If you cut a circle into a finite number of pieces and reassemble the pieces, will the resulting shape always have the same area as the original circle? Yes.

If you cut a sphere into a finite number of pieces and reassemble the pieces, will the resulting shape always have the same volume as the original sphere? NO !!!

The Banach - Tarski Theorem states that it is possible to dissect a ball into finitely many pieces (in fact five will do) which can be reassembled by rigid motions to form two balls each with the same size as the original !!!

To give a simplistic insight in how you can end up with more than you started

... Imagine a perfect dictionary (without definitions !!!) containing every possible word (permutation of letters) however long. It would contain a countably infinite number of entries. (see countable sets).

It would contain:

 AACAT BACAT ... ZACAT ACAT BCAT ... ZCAT ADOG BDOG ... ZDOG AELEPHANT BELEPHANT ... ZELEPHANT etc. etc. etc. etc.

This dictionary could then be cut into 26 identical perfect dictionaries when the first letter of every entry in the dictionary has been ignored.

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