[talanum1 (OP)]

They tell you only in 2 dimensions. To determine if 3 dimensional space is flat you need to show that 2 x flat 2 dimensional planes that start out parallel, stays parallel forever in 2x2 perpendicular directions i.e. all along the x and y axii.

[puppypower]

They tell you only in 2 dimensions. To determine if 3 dimensional space is flat you need to show that 2 x flat 2 dimensional planes that start out parallel, stays parallel forever in 2x2 perpendicular directions i.e. all along the x and y axii.

Cartesian coordinates, which we use to describe space is based on a 3-D grid defined by (x,y,z). In this 3-D grid all (x,y) planes will run parallel by definition. The z-axis simply tells us which parallel planes we comparing, to show that they all will run parallel.

We can draw curves on these parallel planes so the curve moves between two or more planes. But individual planes still remain parallel if we plot them. The plots are pictures we draw on and even between the parallel planes.

An interesting special affect is drawing a 3-D representation on a 2-D plane.  We first choose a 2-D plane that is singular, so z remains constant. However, we can create an illusion of a variable z-axis, using shadows and highlights. I call this 2.5-D or a spatial illusion. It is not exactly 3-D but can fool the eyes. It requires a tactile sense, like touch, to see through it. Touch the screen a prove that the image of the 3-D cone, is not 3-D. However, by convention we call this 3-D.



This 2.5-D spatial illusion affect is very popular in politics. One builds a line of reasoning to get a 2-D drawing of cause and affect on a rational plane. You then add shadows and highlights to make it appear more 3-D to the eyes. The highlights are the valid facts and data that you share, while the shadowing is the denial of the truth in other POV's. Censorship is often a part of the shadow affect. The purposeful gaps in truth, gives deeper shadows that make the image appear more 3-D.

A good example in political science is man made climate change. One is not allowed to talk about natural climate change, without being called a denier. This is very bazaar, since this has occurred many times on earth and it has far more hard geological data than manmade climate change. This denial of truth is needed for the shadowing, to create what will appear to be integrated 3-D truth; only manmade could do this. The consensus of science is a political angle used for the highlights, with a "consensus" implying the whitest light; facts, for your highlights. You use the extra light and the extra dark for a good 3-D illusion.  I am too aware to fall for it.

If we instead of using Cartesian coordinates say we were to use spherical polar coordinates. Instead of (x,y,z) , we now have one distance variable and two angles variables; (r,Φ,θ). Now all the parallel planes become be concentric spheres, curved as a function of two angles (Φ,θ), each with a constant r on the third or distance axis. This is more consistent with Relativity, and stellar and planetary shapes. We can sail around the world using flat maps, using one polar coordinate plane.

[Bored chemist]

A good example in political science is man made climate change. One is not allowed to talk about natural climate change, without being called a denier.

Why tell that lie?

[Bored chemist]

I call this 2.5-D or a spatial illusion. It is not exactly 3-D but can fool the eyes.

That's why it's called "trompe l'oeil".

It requires a tactile sense, like touch, to see through it.

No, it doesn't.
Just look at your phone from a different angle (e.g/ using the other eye).

[evan_au]

In the vicinity of a black hole, spacetime is curved. That's why black holes cause gravitational lensing. And if you could see one close, the positions of stars is badly distorted when the black hole passes between you and the stars.

On large scales, spacetime is close to flat, but on its way to becoming flatter due to the accelerating expansion of the universe.
- This acceleration was discovered by astronomers trying to measure the flatness of spacetime on large scales