The Banach-Tarski Paradox is the consummate example of mathematics behaving badly. It states that, given a perfect sphere, it’s possible to cut it up into 5 pieces, rearrange those pieces rigidly (no stretching, twisting, rescaling, etc.), and end up with 2 copies of the sphere you started with. We will prove the Banach-Tarski Paradox. The proof is long, intricate, and often fascinating, so we’ll skip some boring parts to save room for digesting the main ideas. Along the way we’ll see rudiments of abstract algebra, talk about some very deep set theory, and come out with an understanding of why cutting up mathematical spheres is not the same as cutting up apples. This class is the first of two parts; make sure to catch the second as well if you want the whole proof!

Wouldn't it be great if Splash happened more than once or twice a year, in more places than just MIT? In fact, it does! Over the past four years, the scope of Splash, Spark, and other similar programs has expanded tremendously--there are now 14 colleges around the country where groups of undergrads have gotten together and decided to put together Splash programs. This class will be an informal discussion about the possibilities and prospects of establishing new Splash programs at colleges everywhere. More specifically, how you could create a Splash when you get to college.

The Banach-Tarski Paradox is one of the crown jewels of “weird mathematics”—the art of using standard mathematical tools to come up with truly bizarre results. In this case, the result we’ll come up with is this: it’s possible to slice up a sphere into five pieces, rearrange them using rotations and translations, and end up with two spheres of the same size as the first. Along the way, we’ll see plenty of math that’s interesting in its own right—groups, Cayley graphs, uncountable sets, and a little hyperbolic geometry. By the end of this class, you’ll understand that math is a beautiful and strange beast.

The Riemann zeta function is one of the most famous functions in mathematics because of its connections to number theory (and the million-dollar unsolved problem that bears its name). But it's also the source of some wonderful identities, which is what this class is about. Come learn about the only time $$\pi^2$$ is important, at least three completely different things called "Euler's formula", and why infinity is equal to -1/12. Seriously. Well, mostly seriously.

How do you know Manny Ramirez was a great hitter? Well, he hit lots of home runs, of course. But there’s a lot more to baseball than just batting average, home runs, ERA, and strikeouts. So much more, in fact, that lots of mathematicians make good money telling baseball teams how to win by looking at the right numbers. In this course, we’ll dive into the world of sabermetrics, or the study of baseball through numbers. We’ll talk about how to figure out if a power hitter is really good (Albert Pujols) or just plays in a ridiculous ballpark (Carlos Gonzalez); if a pitcher wins a lot because he’s got a great defense behind him or because he’s actually that good; and how to come up with ways to rate defense other than fielding percentage (hint: Derek Jeter has never been a very good shortstop).

What does it take to pull off a big event? And just how crazy do you need to be to try? In this class, we'll examine what goes into organizing events for hundreds or thousands of participants like Splash, but also academic conferences, quiz bowl/Acadec tournaments, career fairs, and so on. We'll cover everything from very early planning to event-day insanity to post-event evaluations. Finally, we'll discuss how you can personally take charge of a Splash of your own, using the resources of a national network of Splash programs around the country.

How do you detect a bomb without a metal detector, x-ray equipment, or any kind of search? The answer lies in chemical sensors, which are extremely sensitive devices that can pick up traces of TNT, nerve gas, or other dangerous chemicals from several meters away. We’ll examine the inner workings of chemical sensors that rely on polymers that conduct electricity, which currently give the most sensitive equipment known to man. If you like chemistry, you’ll like this class.

Ever wondered what goes on behind the scenes to make a Splash happen? Come see a completely accurate portrayal of exactly what we do to make Splash happen every year, presented by the directors of Splash 2009.

The Banach-Tarski Paradox is one of the crown jewels of "weird mathematics"--the art of using standard mathematical tools to come up with truly bizarre results. In this case, the result we'll come up with is this: it's possible to slice up a sphere into five pieces, rearrange them using rotations and translations, and end up with two spheres of the same size as the first. Along the way, we'll see plenty of math that's interesting in its own right--groups, Cayley graphs, uncountable sets, and a little hyperbolic geometry. By the end of this class, you'll understand that math is a beautiful and strange beast.

As you may know, the 2010 Nobel Prize in Physics went to Andre Geim and Konstantin Novoselov for the discovery of graphene. Graphene, a sheet of carbon atoms just one layer thick, is one of the most interesting and most-studied materials in modern physics. In this class, we'll examine how to make it, what it could be used for, and why all its wonderful world-changing applications are still a little ways off.

What would life be like if you lived on a torus? What about a projective plane? How about a hyperbolic plane? It turns out that some things we take for granted, like area, volume, and whether or not you can go back in time, get warped in different geometries (ha, ha). In this class, we'll explore all the myriad ways you can fold, twist, and glue together space to get mind-bending results.

