On Tue, Jul 16, 2013 at 10:36 AM, Robert Hansen <email@example.com> wrote:
> > On Jul 16, 2013, at 12:04 PM, Joe Niederberger <firstname.lastname@example.org> > wrote: > > Or is there a nice one for the tetrahedrons that I'm just not seeing? > > > The problem I see there is that the big tetrahedron is not solid. It has > voids, unless I am mistaken. Thus, it isn't a mapping of volume like the > cube. It is the count of its pieces that follow the cubes of 1, 2, 3, .... > While there is of course a mathematical reason for this, it isn't intuitive. > > Bob Hansen >
No, you got lost with those holes.
They were just to show the anatomy of the bigger tetrahedron which consists entirely of smaller tetrahedrons AND octahedrons.
*All* the volume gets filled in (counted) at every layer, there are no holes of any kind.
That's why I switched to talking about pouring water, to get away from your fixation on "holes".
Here's another related demo. All it really requires is being able to imagine a tetrahedron "upside down" i.e. with its base pointed skyward, so we can pour water into it, and yet it doesn't tip over and spill. Think of how TV towers are anchored by guide wires and maybe that'll work.
Start with an upside down tetrahedron with edges 1 foot. That's what we can call our "cup of water". Now there's a completely empty tetrahedron next to it with edges 2 feet. All the edges are twice as long. What to say about volume? We know "one cup" of water fills the first. It turns out eight cups of water fill the second.
And if we make the edges three feet? Then 27 cups.
Four feet? 64 cups.
And so on.
What's true is *any* shape that grows in a self similar fashion i.e. all surface and central angles held invariant, grows as a 3rd power of the change in linear dimensions. True of a complicated shape too, like a sewing machine.
However the tetrahedron is the shape we focus on because:
(a) it's topologically simpler than the cube, so a logical candidate for a unit of volume and (b) the tetrahedron plays well with others, in terms of giving students a lot of whole number volumes to play with, for other related shapes (like the octahedron of volume 4)
Indeed, there's a large and growing accessible literature (branch in math) that just assumes the regular tetrahedron is a unit volume.
This used to be something you'd only learn at MIT, like in Dr. Loeb's class, but now, in 2013, it's no longer tenable to avoid all this development even at the high school level -- IF, that is, you're using one of the better, more enlightened curricula (might be from Russia who knows -- not saying competition with Common Core can't come from outside the US).
Today, when people say "cubed" unconsciously, because that's what they learned in school, they don't have any sense of a branch, an alternative. This will not be so true of people coming to adulthood today. More and more of them are getting a better STEM education than their parents did, or at least they have that potential.