Looks to me like this simple humanities person is watching a lot of maths videos and is slowly also developing into a maths person :)
Very nice as always!
Amazing class!!!!!!!! Unforgetable! (and the final music is chilling:)
So the "2" in "x+2" came from the fact that an edge/vertex is defined by two points and a line/edge between them. Is there a mathematical entity where you would use "x+3" or any arbitrary "x+N"?
Since there is no connected graph with one edge and more than two vertices, x+3 would correspond to two vertices connected by an edge, and a third unconnected vertex. If you "square" that, you get one square, two unconnected lines, and one unconnected point.
It probably makes more sense if you think of it as ((x+2)+1), where ((x+2)+1)^2 = (x+2)^2 + 2*(x+2) + 1
Yes! They're the "generalized hypercubes", existing in complex space. en.wikipedia.org/wiki/Hypercube#Generalized_hypercubes
The tetrahedron family is almost x+1.
To make this family we start with one vertex in dimension 0. To go from dimension n to n+1 we add a new vertex in the higher dimension and connect that vertex to all existing vertices. This creates a new m+1 dimensional object for each m dimension object as well as having one new vertex and all the existing objects.
So P_{n+1}(x) = 1 + P_n(x) + xP_n(x).
This has the solution P_n(x) = ((x+1)^(n+1) - 1) / x. That is, we expand x+1, remove the final 1 and shift everything down a dimension.
0 dimensional = 1
1 dimensional = 1 + 2
2 dimensional = 1 + 3 + 3
3 dimensional = 1 + 4 + 6 + 4
etc.
@@jay_13875 So an x+1 would correspond to 1 point connected with an edge, like, a curve? If the edge has to be straight, would this work in non euclidean spaces?
@@FunkyDexter You do not need need any particular spatial embedding. Euler formula works in topological structures, not strictly geometric polytopes. And in topology, everything is rubbery and infinitely elastic. For example, for a sphere the Euler characteristic is still 2, as it is for a cube, although it has neither faces not edges. You simply pump air into a cube till it blows up into a sphere; QED. In other words, it this extended sense, it's not a property of a polytope in R^3, but rather property of any planar graph living in a 2-sphere. Remember that then Burkard projected a cube on a flat sheet, the top face corresponded to the infinite "everything outside" the planar graph. But if you project (strictly, declare equivalence of) all points on the sheet of paper to be a single point, the "specialness" of this projection disappears. You can deform the sheet so it becomes a sphere (in topology everything is infinitely "elastic"; only tears and creases are prohibited), this top face's projection is no longer "special", as there is no longer "outside" of the graph, the outside is also encircled by it, as any "inside" projection is. The problem gains more welcome symmetry tho. You can try to draw the graph with a marker on an inflated balloon to see how really more symmetric it looks. Then take another balloon, and draw approximately equally spaced vertices on it (imagine a cube inside the balloon to get some precision), and connect them with 12 edges on the surface of the balloon. See that the two graphs are actually same, only their embedding ("layout") is slightly different. You can drag all vertices to the new positions, and temporary bend edges so that the graph stays planar (you may "straighten" them later, remember, everything is elastic). What you'll get is a shadow of a cube sitting inside the balloon and a light source in its dead center. Much more symmetric embedding than on a sheet of paper! Note that the graph is still planar, as the surface of the balloon is still topologically flat.
But for a 2-torus, the characteristic is 0: this weakly corresponds to the (much stronger) condition that only convex polytopes qualify for the Euler characteristic of 2. This is a reason why the famous "3 homes must be connected each to to 3 utilities" problem cannot be solved on a plane (or surface of a sphere), but can on the surface of a 2-torus: The Euler characteristic _is that of a graph, not a polytope!_ Someone (3blue1brown?) ordered mugs with the same homes and utilities on it, and gave it to other RUclipsrs. The solution exists, but necessarily involves drawing some utility lines over the handle. The handle is a key: a glass without a handle is still topologically same as sphere, or same as a sheet of paper with "everything else" belonging to one face, so no drawing through the inside of the glass will help, it would be still impossible.
“A topologist does not know the difference between a coffee mug and a doughnut” is a very true joke! :)
Very satisfying and beautiful!
btw, I love binge-watching your videos:)
A wonderful video. Thank you for sharing it. The only additional bit that I would request is a few lines that might give some intuition about why (X+2) is the the magic formula for this. (Why not (X+1) or (2X+2) etc.)
Just wanted to say this is one of the most interesting and entertaining channel I'm following. Each video brings out my curiosity and a smile on my face. Thank you!
And as for this specific video I happen to have a copy of the book "Euler's gem" on my nightstand, a real spoiler 😆
Yes, a great book :)
Incredible as Always!
"How satisfying was that?" .... Very! That was such a perfect full-circle moment!
