This is the best math channel on KZfaq.

Mind blown when you put the proof together at the end.

I come back to this video every so often to just appreciate the beauty and motivate myself to work harder.

Wow. In my studies (computer science) we did this proof. It took more than an hour and I was totally confused. Now I understand it after only seven and a half minutes. I guess I'm more of a "visual and brief"-guy and less of a "proof by contradiction using induction and ten different laws"-guy. I wish I could retake all my math courses, learning from a professor like you.

Beautiful. Far-and-away my favorite Math youtube channel!

PLEASE THANK YOU FOR USING CHARACTERS IN THE MOVIE TRADING PLACES IN YOUR EXAMPLES. Such a pleasant surprise to break the monotony of maths.

damn randolph's legs are creepy as hell

That's a neat proof actually. I remember a different one that was quite intuitive that I found in some book, but I don't remember the details anymore. They were treating the graph as some sort of a field, growing rice or something, surrounded with water and the edges were preventing the water from pouring inside and if I remember well the goal was to destroy several of these walls to flood all of these fields and water the rice while staying connected in such way the farmer could still walk along the remaining walls to reach everywhere. So I guess basically they were also making a tree that way. And the water that was filling the sectors as the walls were being taken down was something similar to Mortimer from this video. Traveling in the dual graph is like water pouring into each region. So in the end it was most likely more or less the same proof, just visualized like that, but I can't remember everything exactly now.

Ahh, now that I've finished my math degree youtube is recommending these old but gold 3b1b videos, some that I've even seen before but went way over my head at the time (I bet I will be saying that again eventually), and some that I needed just now as I continue my journey beyond the halls. Thank you, again and again.

You just reminded me why graph theory is so cool and beautiful; thanks for rekindling my interest in one of my favorite subjects.

Wow! This makes WAY more sense than whatever I read on Wikipedia. Suddenly I'm not afraid of graph theory

Beautiful. I have pondered this proof for several times since I learnt it as a student. Even after a course on graph theory, I did not realise it. Even if I was told to use dual graphs, I would not realise it. Thank you for this

Beautiful elegant proof, beautifully and elegantly presented!

Awesome visualization and proof, thank you :) Having come across your video, I remembered another proof, which appeared intuitive to me those days. The idea of it was to modify arbitrary graph to a single vertex by removing vertices and edges so that V-E+F does not change. And such resulting graph would obviously have V-E+F equal to 2.

I'm in eighth grade and I'm preparing for next year (to do AP exams). We learnt this formula, but the teacher couldn't explain it. Me being very theoretical, I did a long search to find an explanation, however, I only found examples. Thank you for clarifying this.

These videos are excellent. Keep up the good work :)!

The elegance and beauty of math at its finest. Thank you, you made my day

Please don't ever stop making these superb videos

This is the best channel, no subcategories, on KZfaq.

Completly mesmerized. Thank you sir!

Im gonna quit school and start watching every single video that you post.

I never saw before the difference between the quality of your old videos to the new ones, they are very good, but its amazing how you progressed...

I am astonished by this proof

I'm one of the people that was totally fine with the original proof of this, as the original seemed to make logical and intuitive sense, but yeah, this one is super elegant. Great proofs have this quality where the connection at the end feels like it just naturally slides out of the steps preceding it, totally friction-less in form. This was definitely one of those. I had absolutely no idea where you were going with the part about dual graphs, and then it all made perfect sense in this one amazing moment. Your videos are great, is the point.

This really helped my research on identifying null-spaces on discrete exterior calculus applied to fluids. Thanks!

Amazing presentation of this excellent proof!

Very nice proof and animation . Explained very nicely.

Man, plz. Never stop making videos.

