
More mathcasting about MITEs
Posted:
Feb 15, 2007 12:59 AM


So in the foreward of that jabberwocky book that Haim can make no sense of, we have a list of accomplishments, with this among them:
""" Modeling of all geometric developments of energetic synergetic geometry: Including tensegrity models of all geometrical structures and the hierarchy of primitive structural systems. The minimum, allspacefilling module. The foldable, seven unique greatcircle models. The tetrahelixes. Discovered, 1927; demonstrated, 1936. """ [ http://bfi.org/?q=node/406 ]
What's this "minimum, allspacefilling module"?
Well, to be "minimum" we want it to be simple, and in space filling terms, that means a tetrahedron if possible, because spacefilling shapes don't come with fewer edges, unless you're trying to get away with "blobs" by saying they have *no* edges.
OK, so assuming nonblobby shapes, with edges, you'll be wanting a tetrahedron then, not a cube, and not something flat like a triangle (since filling space is our goal).
The minimum number of edges for a sticklike thing, like a polyhedron, is six. That gives you four windows into the same shared interior. You can't have fewer windows, without resorting to blobby ideas, like hemmed pillows.
Ah, but regular tetrahedra don't pack to fill space. They leave all kinds of gaps, no matter what you try.
I suppose you could just take one, make it very very big and say "there, that fills space". But this isn't what we're looking for. We're looking for something like a brick, that you can clone over and over, and compose with itself, to fill volume with no gaps, no spaces. Like cubes. Like rhombic dodecahedra.
And this is where the MITE or MInimum TEtrahedron comes in. It's a right tetrahedron i.e. three right triangles meet at one corner. Two of the "wings" are identically shaped, and then you have two different isosceles triangles, for a total of four windows (the minimum number).
The MITE is typically dissected however, into two sub shapes, also both irregular tetrahedra, the A and the B.
Whereas MITE, being so symmetrical, doesn't have outward handedness, the As and the Bs do. An insideout right handed A is a left handed A and vice versa, and so for Bs too.
A MITE is comprised as follows: a left *and* a right handed A, plus a left *or* a right handed B. In that sense, you could say MITEs have handedness, as if you could see the internal As and Bs, you'd have reason to assign a "sign" (e.g. + or ).
To tie back to that earlier thread about a Peace Treaty [1], our geek subculture would of course *not* want to give up teaching the above to baby geeks in our K12 pipeline. We deem this knowledge essential and share it with like 2nd and 3rd graders, though we don't necessarily push kids that young to derive all the edges and angles for scratch or anything. That comes later.
At the teacher level, our gnu math faculty band together in cyberspace and shoot the breeze about this kind of stuff, pretty much routinely. For example, just this evening I was posting code cued by another math teacher giving out A, B and Mite modules in Chakovian Coordinates (as yet hardly heard of outside our cliquey venues).
I'll end with a link, for those few of you studying to learn a new trade (that of gnu math teacher  few and far between, and therefore highly payable).[2]
Oh, and what's a tetrahelix? For another time perhaps, or maybe you already know.
Kirby
[1] http://mathforum.org/kb/thread.jspa?threadID=1536382&tstart=0
[2] http://www.4dsolutions.net/ocn/python/quantamods.py

