On Sun, Jul 14, 2013 at 7:29 PM, Wayne Bishop <firstname.lastname@example.org>wrote:
> My apologies, I thought you said something about equality because pouring > liquids matched as opposed to mathematical verification of equality of the > volumes. >
> > Knee-jerk redneck bigoted academician that I am, would you at least agree > that, at least historically, "cubing" has something to do with geometric > cubes as in, say, the ancient problem of "duplicating a cube"? (I readily > acknowledge that the word is used in its algebraic form in many other > mathematical situations but we are talking about physical models.) Would > you, for example, explain the idea of duplicating a cube using (as your > primary model) a regular tetrahedron (where the impossibility is exactly > the same) to your new generation of exceptionally creative and unblinded > students? > > Wayne > > I think you still misunderstand. I am not saying there's no connection between 3rd powering and cubes or that this connection should be downplayed.
I'm saying that the tetrahedron may also be used as a model of 3rd powering i.e. you have a dial you rotate clockwise / counter-clockwise to boost-diminish the edge sizes of a tetrahedron from between 1 and 3. The thing grows and shrinks from volume 1 to 27 and every value in between, and its volume is precisely a 3rd power of the edge length. Why? Because we *define* a unit edge tetrahedron to have volume 1. Can we do that? Yes.
Is there a reason to *never* show this in the classroom? I'm saying the Common Core gives us an opportunity to brand curricula as not just "aligned" but extending "well beyond".
Those parents and students who want something more than mere mediocrity will find out about the unit volume tetrahedron whereas those sticking to the standards will grow up with a mediocre math education more like that of their parents.
As to how to duplicate a tetrahedron with a ruler and compass in 3D if that's what's asked, it sounds doable, but not in a way I'd want to try typing out in words here. Sounds like a CAD challenge of some kind.