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Banach-Tarski Paradox

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.

Khan Academy Drunken Walker and Fly -- Math Fun Facts Imagine a drunken person wandering on the number line who starts at 0, and then moves left or right (+/-1) with probability 1/2. What is the probability that the walker will eventually return to her starting point Answer: probability 1. What about a random walk in the plane, moving on the integer lattice points, with probability 1/4 in each of the coordinate directions? What's the chance of return to the starting point? Answer: also probability 1. OK, now what about a drunken fly, with 6 directions to move, probability 1/6? Presentation Suggestions: Try to give a little insight by illustrating a random walk on the line for several steps. The Math Behind the Fact: A probabilist would say that simple random walks on the line and plane are recurrent, meaning that with probability one the walker would return to his starting point, and that simple random walks in dimensions 3 and higher are transient, meaning there is a positive probability that he will never return!

Integral Table Layman's Guide to the Banach-Tarski Paradox Preliminaries First of all, let's nail down what exactly we're talking about so that we're all on the same page. First and foremost, we're talking about a mathematical sphere, not a physical sphere, although I'd like to use an analogy with physical spheres to describe one possible way to intuit the Banach-Tarski Paradox. S = {(x,y,z) | x2+y2+z2 <= 1 } One important difference between S and a real, physical sphere is that S is infinitely divisible. 2Or, to be precise, c points, where c is the cardinality of the continuum. In fact, if we assume that spheres are not infinitely divisible, then the Banach-Tarski paradox doesn't apply, because each of the "pieces" in the paradox is so infinitely complex that they are not "measurable" (in human language, they do not have a well-defined volume; it is impossible to measure their volume). Now let's move on to the paradox itself. The Banach-Tarski Paradox How can this ever be intuitive??! Now, we have successfully built two (physical!) Bingo! Epilogue

Take It to the Limit In middle school my friends and I enjoyed chewing on the classic conundrums. What happens when an irresistible force meets an immovable object? Easy — they both explode. But one puzzle bothered us: if you keep moving halfway to the wall, will you ever get there? Questions like this have always caused headaches. But Archimedes, the greatest mathematician of antiquity, realized the power of the infinite. In the coming weeks we’ll delve into the great ideas at the heart of calculus. If you’re a careful thinker, you might be worried about something already. Imagine using a photocopier to reduce an image of a circle by, say, 50 percent. Of course, this doesn’t tell us how big pi is. Before turning to Archimedes’s brilliant solution, we should mention one other place where pi appears in connection with circles. Here A is the area, π is the Greek letter pi, and r is the radius of the circle, defined as half the diameter. Yes, it is. What makes the problem difficult is that circles are round.

Online Mathematics Textbooks Professor Jim Herod and I have written Multivariable Calculus ,a book which we and a few others have used here at Georgia Tech for two years. We have also proposed that this be the first calculus course in the curriculum here, but that is another story.... Although it is still in print, Calculus,by Gilbert Strang is made available through MIT's OpenCourseWare electronic publishing initiative. Here is one that has also been used here at Georgia Tech. Linear Methods of Applied Mathematics, by Evans Harrell and James Herod. Yet another one produced at Georgia Tech is Linear Algebra, Infinite Dimensions, and Maple, by James Herod.

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! How do we find pi? Pi is the circumference of a circle with diameter 1. Say pi = 3 and call it a day.Draw a circle with a steady hand, wrap it with string, and measure with your finest ruler.Use door #3 What’s behind door #3? How did Archimedes do it? Archimedes didn’t know the circumference of a circle. We don’t know a circle’s circumference, but for kicks let’s draw it between two squares: Neat — it’s like a racetrack with inner and outer edges. And since squares are, well, square, we find their perimeters easily: Outside square (easy): side = 1, therefore perimeter = 4Inside square (not so easy): The diagonal is 1 (top-to-bottom). Squares drool, octagons rule Cool!

Quotient space (linear algebra) In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero. The space obtained is called a quotient space and is denoted V/N (read V mod N or V by N). The equivalence class of x is often denoted [x] = x + N since it is given by The quotient space V/N is then defined as V/~, the set of all equivalence classes over V by ~. α[x] = [αx] for all α ∈ K, and[x] + [y] = [x+y]. It is not hard to check that these operations are well-defined (i.e. do not depend on the choice of representative). The mapping that associates to v ∈ V the equivalence class [v] is known as the quotient map. Let X = R2 be the standard Cartesian plane, and let Y be a line through the origin in X. Another example is the quotient of Rn by the subspace spanned by the first m standard basis vectors. More generally, if V is an (internal) direct sum of subspaces U and W, An important example of a functional quotient space is a Lp space. Let T : V → W be a linear operator.

Free Calculus The Calculus Here is a free online calculus course. This is essentially an ordinary text, but you can read it online. This text is somewhat unusual for two reasons. The text is rigorous. Both points are no doubt controversial, but conceptually the approach gives a kind of clean synergy which generates important examples and unifies calculus to a great extent. Unfortunately, there are no doubt uncorrected typographical errors and logical errors. This text © 1993, 2001 Copyright William V. Table of Contents