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작성자심바 조회 61회 작성일 2021-10-23 00:41:27 댓글 0

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

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Proofs for two theorems used in this video:
► Polygon triangulation: https://www.youtube.com/watch?v=2x4ioToqe_c
► Area of a spherical triangle: https://www.youtube.com/watch?v=Y8VgvoEx7HY

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
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]
BOSTASH : "we will use two theorems proven in the previous two videos"

watches all the videos this channel made
Mathemaniac : Woah! This proof is really unexpected! I have only seen the proof by induction, but this is actually quite a creative alternative proof.
Tim H. : 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.
Jorge C. M. : 4:40 Consider a spherical cow

The second most beautiful equation and its surprising applications

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Zach Star : I've gotten a few comments about this so just wanted to address it here. At the 14:00 min mark I start talking about negative curvature and how my floating globe stand actually has no curvature to which some have asked "what if you just rotate the principle curvatures like 45 degrees, then you'll have two segments with non zero curvature and you wouldn't get zero overall." I didn't mention this in this video but in the other one I did.... those principle curvatures can't just be any segments you want. At any given point you have to pick the segment with the most curvature and the one with the least then multiply those (and those two will always be perpendicular). So for the stand you can't rotate from the two I mentioned or else you wouldn't have the max and min values.
Rocky Robinson : Euler goes blind and said "Now I will have fewer distractions.".
A Wallach : Vigorous proof mathematicians have a sense of humor That was good.
The Dark Sword : my friend: how to solve this problem?
me: use the euler formla
him: which one?
MauLob : This video is sponsored by Leonhard Euler

Euler's Formula and Graph Duality

A description of planar graph duality, and how it can be applied in a particularly elegant proof of Euler's Characteristic Formula.

Music: Wyoming 307 by Time For Three
Ryan Tamburrino : This is the best math channel on YouTube.
Double Bob : 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.
Powersource : Mind blown when you put the proof together at the end.
TheWolfboy180 : damn randolph's legs are creepy as hell
Maks Rosebuster : 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.

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