This is at first a ridiculous question but on the other hand I'm not sure how to prove it.
Take a solid 2D shape, $A$, and duplicate it to give $A'$. The shape $A$ together with the $A'$ make a new region called $B$.
Can we take the shape $A$ and using only translation and rotation make it fill exactly the region $B$ ? (That is to say it now covers the original position of $A$ and it's copy $A'$)
Obviously if $A$ is a shape of finite extent and area this would be impossible because we are asking a shape to cover twice it's own area. But it is not so clear to me that this would be impossible if $A$ were an infinite shape, perhaps some sort of infinite wedge whose thickness follows an exponential.
Another way of saying the same thing is can we split a shape $B$ into two parts such that they both are translated, rotated copies of the original shape $B$?
If not, how can we provide a proof? (By the way without using such obscure things as the Banach–Tarski paradox. ).
I can see how this might be done with a fractal (such as Sierpiński triangle) but I am thinking about solid shapes of fractal dimension 2.