Undergrad here, trying not to get lost in my Operations Research course. This video and the linear algebra ones are helping me understand the subtleties in the various simplex and network algorithms. Thancc, gotta spread the love

+3Blue1Brown First, thank you, your videos have a quality and elegance that are almost impossible to find on youtube. For instance, I watched your "Crash Course on e^x" a number of times (if only they showed that approach at my uni), and I think "What does it feel to invent math?" is beautifully encouraging. I really hope you continue to make videos like these. I had watched this video before, but today I revisited it and started playing with dual graphs and I have learnt that they are not unique in general, since they depend on the embedding of the graph, which may not be unique. This means that the assertion "original graph Dual of dual graph" is not true in general. For example, the dual of the dual of a tree is easily dependent on the embedding of the dual. I proved that if it is 3-connected (implying a number of nice things among which I used that its dual is unique), then the dual of its dual is indeed the original. Now, some questions come to my mind: Is it always possible to find an embedding of the dual such that the dual of the dual is the original? Is it possible to find a sequence (original, dual(original), dual(dual(original)),...) such that the period is different than 1 or 2? It certainly cannot be different each time since the number of edges is constant and the number of graphs with k edges is finite. And the most general question, What conditions are necessary and sufficient for the dual of the dual graph to be unique and equal to the original? Anyway, rather than correcting one second of your video, I wanted to show that your videos are very inspiring, so please keep it up.

One day, I will have watched and completely understood all your videos. 🤗

Awesome proof! Love it

I can finally remember this formula for the first time in my life.

I came home from spain. And there was a new 3Blue1Brown video. Best day ever.

5:10 Randolph and Mortimer adventures, Mortimer. A hundred years *burp* Randolph and Mortimer!

This video is a literal embodiment of the Pierre Deligne quote used in the "Essence of Linear Algebra" series: "I have learned to not take pride in the difficulty of a proof. Difficulty means we have not understood. The point is to paint a landscape such that the proof is obvious." Sure the inductive proof works, and is beautiful in its own way when considering the algebra of planar graphs. However, it lacks the unifying power of this proof, which uses most of the core concepts in graph theory, much like Euler's identity and group theory.

Thank you for the most BEAUTIFUL video c:

Great presentation and excellent explanation. This is a great example of astute teaching. Also, the material is interesting and it makes great sense. I needed just this to help some of my Olympiad Math students (Momentum Learning Academy near Houston, TX) understand this theorem by "Oiler". =)

Do more Graph Theory proofs please :). love your stuff

OMG !! Who are you. All of this is awesome. Expecting some more videos from you.

Somehow every one of your videos is mind blowing

Awesome ❤, impressive how video making went from this to what we see today, very very very interesting video❤

Its amazing what humans can come up with i feel like a spectator in awe

Great material! Good job!

This spanning tree explanation is good. As for myself, I reasoned Euler's Characteristic Formula by looking at a tetrahedron, and then considering what would happen if an edge got a new vertex inside (it would be accompanied by the original edge splitting into 2 edges, and then there being a new pair of edges within each of the 2 face adjacent to the original edge and those 2 faces splitting into a pair of face), or if a new edge was inscribed in a face (the face would split into a pair of faces) , or if a face got a new vertex inside (it would be accompanied by a new set of edges with the number being the face rank (M), and the original face being split up into a new set of faces with the number also being the face rank), with the net result being the formula.

AMAZING! A non inductive proof of on of the most stunning equations in math

This was so beautiful! =)

favorite math youtube channel :)

now u got it . good job u explained this one much better than the last there . just dont lie next time

I DEFINITELY LOVED THIS...

That was fantastic!

damn that turn in the end....awesome video and awesome presentation :)

1:31 Here we see the first member of the beloved 3b1b Pi creature species, a blue specimin named Randolph, a likely ancestor of the now popular Randy descendant

A good example of how it is and always will be for calculations without wondering why it should be by practical experience.

I believe that I am a little bit dumb since I understood this video after I watched it twice. However, it does not prevent me from commending this video! Awesome! This is much easier to understand than my Math class! Thank you.

Great stuff. Thanks for the video

Thank you for letting me see this after getting very frustrated with the induction proof, which didn't actually say anything about the "nature" of the problem...

Best video on graph theory.

Beautiful.

That's really cool! Thanks a lot

fabulous video! thank you!!

I love Mathematics but I am not very good at it. I adore the way you made this so easy to understand.