Wormholes, cosmic strings, the Casimir effect, and other ways to go back in time without violating any laws of physics. Maybe.

Detect TNT! And nerve gas!

Like Mystery Hunt, only smaller. Much smaller. And shorter. And easier. But more fun, because you get to write it!

Take the 40-hour marathon of MIT Mystery Hunt, then subtract all the puzzles and distill it down to the purest rush of adrenaline and competition there is: the runaround!

The world is filled with danger. Explosions, poisonous gases, fires, and even pollution can really spoil your day. In this class, we'll talk about the chemistry behind all of that--what makes TNT explode, what makes nerve gases so dangerous, all the bad things that can happen to you in a lab, and more. We'll also discuss things like how my friend accidentally made nitroglycerin and how you could, with a lot of patience, enrich uranium. Basically, this is everything your chemistry teachers aren't talking about. Two notes, though: first, there are no live demonstrations in this class; don't expect to see me blow up the room. Second, this entire class comes with a big Don't Try This At Home sticker--this class is not about doing dangerous stuff, it's about understanding it.

How do you detect a bomb without a metal detector, x-ray equipment, or any kind of search? The answer lies in chemical sensors, which are extremely sensitive devices that can pick up traces of TNT, nerve gas, or other dangerous chemicals from several meters away. We’ll examine the inner workings of chemical sensors that rely on polymers that conduct electricity, which currently give the most sensitive equipment known to man. If you like chemistry, you’ll like this class.

The twentieth century was the century of physics. So where are we now? In this class, we'll explore the awesome phenomena and techniques of modern physics. Among the topics we'll cover: superconductors, Bose-Einstein condensates (a collection of atoms collapse to look like a single atom), extremely low temperatures and how to get there, the accelerating expansion of the universe and dark matter, and much more! If you have specific topics you want discussed, bring them to the first class and I'll try to include them. Note: this will be taught at a fairly high level; you should have a year of high school physics under your belt, and you'll get more out of the class if you know some calculus (though I'll try not to use it too often).

How do you detect a bomb without a metal detector, x-ray equipment, or any kind of search? The answer lies in chemical sensors, which are extremely sensitive devices that can pick up traces of TNT, nerve gas, or other dangerous chemicals from several meters away. We'll examine the inner workings of chemical sensors that rely on polymers that conduct electricity, which currently give the most sensitive equipment known to man. If you like chemistry, you'll like this class.

What if parallel lines could diverge, rather than staying the same distance apart? This kind of thing happens in the hyperbolic plane, where concepts of distance and area are not what you would expect. We'll inspect the hyperbolic plane from the point of view of symmetry and geometry, using the powerful tools of abstract algebra. Pretty pictures and rather advanced math will both be present.

Sometimes you can get something for nothing. At least, you can when you manipulate a sphere in strange—but volume-preserving!—ways to make two spheres that are exactly the same as the one you started with. This is the essence of the Banach-Tarski Paradox—a mathematically tricky way to rearrange a sphere into two. In the process, we will explore the Axiom of Choice, rearrangements of sets, and some mildly mind-blowing math.

What would the world be like if the speed of light were only 1,000 miles per hour? If Planck's constant were only $$10^{-5}$$ joule-seconds? In this class we'll talk about the parameters of our universe, and investigate what would happen if they were different. In this class we'll see teleporting snowballs and time-traveling cars, and give new meaning to the glass that is both half full and half empty.

Does high school math seem too simple? Bored with trigonometry and logarithms? Then this is the course for you. Starting with some fairly simple notions like operations and properties of sets, we'll explore the beautiful area of math called abstract algebra. In particular, the course begins with groups and their properties, and then moves to rings and fields, finishing up with a bang: a proof that it's possible to move a sphere around using only rotations to get two copies of the same sphere you started with. Background: This will be a pretty hard course. Fluency with high school algebra and familiarity with functions, trig, and some matrices is strongly suggested.

What do Einstein's theory of general relativity and evolutionary biology have in common? Almost nothing, but at least I've got your attention. In this class, we'll discuss recursion, self-consistency, and attainability, a set of concepts that relates to most areas of science. Mostly, it will be an excuse to talk about a lot of cool things: P vs. NP (computer science), the Banach-Tarski paradox (math), evolution of complex structures (biology), warp drive (physics), and predestination paradoxes (science fiction). If any of these sound interesting, take this class! Background: You'll really just need the ability to think clearly, sometimes about challenging concepts. A little math and science knowledge will occasionally come in handy, but I'll try to explain background concepts as we go along.

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