## Sunday, February 22, 2009

### MITE Fight

I donned my "know everything glasses" to tell this one, family scene so let's keep it PG:

Once upon a time, people cared about space-filling, one of those geometry topics accessible to kids, i.e. what shapes fill space. Obviously "the cube" is a right answer or "any brick" (we don't care about the tank's walls -- this is like tiling (like a bathroom floor -- hexagons maybe?), but in volume).

However, cubes and bricks are all hexahedra, which are topologically less minimal than tetrahedra, having six sides instead of just four. So if there's a game to find the "minimum" space-filler, in terms of having the fewest edges etc., then of course you'd be looking for a tetrahedron. The reasoning goes something like this: "if there's a minimum space-filler, then it has to be a tetrahedron, but the regular tetrahedron isn't a space-filler, so it would have to be an irregular tetrahedron."

The plot thickens at this point, because irregularity is often associated with "handedness" i.e. you'll have an R and a backwards R, a left and a right. We have lots of left and right space-fillers, as well as complementary space-fillers e.g. tetrahedra and octahedra in complement (both regular). At this point, a lot of kids are starting to tune out, adults too, as we're piling on the caveats.

So at the end of the day, Bucky Fuller points to this minimum tetrahedron that's neither left nor right handed, though we can create that under the hood with A & B modules if we like. We've got our holy grail in other words, our minimum space-filler, which he cleverly named a MITE (for MImimun TEtrahedron -- kind of GNU-like in its cleverness). Other space-filling tetrahedra (called Sytes in Synergetics) may be assembled from MITEs, making MITEs the most primitive.

What happened next is for the computers to analyze, but my impression is space-filling immediately became a quasi-verboten topic, at least in any contemporary form. It's OK to tell the story up to, but not including, the MITE. Why? Because Fuller had no credentials as a mathematician and a hyper-inflated sense of his own importance, per Coxeter. In other words, it's more fun to engage in character assassination than to actually talk about math, which we all think is boring anyway.

The upshot is if you go to the indexes, you'll see the effects of The Spanish Inquisition, as we translate for Monty Python aficionados. "The Gulag" (university-based professoriate) has turned its back on our heritage and refuses to promulgate any geometry with a "not invented here" brand, no matter how useful.

The goal is to keep anything so accessible as the MITE under lock and key, so that impressionable young eyes don't start asking uncomfortable questions, seeking inconvenient truths.

So far, the program has been pretty successful, just check any K-12 geometry text book and you'll see what I mean. Or study the MAA archive. Remember: you're looking for what's missing, not for what's there (Sartre has us practice in Being and Nothingness, with a busy coffee shop for background).

One of my audience was a medical ethicist in this case. It's true I believe there are ethical implications, the business of philosophers to expound upon. We'll be getting a lot of mileage out of this story, especially given all the advances in ontological studies (OWL, DAML... RDF). We'll get a lot of detailed views to substantiate our model of how "the Gulag" works (or doesn't, as the case may be).

Note: the "orthoscheme of the cube" in Regular Polytopes (Coxeter) is handed, left or right, is what we'd all a "half MITE" or "SMITE" (S for Semi, per Koski). The "tri-rectangular tetrahedron" from pg. 71 (ibid) was named otherwise by Coxeter, but is Fuller's to analyze into A and B modules (see section 950.12 in Synergetics, with accompanying figure).

Followup: D.M.Y. Sommerville documented four tetrahedral space-fillers in his 1923 paper, Space-Filling Tetrahedra. The Mite, and the two tetrahedral Sytes it assembles, were three of four of them.

However the Rite also fractures into four tetrahedral space-fillers that are duplications of one another, and not made of Mites. Fuller doesn't mention it, let alone give it a name. Suggestions?