This is like magic... Thanks for sharing:-)

Didn't ever think it could be so significant

Great video!

I would really love to see you do a video about the duality of points and lines.

This is great!

Incredible explanation

really like your videos, keep it up! :)

Randolph and Mortimer Good one

Beautiful

Randy has grown up so much since this!

3Blue1Brown this is one of the most brilliant things I ever watched here on KZfaq. Continue to do this! Do you intent to give entire lectures in some branches of math like you did for elementary linear algebra?

Graph duality? More like "Gee, this has got to be" one of the best videos I have ever seen on the topic. Thanks so much for making and sharing it!

this is awesome

Damn that's an elegant proof.

Lovely proof.

Y NO PEOPLE SUBSCRIBE TO U? Seriously, your videos are _amazing_!

Holy shit this is amazing!

A nice explanation that I think everyone can consume it

Astonishing proof

thanks man.. that was really awsm

amazing. thank you.

That is really nice

This video needs a re-upload. It's a beatiful proof, that needs a fair treatment, of being high quality and of being seen by Grant's new and bigger audiance

very fantastic

i love how my computer science background literally dances with mathematics in a waltz. mathematics takes the lead and computer science follows!

Thank you very much.

To check whether a Truss/frame has equilibrium or not there is formula 2j_n=3,where j stands for joints(vertices), n stands for number of members (connecting two vertices ). If the equation is satisfied by the Truss or frame one can solve using Graphic analysis without performing rigorous calculations to fine member forces.

cool proof!

Yup gonna have to watch this at least two more times to understand

Thank you sir... it's more effective

Oh.My.God what a proof!

Interested persons might like to look at Imre Lakatos's Proofs and Refutations which considers Euler's Theorem exceptions, generalizations, etc.

Best "Trading Places" reference ever :p

I had to watch it twice to actually understand it, got a little confused at first with the Dual Graph. It's a pretty cool proof, I have to say.

I have a master degree on maths and thought I was not that bad on maths.Mindblown!

Euler's characteristic equation. Polyhedron shapes. Vertices (dimension) - Egdes (connections)+ faces (including outer region for graph) = 2. OR Dots - lines + region = 2 D6. 8 - 12 + 6 = 2 D4. 4 - 6 + 4 = 2 Graph (no intersections - planar) duality. Cycle - path at ending at starting vertex (path - no back tracking. Spanning tree - touches all vertex without cycles Dual graph - make vertices

godamnit so brilliant

The intuition taught me in High School about lines extending to infinity, was to imagine them as great circles on the (Quantum) sphere of infinity, ie also like parallel lines meet at infinity, so do radials, here-now in actuality. It's one of those answers of the "Teacher knows best" and "this is all you need know for now" type. (But empirical visual knowledge is almost a complete obfuscation of the purely mathematical connection concept, equivalent to "I think therefore I am", the converse) Timespace quantized is Spacetime, but the visual perceptions of spacetime obscure the "Clock" quantization = timing-spacing cause-effect. In conjunction with this explanation of the materialist view, we were also told that the same extension of drawn lines in Geometries and Perspective continued "forever", so by default implication, the Great Circles of on the potential sphere of infinity, encapsulated the universe of all time here-now, a very old idea-experience so deeply familiar that we cannot see the actual forest for the empirically familiar local trees. From this start, hidden under everything else of more defined empirically valuable and practical educational usefulness, the little child's question "why?" requires an analysis of the, "sum of all history", (which is encapsulated in the concept of e-Pi-i eternity-now, and original "Great Circle of Being"), of Classical academic education, back to the learning by doing origins in basic Navigation by the Cosmos. The point being that a really competent Mathematician probably covers more mathematical material more effectively by minimising empirical evidence in favour of recent developments in applications of pure thinking techniques (?). Working backwards through to the "point of departure" conceptually, is about 53yrs elapsed time personally, just to recall my own WHY, I now understand the thousands of years in which Eternity Now was the real and perceived environment, (because only the temporal experience of here-now in some proportion was relevant), and Pure Mathematics was "aberration", (Sorcery?). Fun to imagine.