A Surprising Appearance of Euler's Famous Formula: V- E + F = 2

Рет қаралды 1,341

3 жыл бұрын

For a convex polyhedron, its count of vertices V, count of edges E, count of faces F satisfy V-E+F = 2. In the "How Round of a Cube?" video ( • How Round is a Cube? (... ) we showed that this equals the "total pointiness" of the polyhedron - surprising! In this video I explore another surprising interpretation of the famous equation.
This video is long. (Sorry!) Here's its breakdown:
00:10 : A Charming School-Geometry Problem in 2 Dimensions: Pushing it to other dimensions.
06:45 Proving the 2-D Puzzle.
FEEL FREE TO STOP HERE
10:17 Making Sense of the 3-D Version of Things.
16:57 On the Areas of Spherical Polygons
24:58 Bringing it All Together

Пікірлер: 28

@12jgy 3 жыл бұрын

Oh my god, this is wonderful, simply mind blowing! It was quite the surprise when Euler's Formula just appeared out of the blue! I found this channel through the MegaFavNumber colab and I was taken by the cheerful enthusiasm of the presentation, it's really quite contagious, and this video is no exception. My mood was kinda down, but this video managed to pick it right back up, so thank you!

@JamesTantonMath 3 жыл бұрын

Wow! Thank you! Math can be mighty joyous and uplifting for sure! You might enjoy too Exploding Dots at www.gdaymath.com/courses.

@gucker 3 жыл бұрын

Wow, this is grand! Thank you very much!

@nikolausbarlow267 3 жыл бұрын

i watched all of the exploding dots lessons exeped the lesson 10 part 2.

@geraldillo 3 жыл бұрын

Very nice

@nikolausbarlow267 3 жыл бұрын

i like your 3-d shapes

@oldreddragon1579 3 жыл бұрын

If a Torus has an inner dimension (hole) of 0 is it V-E+F=1. ? That is to say a sectional view would look like a pair of circles joined at a point on their circumferences with bars top and bottom or a Sphere whose poles are pulled in to one point to create a Torus.

@JamesTantonMath 3 жыл бұрын

Actually, for a torus, V-E+F=0. And indeed, take a solid sphere and punch a hole through it, then the surface of that is a torus. Or ... take a circle and rotate it around a line and it will trace out a torus and give the cross section you so perfectly describe!

@oldreddragon1579 3 жыл бұрын

@@JamesTantonMath the torus has a hole of 0. Which is the removal of 1 vertex from a Sphere.

@oldreddragon1579 3 жыл бұрын

@@JamesTantonMath If you have a 3D program like Blender try counting the Vertices, Edges and Faces. The center is a single Point. I think people are counting it as 2 points ( Top and Bottom) but they are the same Vertex. Thank You for replying.

@JamesTantonMath 3 жыл бұрын

@@oldreddragon1579 Euler's Formula presumes the surface of the polyhedron has been divided up into triangles (or if not triangles, polygons with no holes) and then V = number of triangle corners in total, E = number of triangle edges in total, and F = number of triangles. (But you can loosen this up a bit and make some faces quadrilaterals, for instance, by combining two triangles that share an edge, etc.) So, you need to image a torus with surface divided up into triangles. Think of a torus as a cube with a square hole cut through it. Then divide the faces you see into triangles. Most of the faces are rectangles or squares and can be made into two triangles, but there are two faces with hole in it. They each need to be made into 8 triangles, say. (I wish I could draw a picture!) Then V-E+F = 0.

@oldreddragon1579 3 жыл бұрын

@@JamesTantonMath I do understand the triangles. The point I was trying to make is that this specific Torus should be V-E+F=1 If a Sphere is equal to V-E+F=2 And a Torus with a hole is V-E+F=0 Then this particular Torus has one less Vertex than a Sphere because if you transform the Where it loses 1 Vertex. Try it on a 3D package making sure to remove Double Vertices.

@nikolausbarlow267 3 жыл бұрын

the 3 d shapes are good.

@nikolausbarlow267 3 жыл бұрын

noice.