Euler's Formula V - E + F = 2 | Proof

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Proofs for two theorems used in this video:
► Polygon triangulation: • Every Polygon can be T...
► Area of a spherical triangle: • Spherical Geometry: De...
Euler's polyhedron formula is one of the simplest and beautiful theorems in topology. In this video we first derive the formula for the area of a spherical polygon using two theorems proven in the previous two videos which are linked above. This result is then used to prove the fact that V-E+F = 2 is true for all convex polyhedra by projecting the polyhedron on the surface of the sphere and doing some algebraic manipulation.
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#mathematics #geometry #Euler

[@mathemaniac]

Woah! This proof is really unexpected! I have only seen the proof by induction, but this is actually quite a creative alternative proof.

[@arikwolf3777]

Also why V - E + F = 2 doesn't always work for concave polyhedrons. The three-section arch in the concave example works because it can be morph into a convex one an therefore be mapped to a sphere. But a N-section ring, (a torus,) cannot.

[@Poklaz1]

@@arikwolf3777 maybe works only for starry polyhedra, i.e. which can be mapped on a sphere

[@ThighFish]

How even would this be possible to prove inductively?

[@bostash8442]

"we will use two theorems proven in the previous two videos" *watches all the videos this channel made*

[@geekjokes8458]

*again*

[@JorgetePanete]

4:40 Consider a spherical cow

[@sourav_kundu]

Here is an alternate proof. Imagine the polyhedron as a planet hanging in space. Imagine that there is a hollow in every face, and every vertex is a mountain. Imagine that every hollow is filled with water. Now imagine it starts to rain on the planet, and the water level starts to rise. One by one the water crosses the edges, until the planet is one entire ocean with V islands sticking up. Whenever the water crosses an edge, there are two possibilities. Either: (a) two bodies of water have joined into one (number of lakes decreases by one, number of landmasses stays the same); or (b) a body of water has joined up with itself, encircling a new island (number of lakes stays the same, number of landmasses increases by one). Initially, there are F lakes and 1 landmass. At the end of the flooding, there is 1 lake and V landmasses. Therefore, there must have been (F-1) edge crossings of type (a), and (V - 1) crossings of type (b). Every edge got crossed exactly once. So E = (F-1) + (V-1), or V - E+F= 2. [Credit : Unknown]

[@sterlingveil]

Vote this up people, this comment was as cool as the video itself!!

[@MeesBorg]

Damn, im impressed :) So can there be a body of water crossing a side of the polyhedron? Because if *not* : then I dont get how point 2)" when water level rises new waterbodies are circling an island.. " is made. Cheers!

[@sourav_kundu]

@@MeesBorg yes, the water can cross any edge. The edges have a lower height than the vertices, and the face centers have an even lower height than the edges. So, the water pools originally occupy the face centers and then cross some of the edges and join up the water bodies. Hope I could explain it.

[@imadhamaidi]

how can you save youtube comments? this proof is just amazing

[@VaradMahashabde]

Isn't this the inductive proof? Doesn't matter, the visualization was amazing

[@timh.6872]

Wow. Very satisfying analytic proof as opposed to the technically correct but somewhat cumbersome induction proof for planar graphs. Working directly with the polyhedron as a preexisting whole made of parts instead of constructing it piece by piece feels so much more satisfying. I knew it was true because of the inductive proof, but now I know _why_ it's true.

[@JM-us3fr]

Great stuff man. Such a simple argument. I only knew of the induction proof and the dual graph proof. This was also very elegant

[@samdob8494]

Truly amazing proof. It blew my mind when I saw the last two videos were building up to this one. Keep up the great work!

[@ThinkTwiceLtu]

Thank you for the support!

[@antoniolewis1016]

That was so brilliant it made me cry. Thank you!

[@ThinkTwiceLtu]

Thank you for watching:))

[@rodrigo-vl7bi]

Awesome as always, the fact you can explain math almost without words is mind-blowing

[@Magnasium038]

Amazing. That you utilised the previous two as build-up for this beautiful proof of a graph theory theorem without really using a graph theoretical proof is amazing.

