Euclidean slices of constant r

[TrickyDicky]

Hi, I was wondering if someone can set this right , I'm discussing this with another person that says that If (working in spherical coordinates) we make r constant in a Euclidean 3d space, in the resulting slice (phi-theta plane) we can define 2-spheres. I say that in my opinion you can't have 3-dimensional objects such as 2-spheres in bidimensional space, but he says it is obvious we can.
I would appreciate it if someone clarifies this seemingly easy problem.

 

[TrickyDicky]

anyone?

 

[Feeble Wonk]

I have very similar confusion, and it relates directly to the concept of topological bending, expansion or contraction of finite 3D space in the absence of at least a 4th spatial dimension.

The classic example offered is the balloon analogy, wherein the balloon expands - with all points diverging - but without a specific center from which expansion occurs. My objection to that analogy has always been that the center is "INSIDE" the balloon, not on it.

The good people in cosmology suggest that this is flawed thinking, and insist that the center of the balloon DOES NOT EXIST. They refer me to the topological concept of a 2D torus existing in three dimensions, in a universe that DOESN'T HAVE THREE DIMENSIONS!

It seems to me that this is an arbitrary set selection of dimensional space... simply defining a limited area of 3D space with 2D specificity. Can anyone please explain this in terms that I can understand?

 

[TrickyDicky]

To make the question more specific, I understand that intrinsic curvature doesn't need a higher dimension embedding to be computed and in this sense you only need 2 dimensions to have a curved surface, and this 2d surface could be spherical, and inhabitants of the 2d world could perceive this spherical curvature, what I'm saying is that without the 3d embedding, one can't discern , or observe 2-sphere objects, the maximum the 2d inhabitants can aspire to is to perceive they live in a spherical world. Is this reasoning right?

 

[Office_Shredder]

Can you specify what extra information they can discover if their world is embedded in 3 dimensions?

 

[TrickyDicky]

Can you specify what extra information they can discover if their world is embedded in 3 dimensions?

Well' I'd say they would be able to actually see and touch 3-d objects such as spheres, cubes and pyramids, while in their original 2d world they can conclude their world is spherical by doing certain measurements in their world, but of course they can only see circles, triangles, etc, they haven't really ever see a cube or a ball, etc so they could only try to imagine what they are like.
It is a similar situation in our 3d world, we can try and imagine what it would be like a 3-sphere(hypersphere) but we can't see them, they don't exist in our world, even if our universe turned out to be spherically curved, a hypersphere itself, since we can't see it from a higher spatial dimension.

 

[zhentil]

Can you define a sphere in a sphere? Sure. You certainly can't embed a sphere in a plane.

For the question of "defining a sphere" the first step is realizing that the question is silly. You can define anything to be anything. The question is whether such a definition would make sense. Let's ask instead an easier question. How would you define a sphere in four dimensions? Five? The concept of a sphere certainly makes sense there.

 

[TrickyDicky]

Can you define a sphere in a sphere? Sure. You certainly can't embed a sphere in a plane.

For the question of "defining a sphere" the first step is realizing that the question is silly. You can define anything to be anything. The question is whether such a definition would make sense. Let's ask instead an easier question. How would you define a sphere in four dimensions? Five? The concept of a sphere certainly makes sense there.

Yes, of course that "define" it is possible,it was possible a bad choice of words for the meaning I had in mind so I have made a distinction in the following posts of two different meanings of "defining" a 2-sphere and my question-confusion is solved.