Science Porn

[SteelCurtain]

Lol, there was a some information missing there...

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Three 90° Angles in Curved Space


Evolution of the moon




A gorgeous shot of Enceladus. What lies beneath its icy surface?

[Alex K]

Haha Well, if you can pass a straight bar through an extremely bowed hole, I should be able to get that couch into my daughters room.

You might want to read Dirk Gently's holistic Detective Agency first, and in case of continuing nagging doubts about the endeavour, employ an electric monk.

[Alex K]

Three 90° Angles in Curved Space

This is an awesome visualization of a corrolary of the Gauss-Bonnett theorem which in Wikipedia is

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.

[Alex K]

LOL from the Wiki article on T. Egregium

An application of the Theorema Egregium is seen in a common pizza-eating strategy: A slice of pizza can be seen as a surface with constant Gaussian curvature 0. Gently bending a slice must then roughly maintain this curvature (assuming the bend is roughly a local isometry). If one bends a slice horizontally along a radius, non-zero principal curvatures are created along the bend, dictating that the other principal curvature at these points must be zero. This creates rigidity in the direction perpendicular to the fold, an attribute desirable when eating pizza, as it holds its shape long enough to be consumed without a mess.

The gaussian curvature is the ordinary curvature in one direction times the ordinary curvature in the other. For an intrinsically flat object, that product must be zero, i.e. if you bent it in one direction, the curvature in the other must vanish.

[SteelCurtain]

Question: what happens if you extend your triangle to cover a quarter of the sphere?

I think my mind just broke a little bit.

Visualizing this it seems that the sum of the interior angles for a quarter sphere triangle would be 360° (Two 90° angles and a 180° angle), and thus the additional angle would be ᴨ radians. Which is fleshed out with the equation:

¼ · 4ᴨr² · 1/r² = ᴨ

Which is fucking awesome.
*nerd mathgasm

[pocaracas]

Now do it for half the sphere.

[Alex K]

Now do it for half the sphere.

Steel Curtain is right - we get a degenerated triangle with one invisible 180 degree angle and a sum of 360. For half the sphere, well...

[SteelCurtain]

Somebody posted something in another thread that made me think of this.

Crosswind Landings or "crabbing" show some pretty dynamic physics really well. The plane's velocity vector differs (sometimes pretty dramatically) from its thrust vector because of wind perpendicular (or close to perpendicular) to the thrust vector. The pilot has to point the plane's nose in such a way that the velocity vector "points" down the runway.

Here's some awesome shots of planes crabbing and landing in Düsseldorf on an icy runway:

[Alex K]

That's my local airport

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Now do it for half the sphere.

Ok if noone is going to bite - for half the sphere the triangle degenerates completely and becomes identical to the equator! It's got three invisible 180 degree angles which corresponds to 3 pi. Incidentally, the integral over the half sphere is 2 pi, and according to the theorem, pi + 2 pi = 3 pi is the predicted sum. Math works even in the weird limiting case where a circle is a triangle!

[Anomalocaris]

Somebody posted something in another thread that made me think of this.

Crosswind Landings or "crabbing" show some pretty dynamic physics really well. The plane's velocity vector differs (sometimes pretty dramatically) from its thrust vector because of wind perpendicular (or close to perpendicular) to the thrust vector. The pilot has to point the plane's nose in such a way that the velocity vector "points" down the runway.

Here's some awesome shots of planes crabbing and landing in Düsseldorf on an icy runway:

The Turkish airline B-777 pilot didn't do a very good job. Ideally he should yaw the plane at the last second to swing the nose back around to point down the runway just as he touch down.

Instead he brought the plane all the way down with the nose pointed upwind with undercarriage tires at an angle to the runway. As a result when undercarriage tires gained traction on the runway at the moment of touch down, it jerked the plane side ways, Adding unnecessary strain on the undercarriage, wear on the tires, and likely giving a fairly rough jerk to the passengers.

The following Lufthansa A-340 pilot did a much better job.