RE: Science Porn
February 17, 2015 at 4:28 am
(This post was last modified: February 17, 2015 at 5:07 am by Alex K.)
(February 16, 2015 at 4:48 pm)SteelCurtain Wrote: Three 90° Angles in Curved Space
This is an awesome visualization of a corrolary of the Gauss-Bonnett theorem which in Wikipedia is
Quote:The sum of interior angles of a geodesic triangle is equal to π plus the total curvature enclosed by the triangle.
(In radians, 360 degrees corresponds to 2*pi)
So, the fact that the sum of angles in this triangle is 270 degrees instead of the usual 180 degrees tells you about the curvature of the surface.
On a flat surface (no gaussian curvature), the theorem tells us we have pi ~ 180 degrees, but in the example in the Gif they measure an additional 90 degrees, or in radians an additional pi/2. How does the theorem predict this?
The surface of a sphere is 4pi*radius^2, and our triangle covers 1/8 of that. The gaussian curvature of a sphere is a constant, k = 1/radius^2. Multiplying that gives us
1/8 * 4 pi radius^2 *1/radius^2 = pi/2
and that's just the additional angle we observe in the Gif!
Question: what happens if you extend your triangle to cover a quarter of the sphere?
This notion of curvature is actually exactly the same that appears in a generalized form in the theory of general relativity, and Einstein cites Gauss (along with his student Riemann) in his paper.
Interestingly, Gaussian curvature is independent of how you bent a surface as long as you don't stretch it (the so called Theorema Egregium tells us that). This means that from the point of view of triangle geometry on the surface, a bent sheet of paper is still intrinsically flat, it is merely *embedded* in space in a crooked fashion. The gaussian curvature has the "egregious" property that it remains unfettered by this and only depends on the fixed properties of the surface. This is for example the reason why you can never map a globe, which has positive gauss curvature, on a flat sheet of paper without changing the lengths of stuff.
The fool hath said in his heart, There is a God. They are corrupt, they have done abominable works, there is none that doeth good.
Psalm 14, KJV revised edition