I assume you're joking, but in case anyone thinks your description is accurate, it's wrong in a significant way.

Point (2) is fundamentally wrong in a critical way. The whole point of the Banach-Tarski Theorem is that you only have finitely many pieces. Of course these pieces are composed of infinitely many points, but it's a finite number of "pieces".

Others may find it amusing, sense of humour vary widely. Usually I just shrug and move on, but in the essence of this "joke" you've genuinely misrepresented the entire point.

I tried to joke about how real objects don't have an infinite number of surface points.

This theorem is often quoted as "mathematically-proven way to deconstruct a sphere and build two identical spheres", but this can't translate to reality in any way, it's an abstract thought exercise. Of course, "mathematically-proven way to deconstruct a sphere and build two identical spheres in your imagination" doesn't sound as cool.

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).