Tiling and Spacefilling

[kirby.urner at gmail.com]

[ slightly improved over Math Forum draft ]

Probably a fault line or cultural divide between acutely
differing schools of thought, is in this area of tiling
or mosaic making. Some schools consider this a stupid
waste of time, others a core topic, whereas a 3rd group
stays more neutral on the issue, and a tiny 4th group has
no idea what I'm talking about.

I'm in the second group (core topic) as tiling is to
spacefilling as figurate are to polyhedral numbers.
We may start in a plane, in deference to some millenia
of pre-NASA reflex conditioning, but we're still on a
runway, accelerating, and before long (way sooner than
in traditional curricula), we'll be airborne, moving
up into higher D. Tetrahedra are most definitely a
feature of K-5. If NCTM doesn't grasp that, it's
missing the boat.

Back to tiling: we have the classifications, the Escher
lithographs (which follow into Part Two), and the tie ins
to Mesopotamian religions, the three biggies especially,
and their respective sacred geometries. I know "sacred
geometry" is taboo in traditional mathematics, but this
is ~M!, Katrina Network type stuff, closer to George Bush
Sr.'s "voodoo economics" (except managed more by geeks
(Bush was more of a jock, not that we can't share power)).

Rolling forward, we get to the Kepler-Penrose aperiodic
tiling studies, various heuristics, again finding echoes
in Part Two (in what some Russians call stereometry),
where we likely look at Hargittai & Hargittai (Hungarian),
plus Chakovian coordinates (quadrays, simplicials,
whatever nomenclature).

I've already made the case for Polyhedra numerous times.
Thanks to breakthroughs in architecture in the 20th
century, we're obligated to teach about the hexapent in
some way shape or form. Katrina Network is strong on
this, but you'll find other sources. Top of my list
would be J. Baldwin's 'Bucky Works' (packed with ~M!).

For a more prefrequency approach, I recommend Cromwell's
'Polyhedra', and its cartoon proof of Euler's V + F =
E + 2 based on the interdigitating trees of Von Staudt's.

And of course we're not about to bleep over Fuller's
volumes tables, nor MITEs, nor the Jitterbug Trans-
formation -- all pure gold as far as we're concerned.

So to recap... Given I'm of a school which regards
tiling as core, I'm putting my shoulder behind two big
initiatives in computer science these days: CP4E and
HP4E.

CP4E was hatched by Guido, BDFL of Python Nation, and
DARPA, and carried forward billiantly (at least one
doctoral dissertation already on file, about our early
beginnings as a think tank slash on-line community [1]).

HP4E was hatched by yours trully, though mostly as a
marketing gimmick in homage to CP4E and Wanderer Glenn
Stockton's global matrix campaign. I certainly take
advantage of Python a lot, using its generators (a recent
feature) to spit out successive terms in figurate and
polyhedral number sequences, e.g. 1, 3, 6, 10... and
1, 12, 42, 92... (see my Focal Points for more details,
plus my CP4E website, one of many curriculum "hot springs"
in our cyberspace-based Python Nation [2][3]).

Kirby

[1] John Miller's PhD dissertation, Promoting Computer
Literacy Through Programming Python (1.37 MB), looks at
the issues around teaching with Python, and explores some
of the threads taken up on edu-sig.
http://www.python.org/community/sigs/current/edu-sig/

[2] Focal Points:
http://mybizmo.blogspot.com/2006/09/focal-points.html

[3] CP4E website: http://www.4dsolutions.net/ocn/cp4e.html

===

For more on ~M!:
http://worldgame.blogspot.com/2007/01/m.html