Just finished watching your quadratic reciprocity video......what a treat. I am a little amateur in mathematics..... would look forward to your video on permutations as you had mentioned there ...it would definitely help in appreciating the complete beauty of the proof (not sure if it's published already)...also sorry for posting unrelated topics to this video.... just wanted to post on an active thread.
Amazing video, resparked my interest in higher dimensions and got me researching again!
Ah, nerts. Still over my head. Videos like this make me WANT to learn more advanced math, though! Thank you for sharing your insights, and thank you even more for sharing your unmistakable and infectious love of the subject! We’re so very fortunate to have people like you, OC Tutor, 3blue1brown, etc who create amazing content which inspires curiosity and imparts knowledge (for free, at that) to anyone who seeks it. Imagine what Euler, Gauss, or Newton could have done with so powerful a means of communication!
Favorite youtube channel.
These geometry videos are always so enjoyable :) Even when the subject is something I thought I understood reasonably well in uni, the videos always help to make it more intuitive instead of just being an algebraic truth. I really hope one day I'll get to see mathologerised versions of Lie groups/algebra. Like it's one thing to sit down and work out that SU(2) and SO(3) algebras are isomorphic but there *ought* to be a way that that's visual right? Given how visual SU(2) and SO(3) are...
Lie algebras and groups are some of favourite playthings. Eventually there may be a video...
What's really nice here, that maybe wasn't stressed very hard in the video, is how the geometrical process of adding a dimension (points become segments, segments become squares, squares become cells) is modeled perfectly by the algebraic process of multiplying by x (thus increasing the exponent). This is a great example of how polynomials are their own kind of object, beyond just a functional relationship between numbers. Once we see the algebraic consequence of the geometrical process, it means we can manipulate algebra symbols and expect that to tell us about geometry. The fact that mathematicians do this is not obvious and deserves to pointed out explicitly.
I was just searching about it and suddenly your video came in notification .what a coincidence
As soon as you pointed out that the number of vertices, edges, faces, etc. of an n-cube sum to 3^n, I had a total "aha moment", where I thought, "of course they do, just as a Rubik's cube has 1 mini-cube for each vertex, edge, face, and (hypothetically) an interior mini-cube for the 1 cell to make 27".
And then I paused the video and worked out a visual proof sketch (of the (x+2)^n coefficients counting the elements of an n-cube). I haven't kept records, but maybe once in 4 videos or so, something early will give me an "aha, I see where he's going" moment and that's always fun. And usually my aha moment pans out later in the video.
But this time you never went in my direction, so I'll present my proof concept (EDIT: Well, Tristan's proof is the same concept, but it treats the vertices and edges directly as algebraic objects instead of using n-volumes like my concept below. I thought my idea was missed because of the Euler tangent.)
If you have a 1+x+1 line segment partitioned into segments of length 1, x, and 1, you can raise it the nth power and get an n-cube with n-volume (1+x+1)^n that is partitioned into 3^n n-cuboids. Each vertex is adjacent to exactly one n-cuboid with n-volume 1^n*x^0=1. Each edge is adjacent to two vertex-cuboids, but also exactly one n-cuboid with n-volume 1^(n-1)*x^1=x. Each face is adjacent to vertex and edge n-cuboids, but also exactly one additional n-cuboid with n-volume x^2. And so forth. (And finally there is one with n-volume x^n in the center.)
It is clearer to start by illustrating for n=1,2,3:
imgur.com/a/VIajEwu
(1+x+1)^2 creates a square with 9 pieces, 4 of area 1 at each vertex, 4 of area x along each edge, and 1 of area x^2 in the center.
(1+x+1)^3 creates a cube with 27 pieces, 8 of volume 1 at each vertex, 12 of volume x along each edge, 6 of volume x^2 in the middle of each face, and 1 of volume x^3 in the center.
Now, I'm not an expert at turning visual proof sketches into proofs, but I think it's very pretty.
Of course, when you went in a different direction than I expected from my "aha", and produced a generalized Euler's formula from x=-1, that's fun too!
Yep, also very nice. Have to also animate that at some point :)
Well, that was exiting. Thank you for providing new videos. This one reminded me at another aspect of Eulers formula: Graph duality demonstrated e.g. by "Euler's Formula and Graph Duality" (3Blue1Brown). We'd love you to talk more (as you indicated) about 'meta-cubes'...
Glad you liked it. This site has the formulae for the n-d simplex and the n-d orthoplex :) people.math.osu.edu/fiedorowicz.1/math655/HyperEuler.html
BSc and MSc in maths here: You just keep outdoing yourself. Always great and interesting content no matter what level your'e at; keep it up.
Very nice animation at the end
Great video as always!
Inspired by the coordinate-proof from the video, here's a proof of "an n-dim cube consists of 3^n bits and pieces":
Consider any bit/piece, its vertices form a subset W of the set V = {vertices of the n-dim cube}.
Now focus on the m-th (1
:)
Every video is beautiful miracle. Thank you The Kind Mathgician ))