[@ranjitsarkar3126]

Absolutely gorgeous. I usually don't complement people directly for being good at something. But for this one, I just can't control myself

[@swankitydankity297]

I've never been able to understand the proofs of this one so when I saw your notification in my feed I knew that I then would finally understand :)

[@ThinkTwiceLtu]

I am so glad to hear that. Thanks for the support!

[@ClaudioDiBiase16]

This is one of the most amazing proofs i've ever seen, the result just comes out of nowhere, it really seems like a magic trick at first glance, but then you realize it has been in front of your eyes for all the time. Thanks for making this videos, you have not just shown me a marvelous proof, you have made my day!

[@ThinkTwiceLtu]

Thank you for very much. Your comment made my day too:)

[@kat5266]

So... When can we see ur proofs for the millennial problems? I'm totally rooting for you! 😇

[@DonReba]

What a good way to present this creative geometric proof!

[@EastingAndNorthing]

But what would a spherical projected concave polyhedron look like?

[@ThinkTwiceLtu]

Depends on the specific polyhedron. With some concave polyhedra the same reasoning would work and you could show that in fact V-E+F=2 would still hold. But it wouldn't be true for every concave polyhedron as the projection of some concave polyhedra in some cases wouldn't cover the whole surface area of the sphere or in other cases the faces and edges may intersect.

[@EntaroCeraphenine]

The formula itself is magnificent while this proof is marvelous as well! I enjoyed both the Spherical Triangle video and the Triangulation video for their own regards but to think that they both combine to generate another marvelous proof is just beyond amazing!

[@mikikaboom9084]

Very interesting approach.

[@HebaruSan]

The animation for projecting the cube onto the sphere was _gorgeous_ !

[@HebaruSan]

Is it obvious that ANY convex polyhedron can be projected onto a sphere such that all of its faces form spherical polygons (i.e. each edge forms a part of a great circle)? I believe you that it's true, but I could see someone wanting to be convinced that there are no corner-case convex polyhedra where the angles of the edges are wrong somehow.

[@ThinkTwiceLtu]

Thanks:))

[@jpalreis]

Very nice visualization indeed! And you keep improving! I'm definitely gonna show my students this and others videos when the time is right. And what do you use to create these animations?

[@ThinkTwiceLtu]

Thank you for the kind words! I'm happy to hear that. I use Cinema4d.

[@jpalreis]

@@ThinkTwiceLtu Thanks for the tip! I'll try to animate something for my students in the near future. Keep the great work, math and artistic visualizations!

[@nishameena8675]

Awesome and thank you for such a explaination

[@rachele7398]

ty for color coding the theorem and the example this is magic, omg

[@halilibrahimkanpak65]

When he said 4π=2πV-2πE+2πF i felt that

[@kaziaburousan166]

When thik twice makes us think🖤🖤

[@GideonEstellaPH]

true hehe

[@HIHIQY1]

twice

[@genericasianboi]

T H I K

[@notsoclearsky]

ThiCC

[@columbus8myhw]

Think once

[@jeff3608]

How are these videos so satisfying! How do you animate these videos?

[@redaabakhti768]

this channel should pop up in my recommended videos more often

[@hemantsingh5537]

Keep making these types of video...too good!!

[@hoodedR]

Love your channel and ita contents! 😊 Never fails to make me happy

[@ThinkTwiceLtu]

Thank you for the support:) happy to hear that.

[@kshitijsharma2200]

Interesting was looking for it yesterday.

[@SemperMaximus]

Mind-blowing! Beauty of maths, visualized! Been watching your videos for a long time and each time is special. Great job and thank you sir.

[@ThinkTwiceLtu]

Thank you so much!!

[@TheOfficialCzex]

Well-produced, as always.

[@ThinkTwiceLtu]

Thank you:)

[@O_Fisikomunista]

bro?? it's an AMAZING proof, i never saw it before.

