The Iron Man hyperspace formula really works (hypercube visualising, Euler's n-D polyhedron formula)
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- Опубліковано 21 тра 2024
- On the menu today are some very nice mathematical miracles clustered around the notion of mathematical higher-dimensional spaces, all tied together by the powers of (x+2). Very mysterious :) Some things to look forward to: The counterparts of Euler's polyhedron formula in all dimensions, a great mathematical moment in the movie Iron man 2, making proper sense of hupercubes, higher-dimensional shadow play and a pile of pretty proofs.
00:00 Intro
01:17 Chapter 1: Iron man
06:05 Chapter 2: Towel man
11:16 Cauchy's proof of Euler's polyhedron formula
17:37 Chapter 3: Beard man
22:16 Tristans proof that (x+2)^n works
26:16 Chapter 4: No man
28:52 Shadows of spinning cubes animation
28:42 Thanks
Here is a link to a zip file with the Mathematica notebooks for creating the cube and hypercube shadows that I discuss at the end of the video in chapter 4.
www.qedcat.com/cube_hypercube...
If you don't have Mathematica, you can have a look at pdf versions of the programs that are also part of the zip archive or you can use the free CDF player to open the cdf versions of the notebooks.
Something I forgot to mention: There is also another purely algebraic incarnations of this process of growing the cubes. It comes in the form of a recursion formula that connects the different numbers of bits and pieces in consecutive dimensions. That recursion formula is also present at the bottom of the "iron man page". Have a close look :) Also, in the Marvel movies the cube that Tony Stark is holding in the thumbnail of this video is called the Tesseract. Probably worth pointing out that "tesseract" is another name for a 4-d cube. I also built an easter egg into the thumbnail that plays on this fact:
imgur.com/a/psIy28k
The formulae for n-d tetrahedra and octahedra can be found on this page;
people.math.osu.edu/fiedorowi...
Here is a link to my video on solving the 4d Hyper Rubik's Cube
• Cracking the 4D Rubik'...
Another proof of Moessner for cubes using cubical shells Anthony Harradine and Anita Ponsaing
www.qedcat.com/StrikeMeOut.pdf
Here is a really nice video on the 120-cell that I only mentioned in passing.
• 120-cell
Noteworthy from the comments:
Today's video was "triggered" by a comment made by Godfrey Pigott on the last video on Moessner's miracle in which he pointed out that (x+2)^n captures the vital statistics of the n-dimensional cube.
Z. Michael Gehlke There is an easy way to see this: (x^1 + 2*x^0) describes the parts of a line; all of the cubes are iterated products of lines: n-cube = (1-cube)^n. Therefore, all cubes are described by iterated powers of (x^1 + 2*x^0)^n. (Me: Nice insight. Of course needs some fleshing out to make this work on it's own, like in the comment by ...
HEHEHE I AM A SUPAHSTAR SAGA I came up with an even simpler visual proof. Take a cube of side length x+2. This cube has a volume (x+2)^3. Now, slice the cube six times. Each slicing plane is parallel to a face and 1 unit deeper than the face. Don't throw away any volume. What you're left with is an inner cube of side length x (volume x^3), 6 square pieces of volume x^2, 12 edge pieces of volume x, and 8 corner cubes with volume 1 each. Adding up these volumes gives you the original (x+2)^3 volume, so it's proven. This works in any dimension.
Here is a link to an animation of this idea that I put on Mathologer 2, as a reward to those of you who who are keen enough to actually read these descriptions. • 3rd proof that the coe...
Typo: The numbers of vertices and faces of the dodecahedron got switched.
Today's music is Floating Branch by Muted.
Enjoy!
Burkard
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.
"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,
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!
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
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*
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.
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.
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 ! 😄
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.
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!
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.
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.
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.
My favorite channel strikes again! I've been going through the Mathologer backlog, waiting patiently
"I'd like to finish off the video" he says roughly half way through the video...
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
Great video! Lots of other UA-camrs 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!
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.
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
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)
Es un placer ver, escuchar y entender! Muy bien logrado Mathloger! 👍 Esperamos el próximo. 😀
The animation at the end is a thing of beauty. It lets me intuitively understand what it means. Thank you.
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
Thank you, Mr. Mathologer. You explain the geometric meaning of mathematical formula precisely. We are happy to see more.
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!
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
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!
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.
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 animation of the rotating 3D and 4D cubes was very illuminating. Thank you for doing these videos.
Love the higher dimensional and geometry based videos!! Very inspiring and helpful!
What a nice surprise! I've been hoping you would release another video soon!
Great work as always. I hope you will show us the astonishing beauty of math for years
Im always fascinated by your discussion of proofs!
Very satisfying and beautiful!
btw, I love binge-watching your videos:)
Beautiful stuff Mathologer!
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.
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.
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.
The video was interesting as usual. And then you conjured Euler's formula out of thin air! Wow!!
That was amazing. Wow! My mind is blown! The feelings & emotions I am experiencing is indescribable.
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! 😎
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 !!!
Woot woot, tidying up my list of things to watch before the year is done!
I love this guy! Keep 'em coming!
woah, it's not often that i upvote a 30 minute video in the first minute, but that cube thing is just too cool!
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.
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!!!
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 UA-cam 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
Amazing class!!!!!!!! Unforgetable! (and the final music is chilling:)
"How satisfying was that?" .... Very! That was such a perfect full-circle moment!
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.
Amazing video, resparked my interest in higher dimensions and got me researching again!
Incredible as Always!
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.
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).
this is the most beautiful video I have ever seen and felt
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!
The video was on point! You've not lost your edge. Let's face it, the video was excellent.
This is astounding, the tying together of something so prosaic as (x+2)^3 to a deep understanding of multidimensional cubes. Plus kittens.
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.
The most beautiful math video I have seen in the web ! Thank you ! Thank you ! Thank you ! 😀
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 UA-cam channel quite often. Thanks a lot for showing me how beautiful that math can be.
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)
God these videos are still great. You're still the best mathematician on youtube, in my opinion!
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 :)
Very nice as always!
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.
Every video is beautiful miracle. Thank you The Kind Mathgician ))
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!
No commercials - You are my hero.
One of the most beautiful videos i have ever seen
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
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.
I have no words. Fantastic stuff! Love seeing all the connections.
Ah, I guess I did have some words after all :P
Always LOVE Your stuff..!!!
I really like spinning shadow of 3D cube over 2D plane at the end. Really well made to be seen as "parallel" to 4D animation.
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.
A stunner as always! I’m still waiting for the Abel Ruffini proof to get the mathologer treatment someday :)
Amazing video SIR!!!🙏🙏🙏 ...as always inspirational ...
The music in this video is great, and also the video is great.
Today's music is Floating Branch by Muted
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 :)
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 :)
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 👍🏻.
ur vids are better than any netflix web series
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
papa flammy pog
Immer einer der ersten :)
Erinnere mich daran - wie war dein ursprünglicher Kanalname?
@@Mathologer Na aber natürlich :)
@@godfreypigott :^)
Saw this video with the non marvel thumbnail a week ago and did a double take now, i love it!
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.
The looping animation at the end is impressive, but the mind blowing thing about four dimensions is that we don't even really understand what it truly looks like. We're still limited to using 3D models to represent 4D.
Cool that i had such idea before it was in math. A space where you can actually move by expanding matter around your and at the same time staying with integrity of linear space. Its really beatifull.
Very nice animation at the end
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_sensz 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 UA-camrs. 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! :)
That spinning tesseract (?) at the end just broke my mind! God damn.
Absolutely fantastic!