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Topic: Pascal's Triangle and Powers of 2
Replies: 20   Last Post: May 16, 2008 3:26 PM

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Kirby Urner

Posts: 4,713
Registered: 12/6/04
Re: Pascal's Triangle and Powers of 2
Posted: May 15, 2008 1:14 PM
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> Truly fascinating!

<<SNIP >>

> Many thanks, Kirby, and others.
>
> GSC


Glad you're enjoying the discussion guy. If you go back
in the archives you'll see where I'm going with those
triangular and tetrahedral numbers, read off Pascal's
Triangle, the former being N(N+1)/2 (sums of consecutives)
and the latter being the running total of triangulars
(a tetrahedron is just a stack of growing triangles,
analogy to cube and squares obvious -- and we exploit
it).

Where I'm going: sphere packing. Six pennies around one
on a table, twelve ping pong balls around one in space,
but with all kinds of interesting caveats, which we don't
ignore. Once you've got balls packing, you've got all
kinds of bridges to scientific subjects, plus ways of
thinking spatially in a generic sense (holodeck
aesthetics, like with XYZ, already a key topic).

Sphere packing wasn't a topic in Dom Rosa's day, not for
kids anyway, but it should be today, given all the
decades of research, including new pedagogy. Thanks to
Kepler, we're well aware that rhombic dodecahedra,
12-sided diamond faced volumes, encase our ping-pong
balls and fill space at the same time. If we assign
them a volume of six (stay with me now), then the
octahedron formed by its long face diagonals has volume
4, and the cube formed by its short face diagonals has
volume 3. The intersecting self-dual tetrahedra that
make this cube: volume 1 each.

How simple, how easy, how accessible -- and how promoting
of fluency in spatial geometric terms. How *not* what's
in any K-12 geometry book today, where polyhedra are
usually tucked in the back with a few volume formulae,
often not reached in the course of the year.

Mind-numbing, idiotic, the kind of thing Wayne had to
put up with as a kid (poor guy), which says a lot about
where MC is today, in the dark ages, and unlikely to
ever snap out of it (not holding my breath).

The fact that today's K-12 math teachers *do not* share
about the 6,4,3,1 simple whole number volume hierarchy
in a sphere packing context is a fact I use as leverage,
behind the scenes, to heap scorn on certain segments of
our economy (textbook publishers, know-it-all professors).

I am rather unforgiving in this regard, paid to be so,
a hard-liner. So in that sense I'm probably worse than
Wayne (as I've warned MPG) in terms of undermining
careers, closing doors, not to kids but adults. You
can't get anywhere close to being a gnu math teacher if
you don't know (and share) these basics.

Kirby



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