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## One Sphere = Two Spheres = Four Spheres =....

Before you continue reading about the websites below, please read the review of this week's resource, the book The Pea and the Sun by Leonard Wapner.

As originally stated in 1924, the Banach Tarski-Paradox claims that you can dissect a solid sphere into six pieces then reassemble them (using rotations and translations) to form two solid spheres of the same size as the original sphere. Odd...impossible...defying physical laws....I know!

These web sites can help you better understand and explore the weird world of the Banach-Tarski Paradox:

• Mudd Math Fun Facts is quite brief, but it does provide an animation (shown here...refresh to repeat it) that is perhaps closer to the truth than most would accept.
• Wikipedia gives a good overview of the paradox and a sketch of its proof...but you will be scrambling (looking up terms, etc.) to understand what is being said.
• Kuro5hin tries to give a layman's overview of the paradox, but again you must swim through a lot of set theoretic terminology.
• PlanetMath provides a "heavy" mathematical statement of the paradox, then does offer some helpful commentary.
• The Pacific Institute for the Mathematical Sciences Newsletter (Issue 2) provides an informative article by Volker Runde, entitled "The Banach-Tarski Paradox or What Mathematics and Miracles Have in Common."
• A last...but not a web site, you might want to read the classic text on The Bananch-Tarski Paradox, a book by Stan Wagon. And while doing that, you can refer (and perhaps even understand) the animation of Jan Mycielski and Stan Wagon provided by Wolfram.
Many other web sites exist....in fact, thousands of web sites mention this paradox. But, as expected, some can befuddle your muddling, rather than clarify. If you have a web site that you think is great, please let me know and I will consider adding it to the list.