
Re: Pascal's Triangle and Powers of 2
Posted:
May 15, 2008 1:14 PM


> 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, 12sided diamond faced volumes, encase our pingpong 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 selfdual 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 K12 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.
Mindnumbing, 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 K12 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, knowitall professors).
I am rather unforgiving in this regard, paid to be so, a hardliner. 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

