The Iron Man hyperspace formula really works (hypercube visualising, Euler's n-D polyhedron formula)

[@Mathologer]

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 :)

[@namantenguriya]

Love❤❤❤ u sir.Stay safe. Pls make video on Collatz Conjecture.

[@mathyland4632]

Do we get a hint of what the next video is about? :)

[@Θρησκόληπτος]

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.

[@WillToWinvlog]

Covid tyranny is all bad though!

[@Θρησκόληπτος]

@@WillToWinvlog Are you violating the rules? You are just prolonging it, you doof!

[@Raging.Geekazoid]

"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?" 😄

[@Mathologer]

:)

[@ikocheratcr]

At least she did not say something more complex.

[@Kastagaar]

Ok, honey. 1.0.

[@livedandletdie]

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.

[@tirocska]

@@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,

[@Robert-jy9jm]

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!

[@Ni999]

This!

[@speeshers]

I couldn't agree more, all I feel is amazement watching this

[@hemartej]

What are you sending to him that is so amazing, wonderful, curious, and entertaining?

[@eeveeofalltrades4780]

Meanwhile school: a + b

[@axiomfiremind8431]

As he hails satan with 6's.

[@TheShadowOfMars]

German mathematician: "Here's another kitten, in a cube. Very cute. Feeling revived?" Quantum mechanics students: "NO ERWIN, PLEASE, NOT AGAIN!"

[@heavennoes]

QM students : "Iron Man, please don't, we know your Erwin in disguise."

[@sitter2207]

Kitten killing lessons were my favorite at math classes actually

[@rickacton7540]

@@sitter2207 ZAP THEM lol

[@francisgrizzlysmit4715]

love cats so a kitten is always good

[@PC_Simo]

@@francisgrizzlysmit4715 Same here. 😻

[@somebodyhere3160]

Ah yes, my favorite mathematicians, iron man and towel man!

[@Mathologer]

:)

[@Robert_McGarry_Poems]

Don't forget to bring a towel...

[@mumiemonstret]

Every mathematician should be as well prepared for galaxy hitchhiking as Euler was.

[@cykkm]

@@Robert_McGarry_Poems 42

[@MrCheeze]

they have a fight triangle wins

[@michamiskiewicz4036]

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).

[@Mathologer]

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.

[@michamiskiewicz4036]

@@Mathologer Thanks for the answer! I didn't think about it, but indeed, the exposition as it is already helps in bridging the gap.

[@bravehen-watch2980]

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.

[@channelsixtysix066]

Another video of Mathologising beauty. The 4D cube rotating in space was a delight.

[@ffggddss]

Yes. And it sort of hints at the 2-nested-tori nature of the hypersphere. Fred

[@fixit4387]

@ss It was only the shadow and not the real one. ^^

[@nzeches]

Plus it’s only a 2D projection of a 3D shadow of a 4d object ! 😄

[@mr_vazovski]

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.

[@TheCloudyoshi]

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*

[@PC_Simo]

”But first, we need to talk about walls, floors and ceilings, for 12 hours.”

[@edskodevries]

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!

[@constexprDuck]

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!

[@andyiswonderful]

I think it was a hyper-kitten, but then all kittens are hyper. Puppies, too.

[@zlodevil426]

Finally a math topic I’ve never heard about! Thank you Mathologer, you’re great

[@DeclanMBrennan]

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"

[@WarmongerGandhi]

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).

[@Ensivion]

@@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.

[@張洪鈞]

Thank you, Mr. Mathologer. You explain the geometric meaning of mathematical formula precisely. We are happy to see more.

[@kokomanation]

this is the most beautiful video I have ever seen and felt

[@docjohnson2717]

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

[@ElTRDG]

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.

[@Icenri]

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.

[@michaeldakos1982]

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

[@speeshers]

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)

[@cutieowl6707]

Great video! Lots of other KZbinrs 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!

[@WhattheHectogon]

My favorite channel strikes again! I've been going through the Mathologer backlog, waiting patiently

[@Piffsnow]

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]

Wow! Which degree did you get at high school?

[@Piffsnow]

@@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.

[@HienNguyenHMN]

The animation at the end is a thing of beauty. It lets me intuitively understand what it means. Thank you.

[@kruksog]

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.

[@ericmckenny6748]

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.

[@bumbo222]

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!

[@kinshuksinghania4289]

10:12 just out of curiosity, how do we differentiate between higher dimension convex and concave polyhedra??

[@Mathologer]

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 :)

[@jessehammer123]

@@Mathologer Well, assuming being on the boundary counts as being “inside” the shape. :)

[@notabotta3901]

@@Mathologer what an elegant definition! So simple, yet bulletproof.

[@helgefan8994]

Although technically, Euler's polyhedron formula also works perfectly for non-convex (concave) polyhedra, as long as they don't have any holes.

