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

[@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. 😻

[@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!

[@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.

[@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,

[@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

[@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.

[@kokomanation]

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

[@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 ! 😄

[@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.

[@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.

[@zlodevil426]

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

[@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!

[@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

[@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!

[@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.”

[@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

[@張洪鈞]

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

[@constexprThoughts]

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.

[@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)

[@WhattheHectogon]

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

[@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.

[@HienNguyenHMN]

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

[@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.

[@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!

[@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

[@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.

[@flytoheights1]

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

[@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!

[@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.

[@CesarMaglione]

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

[@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.

[@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).

[@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 !!!

[@GianlucaDiFrancisca]

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

[@basilharrison3071]

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

[@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.

[@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

[@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 :)

[@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!

[@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 :)

[@gordonstallings2518]

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.

[@bigredracingteam9642]

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

[@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!

[@markjosephbugarin5270]

Im always fascinated by your discussion of proofs!

[@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.

[@franklinjuarez100]

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

[@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

[@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.

[@tsawy6]

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

[@Pajafilm]

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

[@glynnec2008]

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

[@nrpbrown]

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

[@avoirdupois1]

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

[@cashewABCD]

No commercials - You are my hero.

[@mjkluck]

I love this guy! Keep 'em coming!

[@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.

[@accountname1047]

Beautiful stuff Mathologer!

[@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.

[@FlapMeister]

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

[@morkovija]

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

[@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.

[@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.

[@MrBluEDicE]

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

[@Mathologer]

Today's music is Floating Branch by Muted

[@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

[@Nepermath]

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!!!

[@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.

[@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.

[@nathanwestfall6950]

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

[@PC_Simo]

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 👍🏻.

[@ThePerfectKiosk]

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

[@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.

[@pw1169]

One of the most beautiful videos i have ever seen

[@Iamjenishchamling]

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

[@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!

[@jimmy685]

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

[@trtlphnx]

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

[@Mathologer]

That's great :)

[@pranavtiwari_yt]

ur vids are better than any netflix web series

[@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)

[@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! 😎

[@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 :)

[@lucassaito2842]

Very nice as always!

[@jamesderrick1972]

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.)

[@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!

[@mlmimichaellucasmontereyin6765]

Yet again, you provide free support of my proof that Bucky balls (C60), graphene, and regular prolyhedra prove my theorems & metatheorems (of enabling principles, numbers, maths, metamaths, RH, etc.)! Thanks again. And, so, I have yet another link & citation to add to the final drafts.

[@galactusthehungry2681]

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

[@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]

:)

[@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.

[@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 :^)

[@sonine5156]

Incredible as Always!

[@GaneshKumar-vh6ts]

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

[@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.

[@atpugnes]

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.

[@MathEnthusiast314]

Very satisfying and beautiful! btw, I love binge-watching your videos:)

[@jaredshowalter7151]

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.

[@jaredshowalter7151]

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!

[@Mathologer]

Yep, also very nice. Have to also animate that at some point :)

[@rbpompeu1]

Amazing class!!!!!!!! Unforgetable! (and the final music is chilling:)

[@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.

[@ChrisDjangoConcerts]

This video is such a gift ! Are there also formulas for the other Platonic solids ?

[@Mathologer]

Yes there are also formulas for the n-d versions of the tetrahedron and the octahedron. The other two the dodecahedron and the icosahedron only have counterparts in 4d :) Have a look people.math.osu.edu/fiedorowicz.1/math655/HyperEuler.html :)

[@ChrisDjangoConcerts]

@@Mathologer You are a rockstar !!!!

[@sylvainstephant2843]

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 😆

[@Mathologer]

Yes, a great book :)

[@sid6645]

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