Congratulations for being awarded the 2022 Fields Medal!

Really nicely explained (and edited). The modified Pascal's triangle framing is a really fun way to make these topological patterns feel like they pop out of numerical playfulness.

Best handwriting in all of Numberphile. June Huh has remarkable penmanship.

the way he writes the letter "f" is so satisfying

Those F's are fancy as hell

A few years ago I attended a summer school where June Huh was one of the lecturers. It was amazing. He’s the kind of idealistic mathematician who always sees the big picture.

I'm here after June Huh's Fields Medal announcement!!! CONGRATS!!!

"Pick my favourite triangulated sphere in the 17th dimension..." There's just so many, I can never choose just one!

The "unproven" g-conjecture was proven in a paper published in December 2018, just 6 months after this video was posted.

This is the best penmanship I've ever seen on a Numberphile brown paper.

The greatest congratulations to June Huh for having been a recipient of the Medal Field this year

Mathematicians are enormously imaginative.

It’s nice knowing that, as of this filming, June Huh had a bright, bright future. Congratulations on the Fields Metal!

As of 2022, June Huh has been awarded a Fields Medal. Just amazing!

June Huh has a beautifully patient cadence to his presentation style.

Dang, more videos with Dr. Huh. This was one of my favorites. He's obviously passionate about this math, and is very articulate.

This is a very inconvenient time of day for me to watch a 20-minute math video, but I got the notification, so here we are.

Here after June won his Fields Medal. What an amazing mathematician!

who's here again after June Huh has won the 2022 Fields medal?

I watched this video years ago and I never would have thought that he would have won the Fields Medal. Congratulations!!

I love his calm demeanor. What a great guy. Please do more videos with him.

교수님 축하드립니다. 찾다 보니 이 영상까지 보게 되네요. ㅎㅎㅎ

We should rename maths. I suggest calling it "Euler"

4 years later this guy won a Fields medal!! Congratulations Mr Huh!!!

"A 1 dimensional triangle is a straight line" Cool cool

icosahedron? You mean a pentagonal gyroelongated bipiramid?

Great introduction to h-vectors and the g-conjecture by June Huh. You can tell he was careful to provide several examples so it would be accessible to most people.

필즈상 축하드려요!

He just won the Fields Medal!!! 🥇🏅

Fantastic explanation of Euler's Formula. Thinking about it as the alternating sum of the 0, 1, and 2 dimensional faces of a 3d shape really helped me understand it much better than I ever have before.

I love this guy. His explanations are so perfectly clear and direct.

Almost halfway through the video: "And this is our starting point." Oh, ok. This on a Monday. Lol.

His explanation of 4D shapes has helped me understand them better that any of the popular animations that you may see online.

*SUBSTRACT*

"We should start with Euler's formula" Do you have any idea how little that narrows it down?

I liked him

Yeees. Great video. Great mathematician. More from him, please!

There are so many Fields medalists that have been featured on Numberphile, it's quite boggling.

A chunk in this video just helped me understand something I had been struggling with in modern GPU code. Thanks so much for your videos!

This is one of my favorite Numberphile videos. Always telling people that math is actually fun and to check this channel out.

If anyone's interested, Michael from Vsauce did a great video on strictly-convex deltahedrons yesterday. It's a brilliant companion to this one.

This guys is my another numberphile favourite, such an articulated, well explained and inspiring. You can love math because of the way it is presented it to you.

Perfect explanation. Goes inexorably to the point, you have no chances other than nod and agree.

I noticed that with some of the shapes you get parts of Pascal's triangle when you play the subtraction triangle game with them. That's pretty cool.

Dude. Forget everything else. Can we focus on the fact that dude has PERFECT “f”s? That was amazing.

This guy is excellent, I sometimes find these videos hard to follow but his explanation is so clear!

oh my, he's so excited about the thing but so humble about it, I just love him. and the way he writes 8, come on I want more of him, please!

This man is so smart, he deserves the Fields Medal !

Learned about Euler's formula in my math history class this previous semester. Didn't expect to see it used so soon.

Please get this guy on more, he is a wonderful explainer, with great handwriting to boot!

Clearly in my top 5 numberphile videos ever, along with Riemann hypothesis, Glitch Primes and cyclops numbers, All the numbers, and transcendental numbers

June Huh is awesome! I want to see more of him.

I have never been able to understand why the Euler characteristic must flip-flop between 2 and 0. The explanation in this video is very complicated - but all you have to do is include the figure itself to get the same result: pentagon f0(5)-f1(5)+f2(1) = 1, icosahedron f0(12)-f1(30)+f2(20)-f3(1) = 1, 6-orthoplex f0(12)-f1(60)+f2(160)-f3(240)+f4(192)-f5(64)+f6(1) = 1. A pentagon contains 5 vertices, 5 edges, *and 1 pentagon*. An icosahedron is made up of 12 vertices, 30 edges, 20 triangles, *and 1 icosahedron*.

I'm high as faq watching this and it is the most beautiful explanations ever. The thinking behind this is transcendental. I guess.

