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Topic: More mathcasting about MITEs
Replies: 10   Last Post: Mar 6, 2007 4:36 PM

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Kirby Urner

Posts: 4,713
Registered: 12/6/04
More mathcasting about MITEs
Posted: Feb 15, 2007 12:59 AM
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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, all-space-filling
module. The foldable, seven unique great-circle models.
The tetrahelixes. Discovered, 1927; demonstrated, 1936.
"""
[ http://bfi.org/?q=node/406 ]

What's this "minimum, all-space-filling module"?

Well, to be "minimum" we want it to be simple, and in
space filling terms, that means a tetrahedron if
possible, because space-filling 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 non-blobby 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 stick-like 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 inside-out 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 K-12
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



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