[@EebstertheGreat]

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).

[@ThePerfectKiosk]

"I'd like to finish off the video" he says roughly half way through the video...

[@zipohi1191]

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.

[@Mathologer]

Just in case you are interested today's music is Floating Branch by Muted.

[@kleinesfilmroellchen]

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.

[@flytoheights1]

That was amazing. Wow! My mind is blown! The feelings & emotions I am experiencing is indescribable.

[@EntropicalNature]

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 KZbin sure lack proofs and just summarize/explain statements. Hats off to you dear Sir! Hopefully you'll keep on educating us all!

[@quantumgaming9180]

Sounds like you're a cool teacher

[@CesarMaglione]

Es un placer ver, escuchar y entender! Muy bien logrado Mathloger! 👍 Esperamos el próximo. 😀

[@leslie5617]

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 KZbin channel quite often. Thanks a lot for showing me how beautiful that math can be.

[@lobsterfork]

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

[@PapaFlammy69]

[@peacewasneveranoption7645]

papa flammy pog

[@Mathologer]

Immer einer der ersten :)

[@godfreypigott]

Erinnere mich daran - wie war dein ursprünglicher Kanalname?

[@PapaFlammy69]

@@Mathologer Na aber natürlich :)

[@PapaFlammy69]

@@godfreypigott :^)

[@Astromath]

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!

[@ren200758]

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.

[@Benoit-Pierre]

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 !!!

[@matchedimpedance]

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.

[@przemekmajewski1]

Haven't even watched yet, but when YT showed me a brand NEW Mathologer vid, I immediately smiled.

[@publiconions6313]

Damn straight! : ) ... I was at my kids' competition, so couldn't watch immediately.... but a new Mathologer vid is the perfect cherry on top

[@lunalma]

I would love to see a video on the Road Coloring Problem! (Great work with this one, by the way)

[@Mathologer]

Never heard of that one. Very interesting concept. Also just had a look at the proof. Doable :)

[@GianlucaDiFrancisca]

Great work as always. I hope you will show us the astonishing beauty of math for years

[@mjkluck]

I love this guy! Keep 'em coming!

[@basilharrison3071]

Love the higher dimensional and geometry based videos!! Very inspiring and helpful!

[@Iamjenishchamling]

I was just searching about it and suddenly your video came in notification .what a coincidence

[@sofia.eris.bauhaus]

woah, it's not often that i upvote a 30 minute video in the first minute, but that cube thing is just too cool!

[@bigredracingteam9642]

That animation of the rotating 3D and 4D cubes was very illuminating. Thank you for doing these videos.

[@godfreypigott]

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.

[@Mathologer]

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 :)

[@avoirdupois1]

This is astounding, the tying together of something so prosaic as (x+2)^3 to a deep understanding of multidimensional cubes. Plus kittens.

[@nmmm2000]

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.

[@markjosephbugarin5270]

Im always fascinated by your discussion of proofs!

[@ChristopherMcIntosh1]

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!

[@MrBluEDicE]

The music in this video is great, and also the video is great.

[@Mathologer]

Today's music is Floating Branch by Muted

[@orlovskyconsulting]

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.

[@christiansmakingmusic777]

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.

[@jacemandt]

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.

[@nathanmcintosh6004]

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.

[@atrus3823]

There is a similar growing process for tetrahedrons: at each step add a single vertex that is the current side distance from all current vertices, then draw edges between all existing vertices to the new vertex. Octahedrons are a little trickier: basically, draw 2 points orthogonal to the current shape the current side distance from all current points, then edges to them. However, there are some hiccups: 1) the meaning of orthogonal for 0-d, and n-d for n > 3 is tricky, and 2) you sort of need to remove the previous shape each step.

[@PC_Simo]

Well; you can just taper the current shapes, in both directions; with cross-polytopes (”octahedra”). Also; ”Orthogonal” means extending along different axes; also, for x = 3, and even x > 3. For example, you might have a cube, parallel to the xyz-hyperplane, and another cube, parallel to the xyw-hyperplane; and they would be orthogonal to each other.

[@nrpbrown]

Saw this video with the non marvel thumbnail a week ago and did a double take now, i love it!

[@crancowan8020]

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.

[@prometheus7387]

OMG, I legitimately never saw that connection coming!

[@Pajafilm]

WOW, very inspiring. Easy to understand. TOP animations. Thank you!

[@jimmy685]

"How satisfying was that?" .... Very! That was such a perfect full-circle moment!

[@JAK4L]

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

[@benjaminreynolds5733]

You guys should make a video on Euler characteristic of higher dimensions and then relay those topological spaces into algebraic structures like modules or rings.

