One equals two?

A recent xkcd comic featured a puzzling pumpkin-carving outcome: a single pumpkin turning into two!

Like 88% of Randall Munroe’s readers, I went straight to wikipedia to figure out what the “Banach-Tarski” reference in the title-text meant (the other 12% already knew). The Banach-Tarski paradox “… is a theorem in set theoretic geometry which states that a solid ball in 3-dimensional space can be split into a finite number of non-overlapping pieces, which can then be put back together in a different way to yield two identical copies of the original ball.”

This is a fun mental puzzle/paradox because it obviously violates our intuitive sense of geometry. The “trick” is that the pieces into which you carve the pumpkin are rather special. “They are not usual solids but infinite scatterings of points,” says wikipedia. I don’t think the knife that can make those cuts has yet been invented! These pieces are are “nonmeasurable sets”, for which there is no defined notion of volume, so it’s maybe unsurprising that volume might not be conserved. The relevant theorem, though, shows that you can rotate and reassemble these special pieces, without distorting any of them, and end up with two solid full-size copies of the original pumpkin.

A related Tarski-Banach theorem shows that you can similarly carve up any single object and reassemble it into any other object, so your pumpkin can turn into a car or a star or a bacterium (see this very readable summary and analysis).

The final part of the comic punchline arises from the fact that both Tarski-Banach results depend on assuming the Axiom of Choice. This axiom states that given a collection of sets X, it is possible to create a new set Y by picking one item from each of the sets in X. This seemingly straightforward axiom has had a controversial history, partly because it can lead to such unintuitive conclusions. Axioms, of course, are neither true nor false; they are the basic assumptions that one takes as given before proceeding to prove something else. If you include the Axiom of Choice in your base set, then you can proceed to proving the Tarski-Banach theorems. If not, then no matter how you carve your pumpkin, you’ll never accomplish any magic feats of duplication or transmogrification: instead, you’ll only have the same pumpkin you started with!

Still talking when there’s Science to do

I recently had the pleasure of playing Portal for the first time. It’s precisely the kind of puzzle-game I like: progressively more challenging levels that require innovation, and even after you’ve solved a level, there’s often further cleverness to be employed in finding faster or more efficient ways to solve it. I blazed through the first 13 levels in about an hour and a half, and felt a little disappointed when I learned that there were only 19 total. But then I got to level 14, which was the first time that the goal itself had to be divined, not just the way to achieve the goal. And other players’ comments about the levels getting exponentially harder are now starting to make sense. :)

I’ve been fascinated by the process of adjusting to “physics” in a world where you have a teleportation gun. You can open one portal in a nearby wall, and another at a far-off wall, then walk through them to avoid the gaping chasm that lies between. But you can also open a portal beneath your own feet to avoid having to walk to portal 1, or open a portal under some object to make it drop in front of you (instead of having to walk through the portals to retrieve it). And then some crazy stuff starts happening when you pair portals together and bounce between them — or open one in the roof and one in the floor and fall forever between them — or look through a portal and see your own profile from across the room. Mind-bending fun!

Lack a PS3? You can play the flash version, which retains many of the same mechanics but provides a 2D side view rather than a first-person 3D view. The puzzles are different, too.

Portal was first released nearly 3 years ago… and Portal 2 isn’t due until February, 2011. It will feature a two-player cooperative mode! As with so many types of media, I’m glad that my slow adoption rate means I don’t have to suffer through years of waiting. (It may take me a while to solve the last five levels!) Now if only I could finish season 1 of Battlestar Galactica, or season 3 of The West Wing, and catch up with the present!

How hard is that sci-fi?

Many have heard of the Mohs scale of mineral hardness. But did you know that there’s a similar scale for how “hard” a given piece of science fiction is?

Friedrich Mohs anchored his scale with talc (softest, 1) and diamond (hardest, 10). Students of Mineralogy are usually taught to assign hardness based on which of the standard 10 minerals mark the new one, and which don’t. There is also a device called a sclerometer (I’ve never seen one) which employs a diamond head and determines how much pressure is needed to create a visible scratch, or alternatively uses a fixed amount of pressure and measures the width of the resulting scratch.

But now, there is also a scale of science fiction hardness! This page also includes examples of movies that fit at every level of the scale, which ranges from 0 (Barbarella, MST3K, Hitchhiker’s Guide to the Galaxy) to 8 (Real Life). If I were to rate the last few books we read in my sci-fi book club, I’d choose (note the caveat that harder doesn’t equal better!):

  • The Handmaid’s Tale: about 7.5, but is this really sci-fi? Technology isn’t really a player, except as an evil shadow lurking in the past.
  • The Shockwave Rider: a 7; no FTL or broken laws of physics, and eerily prescient about today’s (and tomorrow’s) computer networks.
  • The Atrocity Archives: about 2. There’s no actual space flight, but there are demon-spawned gates to permit travel to other planets (and universes?), with some handwavy handling of pressure differentials and energy conservation. Physics is pretty much entirely broken by the integration with pentagrams and the occult. But hey, it’s fun!
  • Bones of the Earth: right about a 1. Not only does time travel exist, but there’s also a “paradox detector” (never explained).
  • Metatropolis: varied, but about a 7 once you take today and add in various bits of technology and virtual reality overlays. All reasonable extrapolations, without any laws of physics being broken.
  • The Demolished Man: about a 4, if you consider telepathy to be breaking a law of physics. Tenser, said the tensor!

The sci-fi hardness scale page also puts Contact, Avatar, and Ender’s Game at a 3, and Rainbows End and Cryptonomicon at a 5. I’d put Anathem at a 5, too.

How hard is your sci-fi?

Romancing the fig

I haven’t ever eaten a fresh fig. Friends tell me they’re quite tasty, especially here where you can get locally grown ones. Figs with wasps inside? Maybe not so much.

Figs are not actually fruits but a mass of inverted flowers and seeds that are pollinated by a species of tiny symbiotic wasps. The male fig flower is the only place where the female wasp can lay her eggs, at the bottom of a narrow opening in the fruit that she shimmies her way through. The baby wasps mature inside the fig into males that have sharp teeth but no wings and females ready to fly. They mate, the males chew through the special fig pollen holders and drop them down to the females, chew holes in the skin of the fig to let the females out, and then die.

The females, armed with the pollen, fly off in search of new male figs to lay her eggs in. In the process some of the female wasps land on female figs that don’t have the special egg receptacle but trick the female into shimmying inside. As the female wasp slides through the narrow passage in the fig her wings are ripped off (egg laying is a one-way mission) and while she is unsuccessful in laying her eggs, she successfully pollinates the female flower. The female flower then ripens into the fig that you can get at the supermarket, digesting the trapped wasp inside with specialized enzymes!

[From Christina Agapakis, via The Atlantic’s Daily Dish, via Hacker News, via my officemate Ben.]

There is also a PBS special, called “The Queen of Trees”, that includes actual footage of this process (!). Here’s a preview:

These are tiny wasps, only 2 mm long. So their contribution to your protein intake would be minimal. Further, it seems that not all figs use this method of reproduction; some (those most often cultivated in the U.S.) instead use parthenocapy. This is the process of producing fruit without fertilization, which is handy if the plant has been imported without a fertilizing partner (or wasp), or if the desired result is a seedless fruit. New plants can be created “vegetatively” (e.g., putting a stem in water and having it sprout, or grafting one plant onto another). Clever, clever!