The initial description deceived me (e.g. the layman :-)). It used the word "piece" in a very technical way. Piece means contiguous chunk in layman's terms, i

The initial description deceived me (e.g. the layman :-)). It used the word "piece" in a very technical way. Piece means contiguous chunk in layman's terms, it typically does not mean a set of fragments of the object (that would be many pieces). If someone told me you could, mathematically speaking, take 5 piles of fragments of a sphere and combine them into two identical spheres, it would still be slightly mysterious, but it would only take the words "infinite density" to make it clear, instead of three paragraphs explaining what "piece" means. :-) Anyway, I still thought the following explanation was pretty clear. It was probably not the absolute correct mathematical intuition, but what summarization of extremely technical topics are completely correct?