[@redapplefour6223]

Z/n rings are mentioned in kzbin.info/www/bejne/jmeWfoqQfrNmna8 if you haven't seen it yet

[@accountname1047]

Beautiful stuff Mathologer!

[@outputcoupler7819]

You know, Algebra Autopilot would be a great name for an app aimed at teaching algebra. Demonstrate all the algebra basics with these animations, maybe some short clips from our favorite geometer explaining the concept...

[@Mathologer]

Absolutely :)

[@emilejetzer7657]

The `manim` and `sympy` python packages might make such an autopilot a bit easier to build!

[@jonpopelka]

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!

[@rajuhirani9353]

Taking verisatium TM advice about multiple icons and how clickbait helps in getting your message across

[@SoleaGalilei]

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. :)

[@Mathologer]

Looks to me like this simple humanities person is watching a lot of maths videos and is slowly also developing into a maths person :)

[@WildEngineering]

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!

[@cashewABCD]

No commercials - You are my hero.

[@glynnec2008]

The video was interesting as usual. And then you conjured Euler's formula out of thin air! Wow!!

[@hewhomustnotbenamed5912]

2 minutes in I remembered that I actually independently came to a formula for calculating the number of n-cubes in a m-cube. I then ran the binomial formula through my head and found that this trick works for cubes in all dimensions. Cool.

[@franklinjuarez100]

The most beautiful math video I have seen in the web ! Thank you ! Thank you ! Thank you ! 😀

[@morkovija]

Woot woot, tidying up my list of things to watch before the year is done!

[@MrFlaviojosefus]

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.

[@qodeshgraphics]

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.

[@trtlphnx]

Been Watching You Forever, as a Mathematician; You Are My Favorite One On The "Tube" ~

[@Mathologer]

That's great :)

[@FlapMeister]

The video was on point! You've not lost your edge. Let's face it, the video was excellent.

[@kiwanoish]

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.

[@galactusthehungry2681]

Amazing video, resparked my interest in higher dimensions and got me researching again!

[@nathanwestfall6950]

What a nice surprise! I've been hoping you would release another video soon!

[@pw1169]

One of the most beautiful videos i have ever seen

[@profdc9501]

This is a great introduction to generating functions.

[@Mathologer]

I recently did another video featuring generating functions. If you have not seen it yet : kzbin.info/www/bejne/jH3FloN9d7SJm8k

[@profdc9501]

@@Mathologer After your video I looked up generating functions and watched that video also. As a side note, even though high dimensional hypercubes may or may not be "real," they have real use in communications theory. For example, see Hamming distance. Furthermore, to optimize the probability of communicating a sequence of symbols without error, one performs sphere packing in high dimensional spaces to separate encoded symbols so that error is minimally likely to confuse two symbols. This takes advantage of the fact that in high dimensional space, most of the "volume" is near the "surface" of a polytope. See "asymptotic equipartition property." Anyways, your videos are entertaining, insightful, and fun as always!

[@tsawy6]

God these videos are still great. You're still the best mathematician on youtube, in my opinion!

[@sonine5156]

Incredible as Always!

[@Bestape]

Sequencing dimensions with a natural count of Pi divisions at the primary coordinate intersection, as the secondary coordinate intersection drifts away, is a more neutral camera perspective. It also points out when two vertices intersect.

[@drgatsis]

What are the odds? I was giving a lecture today on geometry to some architecture students which included a discussion about Euler's formula. Thanks for the great content. I told my class that they are to watch your video for homework/fun! Still by far my favourite channel on youtube.

[@pranavtiwari_yt]

ur vids are better than any netflix web series

[@yinq5384]

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

[@Mathologer]

:)

[@DeclanMBrennan]

The way of growing squares from lines, cubes from squares and so forth is quite neat. If get the centroid (average of all the vertices each time and then move off along a new dimension until you have reached a unit distance from the current vertices, you get the sequence: line, triangle, tetrahedron, 4D simplex. If you move off in opposite directions from the the centroid, you get the sequence: line, diamond (square standing on its corner), octahedron, 16 cell and so forth. So there are three infinite squences of polytopes. However the dodecahedron and its dual are a special case that only work in 3 and 4D and there's one extra special case (for regular polytopes) in 4D. I haven't fully got my head around this but it might make a good subject for the Mathologerization.

[@Mathologer]

Yes, much more to be mathologerised here :)

[@Xubono]

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! 😎

[@GaneshKumar-vh6ts]

Videos are uploaded only 2 to 3 months once but the contents is really awesome

[@mihaiga]

At 6:16, if you substitute -2 for x, on the first line you get 0^0 = 1 :)

[@SimonBuchanNz]

That's the nice thing about 0⁰, you can get it to equal anything you like!

[@Mathologer]

Finally we know :)

[@sid6645]

That spinning tesseract (?) at the end just broke my mind! God damn.

[@shubhendubanerjee2930]

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)