Posts from Academy of Sciences of Heroesland Auto-translated

But a geoid is, by definition, shaped like the Earth, right?

Geoid.

Obviously, for the three-dimensional case, we have a sphere in 3D, not in 4D space.

But a sphere in 3D space is two-dimensional. It doesn't become three-dimensional.

Okay, fine.

But a sphere in 3D space is two-dimensional. It doesn't become three-dimensional.

Wikipedia in that article about a sphere disagrees with you.
Although, I read the article about the Poincaré conjecture again, and it's written the same way as you said.
But that's not the main point; it's about which terminology to use. I prefer to call the sphere three-dimensional, and the manifold on it two-dimensional, because, as I said again, to describe the sphere we are familiar with, we need three dimensions – it is a three-dimensional figure. Then the Poincaré conjecture would be formulated as follows: a simply connected compact n-dimensional manifold is homeomorphic to an (n+1)-dimensional sphere. But, of course, it's all just a matter of terminology.

I prefer to call the sphere three-dimensional

And I prefer to call it a pot... You are confusing a sphere with a ball, sir.;) A body cannot have the same dimensionality as the surface that bounds it. The bounding surface has a dimensionality 1 lower. Three-dimensional objects have volume, whereas two-dimensional ones have area. I've seen the formula for the area of a sphere, but I haven't seen a formula for its volume.

Yashur, do you believe that two plus two equals four?

I believe.

How would you prove it?

Using this thing. By the way, proving that 2*2=4 is a bit more complicated. :)

But this isn't proof. It seems like something isn't quite right here.

some kind of reptilian technologies.

You're confusing a sphere with a ball, sir. A body cannot have the same dimension as the surface that bounds it. The bounding surface has one dimension less. Three-dimensional objects have volume, while two-dimensional objects have area. I've seen a formula for the area of a sphere, but not a formula for its volume.

Well, I was just saying that I don't take the dimension of a manifold as the dimension of a figure, so I'm not confusing anything.
A figure is like a submanifold. Of course, the dimension of the submanifold doesn't have to match the dimension of the manifold, but a figure is also distinguished by its order in the space in which it is located. In addition to the two angles we can introduce by introducing coordinates on the sphere, the sphere itself has at least a radius (and, in general, also the coordinates of the center, although this is not so important). The main thing is that the radius determines the properties of the sphere. And that's 3D, no matter how you look at it.
But this is just terminology, of course. As you can already see, it's not just you and me who haven't decided on it

Okay, Yashur, I've thought about it a bit more, and you're probably right. Any plane is obviously a two-dimensional figure, regardless of where it's placed: in three dimensions, in four... And a sphere can also be described using the same two coordinates, keeping the third (radius) constant. Just like a plane is described by two coordinates, but to determine the position of the plane in 3D space, you need to add more information. In general, yes, if you call a sphere within a 3D space two-dimensional, there are fewer paradoxes.

A sphere is generally insignificant compared to this thing. :D

Added 2 minutes later

But this isn't proof. There must be something fishy here.

Of course, this isn't proof, but rather a tool for proof. But I personally saw the proof, so I personally have no doubts that 2+2=4. ;)

The Sphere is nothing compared to this thing.

Yeah, it's cute.

Warning, large image!