Euler's Formula V - E + F = 2 | Proof
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- Опубликовано: 30 сен 2024
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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
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"we will use two theorems proven in the previous two videos"
*watches all the videos this channel made*
*again*
Woah! This proof is really unexpected! I have only seen the proof by induction, but this is actually quite a creative alternative proof.
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.
@@arikwolf3777 maybe works only for starry polyhedra, i.e. which can be mapped on a sphere
How even would this be possible to prove inductively?
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]
Vote this up people, this comment was as cool as the video itself!!
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!
@@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.
how can you save youtube comments? this proof is just amazing
Isn't this the inductive proof? Doesn't matter, the visualization was amazing
4:40 Consider a spherical cow
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.
Yooo his RUclips videos are cool and all but have you checked his INSTA though. Asthetics. af. 😩
ฅ^•ﻌ•^ฅ
Juat did. And im like- *damn*
So... When can we see ur proofs for the millennial problems? I'm totally rooting for you! 😇
Great stuff man. Such a simple argument. I only knew of the induction proof and the dual graph proof. This was also very elegant
Monalisa : I m the most beautiful.
Think twice : see this
:D
@@ThinkTwiceLtu please make a video on the banac tarski paradox
@@sasmitarath4312 vsauce has a brilliant, in depth video on that topic:)
@@ThinkTwiceLtu ya, but if not this then make avideo on brouwer's fixed point theorem
@@ThinkTwiceLtu Yeah, I'd love an animation on any set-theoretic topic!
That was so brilliant it made me cry. Thank you!
Thank you for watching:))
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!
Thank you for the support!
Awesome as always, the fact you can explain math almost without words is mind-blowing
When thik twice makes us think🖤🖤
true hehe
twice
T H I K
ThiCC
Think once
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 :)
I am so glad to hear that. Thanks for the support!
But what would a spherical projected concave polyhedron look like?
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.
When he said 4π=2πV-2πE+2πF i felt that
What a good way to present this creative geometric proof!
Absolutely gorgeous. I usually don't complement people directly for being good at something. But for this one, I just can't control myself
Beautiful.
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.
Very interesting approach.
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!
Thank you for very much. Your comment made my day too:)
when I first found this channel I felt like I found a treasure
I stil feel that way
Happy to hear that! Thanks a lot for the support:)
bro?? it's an AMAZING proof, i never saw it before.
Throughout the video I be like:
Okay, ooo, wow, whaaaat, WOW, DAMN
Well-produced, as always.
Thank you:)
How are these videos so satisfying! How do you animate these videos?
What tool did you use to make the video?
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?
Thank you for the kind words! I'm happy to hear that. I use Cinema4d.
@@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!
Very epic bruh!!!! Epic quality
Thanks:>
Amazing bro...and I still wonder why schools and other institutions don't follow these methods...at least they can follow people like you
This is soooooo cool :D
better short video; but this is equal good.
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!
Pls do collatz conjecture next :)
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.
Thank you so much!!
This is inspiring!!!!!!!!
:))
Oh my god... It's beautiful ❤️
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?
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?
Amazing work! Hug from Portugal.
Thanks! :)))
lubly
made my day
Pure magic.
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.
Technically, you only proved you can triangulate an n-gon into n-2 triangles in cartesian geometry. At ruclips.net/video/2x4ioToqe_c/видео.html of your proof of the triangulation theorem, your definition of vertex V using line YZ makes implicit use of Euclid's fifth postulate (also known as the parallel postulate) which is not satisfied by spherical geometry. Don't you need to amend the theorem to work on spheres?
the 12 dislikes are from stupid people randomly clicking buttons
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...
Love your channel and ita contents! 😊 Never fails to make me happy
Thank you for the support:) happy to hear that.
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
Beautiful
eyegasm
Wow....
The Math Speaks For Itself
- Leon Eulermann
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.
math time
fun time:)
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
Awesome and thank you for such a explaination
this video is false advertisement and i will take my legal obligations to remove it unless you do.
please please don't use a music piece which can create irritation while reading whatever on the screen.
ty for color coding the theorem and the example
this is magic, omg
this channel should pop up in my recommended videos more often
That's one satisfying proof.
Thank you, much appreciated.
Me watching funky math videos at 3 am
The animation for projecting the cube onto the sphere was _gorgeous_ !
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.
Thanks:))
Check out "Proofs And Refutations" by Imre Lakatos if you have not already done it. Nice video!
Thank you will definitely check it out:)
Best presentation of Legendre's proof I have ever seen.
Thanks.
thanks for watching:)
The pure joy of induction!
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! 🎉
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?
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.
2:00 -- The two-minute warning
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.
Is it just me or does the background music sound like “Boulevard of Broken Dreams”?
Thanks, to you and Euler
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
If you count each side of every polygon you will double count each side because for every side there are two polygons sharing it.
@@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
does this proof extend to any shape that is topologically equivalent to a convex shape?
Yes
Awesome 😊💖💖
I'm glad that I subscribed to your channel.
:)
not a fan of mathematics, but your animations make it attractive
:>
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?
This is sum of angles around a point, not sum of angles in a triangle.
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.
Beautiful and smart
Sound is way toooo low
Excellent proof
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...
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?
@@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...
Cinema 4D? Amazing video by the way
Yes and thank you:))
Great proof
Continues to be some of the crispiest math on the interwebs.
:0
Wow....
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?
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.
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
Thanks for the support:)
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.
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.
OMG! Your channel is outstanding.please upload video every week.Thank you so much!😊
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:)
0:29 polyhedorn
:(
Interesting was looking for it yesterday.
Can you make it dark mode friendly next time?
Keep making these types of video...too good!!
Which software do you use to make videos?
Cinema 4d
@@ThinkTwiceLtu thanks a lot brother 😇😇
Lovely proof, and awesome animation. What are you using to animate this?
Thanks! I use C4D.