publius: It appears to me that I am the main proponent in this Forum for the model of the expanding Universe based on a four-dimensional hypersphere the three-dimensional "surface volume" (your term) of which is the three-dimensional curved space of the Universe that we think we know. I argued for this model in the thread entitled "The Shape of the Universe" that I started as an argument against the doughnut-shaped universe that Dr. Gay argued for.

I did not introduce the hypersphere model with any thought of belaboring its geometry by devising ways to calculate its properties. My purpose, aside from an argument against Dr. Gay's doughnut-shaped model was to provide an easily visualized answers to questions such as the following:

Where is the center of the Universe?
What is the nature of the expansion of the Universe?
Where is the point that the Universe is expanding away from?
What is meant by curvature of space?

I feel that bringing time into this model would be over-developing it. It's hard enough to visualize the four-dimensional nature of spacetime outside the event horizon of a black hole or even in the spacetime occupied by orbits in the vicinity of the sun where no event horizon needs to envisioned.

I can appreciate that mathematicians so inclined could have a field day practicing their arts in working out methods for analyzing the properties of the four-dimensional Universe that I visualize, but I also feel that it would be a waste of their time.

The statement

However, it can be a very useful visualizing tool to help you understand.

states my sole reason for my having introduced this model.