Banach Tarski paradox on a closed 1D or 2D set embedded in $\Bbb R^3$

Question

Suppose we have a 1D closed line segment, or a 2D closed disk embedded in $\Bbb R^3$. Is it possible to perform a Banach-Tarski type procedure on the object to produce two such objects?

The reason I ask is because Banach and Tarski showed that for a 2D disk in $R^2$, they can produce only one disk with their method using standard Euclidean congruences. But, if we embed the 2D disk into 3D, there are now more types of "rigid motions" permitted, and the resulting groups of motions have different properties. So, is it then possible to perform a Banach-Tarski type procedure?

I am most curious about the simple example of a line segment embedded in $\Bbb R^3$.

Answer 1

While you get more rigid motions by embedding into a larger space, you can't do anything new with them (unless we care about finer details like how the pieces do/don't intersect as we move them around or avoiding reflections). Focusing on the disks case for concreteness, this is because $(i)$ we can always assume that the final product lies in the same plane as the original shape (just move the pieces otherwise) and $(ii)$ we can then just look at the starting/ending positions of the shapes involved, and the "net motions" of the pieces will be rigid transformations of that plane.

That is, we always have to (more-or-less) come back to the same smaller space that we started with, and this means we could have just stayed there from the get-go.

Answer 2

The Sierpinski-Mazurkiewicz Paradox does the job.

The OP asks for a closed set, in this case, the whole complex plane. You will see in the link above that it is unknown whether a 2D bounded set is paradoxical.