[@drewmichael3986]

That was sooo nice I havent seen a proper proof before for this

[@emuman9]

Beautiful.

[@sasmitarath4312]

Monalisa : I m the most beautiful. Think twice : see this

[@ThinkTwiceLtu]

:D

[@sasmitarath4312]

@@ThinkTwiceLtu please make a video on the banac tarski paradox

[@ThinkTwiceLtu]

@@sasmitarath4312 vsauce has a brilliant, in depth video on that topic:)

[@sasmitarath4312]

@@ThinkTwiceLtu ya, but if not this then make avideo on brouwer's fixed point theorem

[@omerresnikoff3565]

@@ThinkTwiceLtu Yeah, I'd love an animation on any set-theoretic topic!

[@iiib2975]

when I first found this channel I felt like I found a treasure I stil feel that way

[@ThinkTwiceLtu]

Happy to hear that! Thanks a lot for the support:)

[@BeesAndSunshine]

So this is a rule that convex polyhedron must abide by, but concave polyhedron can fulfill it too, correct? So how do you determine if a polyhedron is concave or convex if it does fit this rule?

[@sonalichakraborty6830]

Amazing bro...and I still wonder why schools and other institutions don't follow these methods...at least they can follow people like you

[@ryanbell3704]

can someone please explain to me why this same proof doesn’t work for concave polyhedra? I know the whole “projecting onto a sphere” thing is immediate with convexity, but i cannot think of an example of a concave polyhedron that wouldn’t also project similarly onto a sphere

[@vedantverma4797]

Very epic bruh!!!! Epic quality

[@ThinkTwiceLtu]

Thanks:>

[@Eazoon]

Does this work for any star-shaped polyhedron? I feel like convexity was only used when projecting to the sphere, but that doesn't need full convexity.

[@lopkobor6916]

My only question at this point is how do people even figure these proofs out? Just the sheer amount of imagination and creativity that comes into certain proofs is too much for me to handle.

[@DiegoMathemagician]

Very pretty! I spent like 30 minutes watching it so I don't miss any details lol. Is this proof yours? Never seen it before. I did not see the connection between spherical polygons and Euler's Formula until the end, which made it awesome :) I just have one question: how do you know that when the polyhedra are projected into the sphere, the edges become circumference arcs instead of some other curvy segments, i.e. the faces become spherical polygons instead of other closed loops?

[@sayamqazi]

because all of the edges of polyhedron are straight lines and any straight line projected from inside the sphere will become the spherical arc which in turn means all the faces become spherical polygons.

[@andresfernandoaranda5498]

What tool did you use to make the video?

[@chotabacha18]

Its beautiful....i really love watching your videos ,it gives great pleasure and i really appreciate your hard work in creating such wonderful explanatory mathematical videos

[@ThinkTwiceLtu]

Thanks for the support:)

[@barbietripping]

Lovely proof, and awesome animation. What are you using to animate this?

[@ThinkTwiceLtu]

Thanks! I use C4D.

[@shambosaha9727]

Best presentation of Legendre's proof I have ever seen.

[@jlpsinde]

Amazing work! Hug from Portugal.

[@ThinkTwiceLtu]

Thanks! :)))

[@drozfarnyline4940]

OMG! Your channel is outstanding.please upload video every week.Thank you so much!😊

[@ThinkTwiceLtu]

Thanks! It takes a lot of time to produce one video, but I will try to upload more frequently over the summer. See you soon:)

[@matron9936]

Thanks, to you and Euler

[@thatdodude1464]

Pls do collatz conjecture next :)

[@alpharum3141592]

It's a wonderful presentation about graph theory and topology. Also, after watching this video, I planned another problem and discovered a result: If a graph with faces is "isomorphic" to a torus surface (a donut shape), then F+V=E If you prefer, would you proof it geometrically?

[@thomasbaird01]

Thank you, much appreciated.

