For anyone still following, the theorem is perhaps best described like this:

We model spheres as collections of points, and given a collection of points there is a fairly obvious way to model the idea of moving that collection around in a way that reflects moving a physical object. The problem is, given a sphere, we can partition the points into six sets, move those sets around, and recombine them to give us two spheres the same size as the initial one. In a very real sense, in this model we can "cut up" a sphere, then rearrange the pieces to make two spheres. We've doubled our volume.

Clearly this is nonsense, and it shows the limitations of the model. But equally, it tells us something important about the maths we use every day to model buildings, bridges, fluid flow, and more.

All models are wrong, some models are useful. -- George E. P. Box

I would add:

Some things are nonsense, but sometimes the nonsense can tell us useful things about the way the models are wrong.

I don't know if anyone is still following, but thank you for being civil and instructive.

You're welcome, and thank you in return for the constructive dialogue ... it makes a refreshing change, and is very welcome.

Doubling the volume is not so weird, as long as you do not double the mass

But these are of uniform density, so you are doubling the mass as well. To some extent, that's the point of the theorem (and discussion).