This relates to

the "shape" of the universe on Wiki which gives this picture:

Image courtesy of NASA, see WikipediaYes it's confusing all right, because at least two out of three above were always going to be wrong, and because we aren't dealing with the shape of the universe at all. And we aren't dealing with its geometry. We aren't even dealing with the geometry of spacetime. It's more like "the geometry of spacetime at this moment". That's why the Wikipedia article says

*"Cosmologists normally work with a given space-like slice of spacetime"*. But then there's a problem in that people think this is the geometry of space, when it isn't. If spacetime is curved light will curve*, but that doesn't mean space is curved. See

Baez and note this:

*"Similarly, in general relativity gravity is not really a 'force', but just a manifestation of the curvature of spacetime. Note: not the curvature of space, but of spacetime. The distinction is crucial."*Instead it means space is inhomogeneous. See Einstein's

Leyden Address for that:

*"This space-time variability of the reciprocal relations of the standards of space and time, or, perhaps, the recognition of the fact that "empty space" in its physical relation is neither homogeneous nor isotropic, compelling us to describe its state by ten functions (the gravitation potentials gmn), has, I think, finally disposed of the view that space is physically empty."* Also see

http://iopscience.iop.org/0256-307X/25/5/014 which says inhomogeneous space is essentially what curved spacetime is. Now look a bit further down the Wikipedia article. See where it refers to

the FLRW model? Follow the link to the

FLRW article and what you can read is this:

*"The FLRW metric starts with the assumption of homogeneity and isotropy of space"*. It assumes space is homogeneous, so it assumes spacetime is not curved. So when WMAP found that the universe was "flat", it shouldn't have been a surprise. If you have two parallel light beams, they don't curve round in some great circle, and they don't diverge or converge. They just go straight. As for why Einstein though the universe was closed like the sphere above, I don't know. It's as if his confidence and intuition failed him when it came to cosmology. If he hadn't modelled the universe using dust, but instead used space alone, I think he wouldn't have made his "greatest blunder".

As for the

*shape* of the universe, the actual

*shape* of it, well that's a whole different kettle of fish. I think it's spherical myself, with an outer edge, but you won't find many people saying that. Instead there's a tendency to say it's infinite, and airbrush away any issues about how it can get to be infinite in a finite time, or whether it was infinite at the time of the big bang, or how an infinite universe can expand. I expect this infinite universe assertion won't persist, because it's what Phil Plait would call "bad cosmology" that IMHO leaves the door open to unscientific multiverse hypotheses. See

George Ellis on that. To answer you particular questions:

So is "geometry" maths-speak for "shape"?

I'd say it's maths-speak for the disposition of spacetime, but it isn't much to do with the shape of the universe.

And does "flat" quite literally mean flat, like a 2D-sheet?

No. It means light goes straight.

And if that's true, how can the rule of homogeneity apply?

I'm not clear what you mean, but light goes straight when space is homogeneous.

*

actually, light curves when spacetime is "tilted", but it's got to be curved to acquire a tilt. The plot of gravitational potential can't get off the flat and level if there's no curvature.