The Banach-Tarski Dupla-Shrinker recently made an appearance on an episode of Futurama. Using it, Bender got to make two, slightly smaller, copies of himself. The smaller Benders are why they added the 'shrinker' part of the name. The actual Banach-Tarski theorem allows an object to spawn a perfect copy of itself, at its exact size, just by being chopped into bits.

The paradox was first described in 1924 by Stefan Banach and Alfred Tarski. They showed that, if someone were to chop up a solid ball of any size into six very precisely-shaped pieces, those pieces could be rearranged and used to form two new solid balls, each exactly the same size as the original. The pieces would be re-positioned, but not stretched or blown up to larger proportions. Later, another mathematician reduced the number of pieces necessary to do this to five. It gets even stranger. Another version of the theory has a ball the size of a pea being chopped up and reconstructed to form a ball the size of the sun.

This seems intuitively impossible, but as an added bonus, it's theoretically impossible as well. Physics says that mass cannot be created or destroyed with nothing more than a pair of scissors. Anyone who managed to achieve a practical Banach-Tarski operation would essentially be doubling the mass of something, like magic.

It's not magic that drives this paradox. Rather, it's a certain set of assumptions and some tricks with volume. The theoretical Banach-Tarski ball can be cut into fragments that are infinitely small and thin. These fragments are special shapes for a reason. If we wanted to find the volume of a cube, we could measure one side and cube that measurement. If we encountered a rectangular solid, we could simply multiply the base area times the height. But some shapes don't have such easily defined volumes. The shapes that the ball would have to be cut into would be so jagged and scalloped that they would be more like a scatter of points than a solid. As such, their volume is nebulous enough to double if they're put in the right place. At least theoretically.

Via Wolfram Mathworld and Math Fun Facts.

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This week on Game|Life, Wired magazine senior editor Chris Baker took us through the crazy videogame-obsessed brains of the creators of Futurama, the geekiest show on television.

In case you missed any of the in-depth reports on how Futurama and games have and will continue to intersect, here is the complete set of links. Please waste your entire Friday reading them! I will.

• Link

Related:

• Wired Magazine on the resurrection of Futurama
• Futurama’s David X. Cohen Interview
• New episodes of Futurama coming

Cohen has another reason to be happy. The segment he’s watching is from Futurama, the show that he codeveloped back in 1999 with Simpsons creator Matt Groening. (Cohen wrote and produced some of the animated sitcom’s most popular episodes.) With that pedigree, Futurama seemed like a can’t-fail proposition, but it was canceled five years ago. This footage, however, is new: Futurama is back in production, and the unexpected return is as curious as the story of its abrupt cancellation.

• Futurama Is Back! Grab a Can of Slurm and Settle In
• The Geekiest Show on TV
• The Original Futurama
• Gallery: Behind the Scenes of Futurama

It’s the good news everyone has been waiting for! A new season of “Futurama”—the brilliant, animated sci-fi comedy from “Simpsons” creator Matt Groening and executive producer David X. Cohen—is finally being produced for a 2008 release.

It’s still not determined whether the new episodes will debut on Comedy Central or DVD, but we’re too excited to care! We pinned down Cohen to get the skinny on what we can expect when “Futurama” reawakens from its cryo-sleep.

• Link (via Digg)

Related:

• New episodes of Futurama coming

Boston University’s doctoral program in cognitive neuroscience prepares students for a career in brain modeling, robot design, or biomedical engineering—or for winning cash on the television quiz show Who Wants to Be A Millionaire?. Researchers in my department, Cognitive and Neural Systems (CNS), seek to understand the brain’s mechanisms, including three cognitive systems that happen to be essential for a profitable performance on Millionaire: learning, memory, and decision-making. This summer—the start of my final year in the CNS Ph.D. program—I decided to apply my graduate skills to a decidedly practical purpose and auditioned for a turn in the show’s perilous hot seat.

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Since graduating from Brown University in 1997, Oka has worked on more than 30 big-budget Hollywood films at ILM. During that time he has written more than 20 programs and 100 plug-ins for the leading special-effects house. While audiences might not have known his name or face until Heroes, they’ve seen his programming magic on the big screen in films like The Perfect Storm, Star Wars: Episode II, Terminator 3 and the first two Pirates of the Caribbean movies.

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Been eyeballing those sweet new high definition projectors? So have we. We’re not going to feel guilty for it either, we know what we like. But if your wallet is a little light from your last Engadget inspired shopping session you might dig our how-to latest project. In Today’s how-to, we’ll start taking a look at building your own HD projector… from an LCD Monitor. This isn’t one to be missed.

Building your own projector isn’t a new idea. Disassemble a desktop LCD display, put a really, really bright light behind it and add some optics and voila! you’ve got a projector. The real question: is it worth it? Since you might be a little leery of dropping several hundred dollars on parts and gutting a nice display because people you don’t know on an online forum said so, we’re going to do it for you.

• Link (via hack a day)

When Katey Sagal visited The Late Late Show the other day, she told Craig that the show is coming back to Comedy Central in 2008 for at least 13 episodes (I was watching that appearance, too. Must have slept through that part). Don Kaplan of the New York Post confirmed this; his sources told him that all the primary voice actors signed new deals this week. Also according to the article, FOX may have the option to air the new episodes, but probably won’t. The new episodes will also come out on DVD after they air.

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“Caprica” will be set more than 50 years prior to the events of “Battlestar Galactica” and focus on the lives of two families — the Adamas (ancestors of future Galactica commander William) and the Graystones. Humankind’s Twelve Colonies are at peace and on the verge of a technological breakthrough: the first Cylon.

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