[@erfanshekarriz4707]

Yooo his KZbin videos are cool and all but have you checked his INSTA though. Asthetics. af. 😩

[@ThinkTwiceLtu]

ฅ^•ﻌ•^ฅ

[@betabeast12]

Juat did. And im like- *damn*

[@louis2271]

This is soooooo cool :D

[@mitranoarthur]

This is really wonderful. Do you have a video of the area of the sphere?

[@ThinkTwiceLtu]

thanks! yes I do. I believe it's called cavalieris principle.

[@mitranoarthur]

@@ThinkTwiceLtu Oh I thought that one was about the volume of the sphere. That one is cool too.

[@user-tk4zh7wk7h]

I just thought that the n in the sum of the edges was a bit confusing, because it gave the idea of being a constant. Besides that, great video

[@usernameisamyth]

Excellent proof

[@benburdick9834]

That's one satisfying proof.

[@soumitrapharikal5503]

5:52 How can you say the angle is 2pie, since in a curved surface the angles get distorted, won't we need to calculate the Gaussian Curvature here, then Apply the Girrard Theorem?

[@hxka]

This is sum of angles around a point, not sum of angles in a triangle.

[@realedna]

OK, curved edges meeting up at a point on a sphere is like on a plane. There is no curvature involved, if there is no distance from the point.

[@themaverick1891]

I'm glad that I subscribed to your channel.

[@ThinkTwiceLtu]

:)

[@imadhamaidi]

does this proof extend to any shape that is topologically equivalent to a convex shape?

[@ThinkTwiceLtu]

Yes

[@nadiyayasmeen3928]

6:01 I don't understand why it's 2E "The sum of number of sides of all polygon is 2E since each edge is shared by 2 faces." But what about overlap. Shouldn't we subtract the overlapping faces or am I getting something wrong here? Someone pls help

[@ThinkTwiceLtu]

If you count each side of every polygon you will double count each side because for every side there are two polygons sharing it.

[@nadiyayasmeen3928]

@@ThinkTwiceLtu Ah yes. Overlooked that. Amazing video btw. I never looked into the derivation because it used graph theory which I knew nothing about. This is an amazing proof

[@AmanKumar-vd1jc]

Beautiful

[@diegorusso2315]

better short video; but this is equal good.

[@ZReChannel]

Wow, it's pretty neat

[@kaisarakimzhan1927]

Wow, it's well shown. Very understandable and pretty too.

[@ThinkTwiceLtu]

Thanks!

[@MadOokami]

Can you make it dark mode friendly next time?

[@jimmykitty]

Awesome 😊💖💖

[@moisesbello-morales343]

Great proof

[@user-ig6cb6hd6j]

Oh my god... It's beautiful ❤️

[@ThinkTwiceLtu]

[@carminesans90]

Beautiful and smart

[@sigmac30]

Incredible, as always

[@ThinkTwiceLtu]

Thanks!!

[@Fortynienq12]

Which software do you use to make videos?

[@ThinkTwiceLtu]

Cinema 4d

[@Fortynienq12]

@@ThinkTwiceLtu thanks a lot brother 😇😇

[@oosmanbeekawoo]

The Math Speaks For Itself - Leon Eulermann

[@easymarks1637]

Love these videos so much but why don't you show equation simplification visually too? You often jump over a lot of steps with factors and like terms, arriving immediately at the simplified version. It's sometimes difficult to see how you got somewhere, especially if you've rearranged the term order.

[@ThinkTwiceLtu]

Thank you! Yes you're right sometimes I might skip over some simplification steps. However the software that I'm using becomes very slow the more text objects I add into the scene so I always try to deal with as little text/equation manipulation as possible. I understand your point though, I'll keep your comment in mind when making future videos.

[@georgefan2977]

Throughout the video I be like: Okay, ooo, wow, whaaaat, WOW, DAMN

[@iagodantasf]

Cinema 4D? Amazing video by the way

[@ThinkTwiceLtu]

Yes and thank you:))

[@ansonngpersonalgoogleaccou5104]

This is inspiring!!!!!!!!

