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"!
Steven Strogatz is the Schurman Professor of applied mathematics at Cornell University. Among his honors are MIT’s highest teaching prize, membership in the American Academy of Arts and Sciences, and a lifetime achievement award for communication of math to the general public, awarded by the four major American mathematical societies. A frequent guest on National Public Radio’s “Radiolab,” he is the author, most recently, of “The Joy of x,” which grew out of his previous Opinionator series “The Elements of Math.” He lives with his wife and two daughters in Ithaca, N.Y. Take It to the Limit
Prehistoric Calculus: Discovering Pi Pi is mysterious. Sure, you “know” it’s about 3.14159 because you read it in some book. But what if you had no textbooks, no computers, and no calculus (egads!) — just your brain and a piece of paper. Could you find pi? Archimedes found pi to 99.9% accuracy 2000 years ago — without decimal points or even the number zero!
Free Calculus The Calculus Here is a free online calculus course. This is essentially an ordinary text, but you can read it online. There are lots of exercises and examples.