Edit made on February 13, 2009 by GuestEditor at 02:48:56
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HEADERS_END
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