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Mudd Math Fun Facts: Banach-Tarski Paradox

Mudd Math Fun Facts: Banach-Tarski Paradox
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. An alternate version of this theorem says (and you'd better sit down for this one): it is possible to take a solid ball the size of a pea, and by cutting it into a FINITE number of pieces, reassemble it to form A SOLID BALL THE SIZE OF THE SUN. 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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Glass-sphere microscope A Glass-Sphere Microscope Giorgio Carboni, First draft March 1988 Publication in "Scienza e Vita" December 1993 Publication on this website January 1996 Updated on the 30th of December 2010 Translated by: Sabrina Gemelli, Giacomo Giuli, Karen Coleman Acknowledgments to David W. Walker for his helping in the revision of the English text A snowfall, a flower, a puddle – these all seem like normal things, without surprises. Yet, if you could see the beauty of a snowflake, the hidden shapes of flowers, the variety and the strangeness of the tiny creatures that live in a puddle, you would surely be amazed. You will notice that you are surrounded by a fascinating and unknown world. Usually, while attempting to observe very small objects, we realize the impossibility of distinguishing details smaller than a tenth of millimeter by naked eye. This small instrument will give fairly detailed images if you consider that it is essentially constructed from a small homemade lens. [1] C.L.

Crazy stuff from the bible . Have I taken things out of context ()? Meanwhile read on... "For I am merciful, saith the Lord, and I will not keep anger forever." "If I testify about myself, my testimony is not valid." "And Jesus coming, spoke to them, saying: All power is given to me in heaven and in earth." And Jesus said, "For judgement I am come into this world." “Love not the world, neither the things that are in the world. "Let your light so shine before men, that they may see your good works, and glorify your Father which is in heaven." "Jacob said, 'I have seen God face to face, and my life is preserved.'" We should fear God (Matthew 10:28) We should love God (Matthew 22:37) There is no fear in love (1 John 4:18) The Arguments from Confusion and Biblical Defects Foolery "But anyone who says 'You fool!' "You fools!" "But God said to him, 'You fool!' "You foolish Galatians!" Feedback: This "foolery" section has been criticised and is discussed further here. Rejoicing Mercy "Blessed are the merciful" [Matthew 5:7]

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