From our point of view, we're only directly aware of four dimensions, and one of those, being time, is difficult enough to get our heads around as it is. A complete dimensional model really needs to be applicable to n-dimensional environments though, so you shouldn't get too tied up with just four dimensional solutions, or indeed any specific number of dimensions.

Another key factor is change. If you try to think in terms of purely spatial dimensions there's no scope for change, resulting in a totally static universe.

If our universe

*is* closed, then a sphere, or even better still a torus, is an applicable model. However, our universe does not seem to be closed, so a better model would have been that of a four-dimensional elliptic paraboloid, which may be 'open' in both the spatial and temporal dimensions; that the expansion of the universe does not seem to be slowing down at all, but is actually accelerating, indicates that it is open spatially, and although we believe there to have been a start, or origin, of time, but no obvious end, suggests that there's no end to the temporal axis of the elliptic defining curve. In fact, if the acceleration of expansion turns out to be true, then we're not in an elliptic paraboloid shaped universe but a Hyperboloid shaped one, the essential difference being that we're on the opposite surface to the one we thought we were on i.e. outside rather than inside, and instead of the gradient of the curve approaching zero in one dimension it's increasingly deviating from it and actually approaching zero in another dimension, at right-angles to the one we were thinking of.

All fun stuff, but in terms of seeing the curvature of space, you really need go no further than gravitational lensing - have a look at some of the excellent photos that show it. This shows, in a very real way, the curvature of space; the light that we see lensed has followed a straight path through space. Gravitational lensing seems to be a paraboloid type function though

Regarding mass, well logically, if you imagine looking at the 3D spatial matter in our four-dimensional universe from a five-dimensional point of view you'd be seeing a hypermass, which would be the product of the 3D mass multiplied by (our) time. As to whether the hypermass <-> energy equivalence would simply be e=mc^3, or whether the 'c' factor is finite at all, due to time appearing to be unbounded and so possibly even infinite (or infinitely small), is an interesting question.