Math Fun Facts
Did you know that it is possible to cut a solid ball into 5 pieces, and by re-assembling them, using rigid motions only, form TWO solid balls, EACH THE SAME SIZE AND SHAPE as the original? This theorem is known as the Banach-Tarski paradox. So why can't you do this in real life, say, with a block of gold? If matter were infinitely divisible (which it is not) then it might be possible. But the pieces involved are so "jagged" and exotic that they do not have a well-defined notion of volume, or measure, associated to them. In fact, what the Banach-Tarski paradox shows is that no matter how you try to define "volume" so that it corresponds with our usual definition for nice sets, there will always be "bad" sets for which it is impossible to define a "volume"! Presentation Suggestions: Students will find this Fun Fact hard to believe. The Math Behind the Fact: First of all, if we didn't restrict ourselves to rigid motions, this paradox would be more believable.
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