[kmiller755]

When scientists give answers like this, they are not talking about shape in the normal way. What they are talking about is the overall geometry of space. In a spherical geometry, the angles of a very large triangle sum to less than 180 degrees. In a saddle geometry, the angles of a very large triangle sum to more than 180 degrees. In a flat shape, the angles sum to 180 degrees.

Yes, the 'shapes' that are given, such as "saddle" or "sphere" are in analogy with our intuition of 3D geometry.  In the analogy, the space under consideration is 2D but "embedded" in 3D.  For our 3D world, we might imagine that it's embedded in 4D (as Einstein did when he started GR, but discarded it later when he realized the embedding is not necessary mathematically).  The "shape" of the 3-space in the 4-space is what scientists mean by the shape.

PhysBang's reference to the angles of the triangle are one means by which we can measure this curvature of space.  We may set up a triangle by means of three lasers, as lasers travel in "straight lines" (or "geodesics" in the language of general relativity), and then proceeding to measure the angles subtended by each joining pair of lasers.  The sum of the angles indicates the curvature. 

You can analogize by thinking of what it would be like for 2D creatures living on a sphere to shine a beam of light.  The light ray would be forced to travel in the 2D space, tracing out a "straight line" in 2-space but in reality curving in the embedding 3-space.  In this case, a straight line is simply one which minimizes length.  A triangle drawn with light rays on a sphere this way would not have angles that add up to 180.

[Ron Hughes]

My definition of space. It is where energy is stored. There is not a single cubic millimeter of space that does not contain radiation.No radiation no space. Space was blasted into existence by the radiation released in the Big Bang. Prior to the BB there was no space.