It's been said that the universe is flat. How can it be flat if flatness has no volume?
"Flat" means three-dimensional Euclidean.
Maybe "flat" isn't the right term; 2-dimensional would be more accurate (see attached image). What would you call a shape with no thickness? Volumeless?
Those are 2D surfaces. The reason that a lot of diagrams use 2D to represent the universe is that, although you could represent the "flat" universe as a 3D box, you can't draw a curved 3D surface (you'd need 4D for that, which we don't have).
So, these diagrams lose a dimension in order to illustrate the concept of curved space.
It's interesting that you've taken the concept of "flat" so literally.
PS Lawrence Krauss explains this right at the start of his lecture here:
I thought flat was flat.
Krauss said "..it's open, closed, or flat."
No, "flat" in cosmology means 3D but not curved. "Flat" in everyday speech means 2D.
That is an intellectual hurdle you are going to have to jump.
thanks for that video
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