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

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 volume of the resulting shape always have the same volume as the original sphere? NO !!!

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 six pieces which can be reassembled by rigid motions to form two balls each with the same size as the original !!!

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 !!!

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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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http://mathworld.wolfram.com/Banach-TarskiParadox.html

http://en.wikipedia.org/wiki/Banach-Tarski_Paradox