This is my new favorite channel. I can't get enough!!!

this man is awesome!! he is so passionate and so clear :D

I can't believe this was never mentioned, but I just noticed that there's a way (much easier than the pascal triangle thing) to get to 1 every single time. The pattern is defined as follows: count the amount of objects with dimension "x" inside the solid, and take the alternating sum as x increases to d-1, where "d" is the highest dimension that the solid lives in. All you gotta do to get 1 every time (rather than oscillating between 0 and 2) is increase x to d, not d-1. Here's an example, using a 3D simplex (tetrahedron, d=3): Vertices (x=0): 4 Edges (x=1): 6 Faces (x=2): 4 Solids (x=3): 1, because the tetrahedron contains (and is) a single 3D solid. 4-6+4-1=1. Here's the same example with a 4D simplex (d=4): Vertices: 5 Edges: 10 2D Faces: 10 3D Faces: 5 4D Solids: 1 (again, the entire simplex). 5-10+10-5+1 still equals 1. As you can see, this works with all of these solids in all dimensions, assuming the oscillation between 0 and 2 in the original pattern continues indefinitely. The alternating sum happens to work out such that whenever a 2 is reached the 1 is subtracted, and whenever a 0 is reached the 1 gets added- it always ends at 1. Side note: I totally realize that leaving out the final 1 was kinda needed for the purpose of the pascal triangle bit, I just thought that what I found was super interesting. (btw I typed this entire comment on a crappy phone keyboard) TL;DR What this video forgot to do was factor in the entirety of the solid along with its edges and faces, and if it did that, the pattern would be a clean string of 1s rather than an oscillation between 0 and 2.

Amazing explanation, very clear, articulate and easy to understand.

The absolute best math channel to ever exist.❤

Congrats on his fields medal

This is level of content I like to see!

This guy's handwriting is unbelievable. Watching his hand movements while writing formulas is hypnotizing.

June Huh is actually amazing

Love this guy's handwriting.

He was awarded the Fields Medal 3 days ago!!

We need this guy again. Great mathematical insight, even better calligraphy.

Excellent editing job and production values as usual, thanks!!!

I guess you can also see it in the way that eulers formula is missing the sphere itself and that's where the 1 comes from.

This guy writes his 7's like the katakana ワ/ク, which is a great idea I wish I'd known earlier.

June Huh is so well spoken, brilliant mind!

What an incredibly likable guy.

congratulations!

I noticed that 1 3 3 1 was a line on Pascal’s Triangle (a+b)^3. So is 1 4 6 4 1 (a+b)^4. Then I thought about the 1 9 9 1 one and thought that perhaps it’s because that was the next level up in complication (octahedron -> icosahedron) And the tetrahedron was 1111 and is the simplest, so if the “complexity” was given a number like tetrahedron: n=0 octahedron: n=1 icosahedron: n=2 then the h number would be 1^n 3^n 3^n 1^n. I predict that the next level up in complexity would be 1 27 27 1. The same seems to be true for the 4-dimensional objects except it’s the next level down on Pascal’s Triangle 1^n 4^n 6^n 4^n 1^n. I’m sure the real mathematicians already know about this, though it wasn’t stated in the video.

Really enjoyed hearing from June Huh!

This attracted me because it looked like a network mesh. I think the basis of the g-conjecture may be a generalization of a recurrence relation, which seems to be able to be constructed using a function that depends on recursion to instantiate itself in the lower dimensions.

Somehow I feel like Grasshopper absorbing the lectures of Master Po. Master *HUH* talking about the *H* -factor and *palindromic* sequences.

I love the channel and videos and I have a small remark. The sound effects of the video (the ones used for counting) are too loud compared to the volume of the voice. This problem is apparent on other numberphile videos as well but this is one of the most obvious ones. It is hard to watch the video on the phone.. with love. Cheers

I was just reading about June last week, amazing.

Thanks, your video inspired a breakthrough!!!!! Best feeling ever!!!

It’s been proven today!

Anyone else here cause they saw that it's now been proven and they want to understand?

Use the terms in the f-vector to make a polynomial e.g. 1, 8, 24, 32, 16 -> x^4 + 8x^3 +24x^2+32x + 16. Now substitute x - 1 for x and collect terms. In this case we get x^4 + 4x^3 + 6x^2 + 4x + 1 (coefficients are 1, 4, 6, 4, 1) and in general this transforms the f-vector into the h-vector.

허준이 교수님 5년전 영상인데도 얼마전에 찍은 느낌이네요 ㅋㅋ 신기한 체험입니다.

Once again Euler did find a pattern

I thought it was Terence Tao in the thumbnail.

This was an impressively clear and interesting presentation

His handwriting is better than almost all of the people that appeared in number theory. You can see the poetic power behind it

Powers of 11..... 11^3=1331, 11^4=14641.... it’s hidden in Pascal’s triangle too

Decagon infinity opens the door Decagon infinity opens the door Wait for answer to open the door Decagon infinity - ah!

I like this guy. He has a knack for explaining things

What a fantastic speaker. Very enjoyable video

Did I hear the word... *conjecture* ?

Every single one of the h vectors shown was a row of Pascal's triangle with elements raised to a power. Most cases that power was 1 (and the vector was the same as a row in Pascal's triangle), but in the case of [1,9,9,1] and [1,1,1,1,1], the powers would be 2 ( [1²,3²,3²,1²] ) and 0 ( [1⁰,4⁰,6⁰,4⁰,1⁰] ) respectively. Is there a counterexample to this?

such fun to play with mathematics.. thank you so much for the video. Love Dr. Huh

Back here when I realised you won the Fields Medal! Congratulations

Fields Medal FTW!

That f is FANCYYY!!

euler's formula makes even more sense if you count the trivial faces of a polytope P, namely f_-1 = {} and f_n = P then the alternating sum is always 2 in every dimension

You can feel how much this guy loves maths. Great vid

No longer an open problem!