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

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Proofs for two theorems used in this video:

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.

[bostash]

"we will use two theorems proven in the previous two videos"

*watches all the videos this channel made*

[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]

[timh.]

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.

[samdob]

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!

[JM-usfr]

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

[antoniolewis]

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

[rodrigo-vlbi]

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

[swankitydankity]

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 :)

[Magnasium]

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.

[ranjitsarkar]

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

[ClaudioDiBiase]

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!

[kat]

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!

[HebaruSan]

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

[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!

[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?

[sasmitarath]

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

[EastingAndNorthing]

But what would a spherical projected concave polyhedron look like?

[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.