[@ThinkTwiceLtu]

:))

[@sammyskye9498]

I have some gripes with this, for example; Take a cube and translate the vertices randomly. using this proof, it will still be convex even though the definition would say otherwise. Unless this proof is talking strictly about *regular* polyhedra, which wasn't stated anywhere.) then I don't understand how this can be a proof. P.S. Please enlighten me.

[@TechToppers]

The pure joy of induction!

[@oldreddragon1579]

V-E+F=2 except for a Torus which is V-E+F=0. What is it when the Torus has a center of 0? That is if you draw it as as a section view it would look like a pair of circles joined at a point on their circumference. I'm not a Mathematician so bare with me :) Is this V-E+F=1?

[@pool7216]

Pure magic.

[@sasoribi]

Thanks.

[@ThinkTwiceLtu]

thanks for watching:)

[@janherman2073]

When placing the polyhedron inside the unit sphere, you should ensure that the centre of the sphere lies inside the polyhedron. Otherwise the projection won't cover the whole sphere...

[@mediter123]

Can't you assume this possible without loss of generality? Like you're right, but is there a situation where some convex polyhedron can't contain the center?

[@janherman2073]

@@mediter123 No, there is not. You can place the polyhedron such that its arbitrary inner point coincides with the centre of the sphere and scale it down to fit inside the sphere. But my point was that the position of the polyhedron is not arbitrary...

[@carollim5382]

not a fan of mathematics, but your animations make it attractive

[@ThinkTwiceLtu]

:>

[@cosimobaldi03]

You're a fucking genius! Also this makes me wonder if we can prove it by reducing every polyhedra to a polyhedra with triangle faces only and then considering only that particular case...

[@AntoCharles]

Continues to be some of the crispiest math on the interwebs.

[@ThinkTwiceLtu]

:0

[@garrettcredi5561]

Couldn't you also drop the convexity requirement and generalize it to any shape that can be continuously and bijectively morphed into a sphere along some homeomorphism f? Points must go to points so f(Vertex set) must be another vertex set (of the same size since f is a bijection); f(Edge set) must be another edge set since non can be destroyed and if any were created that would imply the image of 2 edges intersect so the original edges must intersect (as f is again bijective); and f(Face set) must be another face set since a face is uniquely bounded by a loop and loops go to loops over a continuous function. Then your proof could proceed as normal?

[@SpencerTwiddy]

Yes you could, that homeomorphism step is the only dependency for concave not working. But I think he stated it this way because saying you proved something for all convex polyhedra is simpler and better for a popular YT presentation than saying you proved something for all shapes “that can be continuously and bijectively morphed into a sphere along some homeomorphism”

[@SpencerTwiddy]

But I agree, that would have at least been a good extension to note at the end, for completeness

[@nommindymple6241]

I wish the people who downvoted this would add comments about WHY they downvoted. It's a geometric math proof. Are they seeing an error with it? If so, it would be nice to know what that might be.

[@farisakmal2722]

eyegasm

[@jiaming5269]

Wow....

[@AlgyCuber]

0:29 polyhedorn

[@ThinkTwiceLtu]

:(

[@hamiltonianpathondodecahed5236]

lubly made my day

[@Victor_Gonzalez98]

I can't understand how this channel doesn't have more followers. This is one of the best mathematical outreach channels on KZbin without a doubt. Great job dude.

[@ThinkTwiceLtu]

Thanks! Appreciate your support.

[@shoam2103]

Brilliant in its simplicity! You're like a magician: * here's a small proof on my left hand, and another on my right👈👉 . * see how combining these 2 almost appears to make this more convoluted🐇🎩? * but voilà, the proof! 🎉

[@studset]

Check out "Proofs And Refutations" by Imre Lakatos if you have not already done it. Nice video!

[@ThinkTwiceLtu]

Thank you will definitely check it out:)

[@dooflesshampoofles9226]

Me watching funky math videos at 3 am

[@yuufgreat9935]

Sound is way toooo low

[@Jacob-qx4bc]

math time

[@ThinkTwiceLtu]

fun time:)