Anonymous at Sun, 20 Oct 2024 12:32:56 UTC No. 16441090

>>16441087
You count angles in a dumb way that's why

🗑️ Anonymous at Sun, 20 Oct 2024 12:33:26 UTC No. 16441091

Zero magnitude.

🗑️ Anonymous at Sun, 20 Oct 2024 12:34:39 UTC No. 16441092

The fact you can stare at this shape as a pic on screen implies zero magnitude.

🗑️ Anonymous at Sun, 20 Oct 2024 12:35:48 UTC No. 16441093

The pic on screen is like farmland to you, if you eat the pic, it counts as more than measuring it.

Anonymous at Sun, 20 Oct 2024 12:40:47 UTC No. 16441098

>>16441092
suppose each side is 2 meters

Anonymous at Sun, 20 Oct 2024 14:23:35 UTC No. 16441218

We substract 180 from every exterior turn for convenience. If we didn't do it, we'd get
[sum of exterior angles] + [sum of interior angles] = 360*p

Anonymous at Sun, 20 Oct 2024 16:59:22 UTC No. 16441376

>>16441087
Imagine walking a path around the polygon: to do one lap, you come back to where you started, facing the way you started moving. AKA, you have to turn 360 degrees.
Internal angles: Can't have quite the same intuition, but every time you add a vertex you need to widen every other angle too (e.g. an equiangular triangle has 60 degree corners, a square has 90 degree corners). As you approach infinite corners, each bend gets progressively closer to being straight, AKA it approaches 180 degrees. The gain from adding another corner is EXACTLY 180 degrees each time, because you widen the other angles (e.g. triangle to square, as the corners become wider the change is less, but compensated because you add another angle of nearly 180 degrees)

>Suppose you have a decagon, one person walks the interior perimeter and the other exterior perimeter. They are 1cm away from each other when they do this. They make more or less the same turns but one is considered to have turned a total of 360 degrees and the other 1440 degrees. Why?
Because you're measuring the angles of the internal guy wrongly. He turns 360 degrees. Remember at a 60 degree angle he doesn't turn 60 degrees, he turns 180 - 60 = 120 degrees. In a decagon, where every internal angle is 144 degrees, he turns 180 - 144 = 36 degrees at each corner, same as the outside guy.

Anonymous at Sun, 20 Oct 2024 20:13:10 UTC No. 16441629

>>16441087
Generalized Gauss-Bonnet theorem is concerns the sum of the exterior angles. If you were an ant forced to move in a straight line, but are allowed to rotate when you hit a vertex, it turns out that in a closed loop (like a polygon), the ant will have to turn around some multiple of 360 degrees, but that multiple depends on the surface itself.

>your problem
So, for a normal flat surface, it's like if an ant is confined to move on a circle, the ant will essentially be rotating 360 degrees when it end up at the same initial position. For an flat polygon (your question), if an ant starts at a vertex and moves along the sides of that polygon, when it ends up at the same initial position, it overall rotates it's body 360 degrees. That represents the exterior angle.

If you start with this, you can easily derive the interior angles formula for a n-sided polygon: each vertex is 180 - 360/n degrees, so the total sum is n(180-360/n) = 180(n-2)

>generalized surface
Shapes on a flat surface has an Euler Characteristic of X=1. This means that for polygon's, (Vertices) - (Edges) + (Face) = V-E+F = X = 1. So for a square, 4 - 4 + 1 = 1. For something different like 3D polyhedra like a cube, 8 - 12 + 6 = 2 = X (this is what Euler discovered). For a sphere, X = 2 as well, which means that if you were to draw a curve on a sphere, you'd notice that an ant forced to move along the curve "facing straight" (parallel transported) would overall rotate 360 * 2 = 720 degrees when they end up in the initial position. For a cylinder, X = 0; for a sphere with g holes, X = 2-2g (a doughnut/torus is g=1).

Using Gauss-Bonnet, you can convert to interior angles, and see that for a triangle drawn on a sphere, the sum of the interior angles is actually always bigger than 180 degrees.

You can look up the full equations on google

🗑️ Anonymous at Sun, 20 Oct 2024 20:19:58 UTC No. 16441636

>>16441629
I tried making it simple but after rereading it's a little lacking.
The 360*X is equal to the sum of the exterior angle, plus the parallel transport of the ant, plus the integral of the Gaussian curvature over the enclosed shape on the surface. By reading my initial comment, it might've seemed like it was just the parallel transport of the ant alone. My mistake

Anonymous at Sun, 20 Oct 2024 20:22:47 UTC No. 16441640

>>16441629
A lot of details are cut, just look it up for more info

Anonymous at Mon, 21 Oct 2024 04:24:50 UTC No. 16442156

>>16441629
Actually, the characteristic is actually associated with the region you're integrating your ant over, not the entire surface on which the curve lies. So for a triangle on a sphere, X=1 (this is true for any simple region)

Anonymous at Mon, 21 Oct 2024 08:15:39 UTC No. 16442282

>>16441087
Because each corner contributes 180, minus the amount of the exterior angle. Those shortfalls sum to 360, which is 180x2
So the sum of interior angles is p*180 - 2x180 = 180*(p-2)